%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: engart.dvi %%Pages: 57 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips engart %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.10.28:1707 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{ CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end} repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N /@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X /yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (engart.dvi) @start %DVIPSBitmapFont: Fa cmtt10 10 18 /Fa 18 119 df<007FB6FCB71280A46C150021067B9B2C>45 D<121FEA3F80EA7FC0EAFF E0A5EA7FC0EA3F80EA1F000B0B708A2C>I64 D<3801FFF0000713FE001F6D7E15E048809038C01FF81407EC01FC381F80000006C77EC8 127EA3ECFFFE131F90B5FC1203120F48EB807E383FF800EA7FC090C7FC12FE5AA47E007F 14FEEB8003383FE01F6CB612FC6C15FE6C14BF0001EBFE1F3A003FF007FC27247CA32C> 97 D<903803FFE0011F13F8017F13FE48B5FC48804848C6FCEA0FF0485A49137E484813 1890C9FC5A127EA25AA8127EA2127F6C140F6DEB1F806C7E6D133F6C6CEB7F003907FE03 FF6CB55A6C5C6C6C5B011F13E0010390C7FC21247AA32C>99 DII103 DI<1307EB1FC0A2497EA36D5AA20107C7FC90C8FCA7387FFFC080B5FC7EA2EA00 07B3A8007FB512FCB612FEA36C14FC1F3479B32C>I107 D<387FFFE0B57EA37EEA0003 B3B3A5007FB61280B712C0A36C158022337BB22C>I<3A7F83F007E09039CFFC1FF83AFF DFFE3FFCD87FFF13FF91B57E3A07FE1FFC3E01FCEBF83F496C487E01F013E001E013C0A3 01C01380B33B7FFC3FF87FF0027F13FFD8FFFE6D13F8D87FFC4913F0023F137F2D2481A3 2C>I<397FF01FE039FFF87FFC9038F9FFFE01FB7F6CB6FC00019038F03F80ECC01F0280 7FEC000F5B5BA25BB3267FFFE0B5FCB500F11480A36C01E0140029247FA32C>II< 131E133FA9007FB6FCB71280A36C1500D8003FC8FCB1ED03C0ED07E0A5EC800F011FEB1F C0ECE07F6DB51280160001035B6D13F89038003FE0232E7EAD2C>116 D<3A7FF003FF80486C487FA3007F7F0001EB000FB3A3151FA2153F6D137F3900FE03FF90 B7FC6D15807F6D13CF902603FE07130029247FA32C>I<3A7FFF01FFFCB514FE14831401 6C15FC3A03E0000F80A26D131F00011500A26D5B0000143EA26D137E017C137CA2017E13 FC013E5BA2EB3F01011F5BA21483010F5BA214C701075BA214EF01035BA214FF6D90C7FC A26D5A147C27247EA32C>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmsl10 10 27 /Fb 27 124 df12 D45 D71 D<013FB512E0A25B9039007FE0006E5AA2147F5DA514 FF92C7FCA55B5CA513035CA513075CA5130F5CA5131F5CA5133F5CA4EBFFE0007FEBFFC0 A2B6FC23397EB81E>73 D<017FB500C090B512C01A80A2010001C0C7383FF8006E48EC1F C096C7FC02FF151C92C85A6060EF03C04DC8FC49150E4A5C5F5FEE01C04C5A01034AC9FC 4A130E5E167016F04B7E010713034A487E151FED3BFE1573EDE1FF90380FF9C1DAF7007F 14FE4A6D7E5C4A6D7E131F4A6D7EA2160F83A2013F6E7E5C707EA2707EA2017F6E7F5C71 7EA24D7E48486C4913F8B6D8800FEBFFC04C5CA242397DB841>75 D83 D<0007B812FCA3903AF8007FE003D80FC090383FC00049163C000EC7127F001E4B131C12 1CA212180038170C003014FF93C7FCA25A1818A2485BC7491400A514035DA514075DA514 0F5DA5141F5DA5143F5DA3147F4A7E0003B612F85AA2363977B83C>I<14FF010713E090 381F01F8903878007C01F8137E01FE7F0001801680A35BEA007090C7FCA4EC0FFF49B5FC 90390FFC3F00EB7FC03801FE00EA03F848485B485A4848137E485A007F150690C7FC15FE 48ECFC0C481301A21403007F9038077C18140E3A3F801C7E303A1FC0F83FF03A07FFE01F C0C69038000F8027277CA52A>97 D<137FEA1FFF5BA212011200A35BA512015BA512035B EC1FC0EC7FF89038F1E03E9038F7801F3A07FE000F8049EB07C04914E04913034914F0A2 000F15F8491301A41503121F5BA41507003F15F090C7FC16E0150F16C0151F481580ED3F 005D6D137E007D5C3979C001F039F0E007E039E0781F8026C01FFEC7FC380007F0253B78 B92E>III<147F903803FFE090380F81F090383E 00FC49137C48487F4848133F0007805B48481480121F5B123FA248C7FCA3B71200A248C9 FCA65A7EA2007E140EA25D6C14186C14386D5B6C6C485A3907E003802601F01FC7FC3800 7FFCEB1FE021277BA525>I<157F913801FFC0913807C1E091381F87F0EC3F0F147E14FC A2D901F813E0ED07C04948C7FCA413075CA5130F5CA20007B512E0A326001FC0C7FC5CA5 133F91C8FCA55B137EA513FE5BA512015BA4487EB512F0A3243B7EBA19>IIII<14FC137F14F8A213071303A314F0A513 0714E0A5130F14C0A5131F1480A5133F1400A55B137EA513FE5BA512015BA41203B512E0 14C0A2163A7EB917>108 D<90270FC03FC0EB7F80D803FF903AFFF001FFE048903BC3C0 F80781F0913BCF007C1E00F826003FDCD97E387F6D485C02F0D93EE0137C4AD93FC0137E 4A5C047F14FE494891C75AA291C7127EA44902FE1301017E4A5CA501FE01011403494A5C A5000102031407494A5CA4486C496C497EB500E1B500C3B51280A202C10283140041257E A445>I<90390FC03FC0D803FFEBFFF0489038C3C0F89138CF007C26003FDC137E6D5A02 F0133E4A133F5C5E4948137EA291C7FCA316FE5B017E5CA4150113FE495CA41503120149 5CA400031407B500E1B512C0A202C114802A257EA42E>II<9039 01F80FE0017FEB3FFC01FFEBF03F9139FBC00F80902607FF0013C06D48EB07E04AEB03F0 5C4A14F81601010715FC5CA5130F5CA41603011F15F85CEE07F0A2EE0FE0A2013FEC1FC0 1780163F6EEB7F0016FE9138E001F890397F7003F090397E3C0FC0DA0FFFC7FCEC03F891 C9FC13FEA25BA41201A25BA2487EB512E0A32E3581A42E>I<90381F807C3903FF81FF48 9038878F80EC8E1F39003F9C3FEB1F3814709138601F00ECE0044AC7FC133F5CA291C8FC A35B137EA513FE5BA512015BA4487EB512F0A321257EA421>114 D<903803FE0C90380FFF9C90383E01FCEBF0004848137C4848133C1538485AA215181538 487E1530D807F0130013FCEBFFE06C13FC14FFC614806D13C0011F13E01300EC0FF01407 003013031401A31238007814E0A3007CEB03C0EC0780127EB4EB1F0038F3C07C38E1FFF0 38C03F801E277DA521>I<1306A4130EA2130C131CA2133C137C13FC5B12031207001FB5 FCB6FCA23803F8005BA512075BA5120F5BA5001F130C1380A4141C003F131813007E1438 EB80301470380FC0E03807C1C03803FF8038007E00183479B220>II<90B538803FFE5A150026000FF8EB0FF06D48EB07C017 801700010314065EA26E5B0101143816305E8001005CA24B5A1503027E90C7FC1506A25D 147F6E5A1538153015E0141F5DA25D140F92C8FC140EA2140CA25C143814305CA2003E5B 127E38FE018049C9FC5BEAFC0EEA701C1378EA3FE0EA0F802F3580A32C>121 D123 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmti10 10 55 /Fc 55 128 df<04FFEB03F003039038E00FFC923A0FC0F01F1E923A3F00783E0F923A7E 01F87C3FDB7C03EBFC7F03FC14F8DA01F813F905F1137EDC01E1133C913B03F00003F000 A314074B130760A3140F4B130F60A3010FB812C0A3903C001F80001F8000A3023F143F92 C790C7FCA44A5C027E147EA402FE14FE4A5CA413014A13015FA313034A13035FA313074A 495AA44948495AA44948495AA3001CD9038090C8FC007E90380FC03F013E143E00FE011F 5B133C017C5C3AF8780F01E0D878F0EB07C0273FE003FFC9FC390F8000FC404C82BA33> 11 DI< EE7FE0923903FFFC7E92380FC03E92381F000F033EEB3FFE4B137F03FC14FC5D1401173D 4A48EB01F8A21703A24A4814F0A21707A2020F15E05D170FA218C0010FB7FCA3903B001F 80001F80A2173F143F92C71300A25FA24A147E147E17FEA25F14FE4A1301A25FA2010114 035CEFF070A21607010316F04AECE0E0A3EFE1C013074A14C3933803E380EE01E7933800 FF004948143C94C7FCA3495AA3001C90CAFC127E133E12FE133C137CEAF878EA78F0EA3F E0EA0F80374C82BA31>I<130FEB1F80133F137FEBFF00485A5BEA03F0485A485A485A00 3EC7FC5A5A12E05A111064B92A>19 D39 D<387FFFF8A2B5FCA214F0150579941E>45 D51 D<16E0ED01F01503A3150716E0A3150F16C0A2151F1680A2ED3F00A3157E A2157C15FC5D14015D14035D14075D140F5D141F92C7FC143EA25CECF81C153E903801F0 7EEB03E014C090380780FE130F49485A133EEB7C01137801F05BEA01E03803C003EA0FFE 391FFFC3F04813FB267C01FF13403AF0003FFFE000601307C71400EC0FE05DA3141F5DA3 143F92C7FCA4143E141C24487DB72A>I<133C137E13FF5AA313FE13FCEA00701300B212 0EEA3F80127F12FFA31300127E123C102477A319>58 D65 D<0107B612FCEFFF8018C0903B000FF0001FF04BEB07F81703021F15FC 17014B14FEA2023F1400A24B1301A2147F18FC92C7120318F84A140718F04AEC0FE0EF1F C00101ED3F80EF7F004AEB01FEEE07F849B612E05F9139F80007F0EE01FC01076E7E177F 4AEC3F80A2010F16C0171F5CA2131F173F5CA2133FEF7F805C1800017F5D4C5A91C7485A 5F49140FEE1FE0494A5A00014AB45AB748C7FC16F816C037397BB83A>II<0103B612FEEFFFC0 18F0903B0007F8000FF84BEB03FCEF00FE020F157FF03F804B141F19C0021F150F19E05D 1807143F19F05DA2147FA292C8FCA25C180F5CA2130119E04A151FA2130319C04A153FA2 01071780187F4A1600A2010F16FEA24A4A5A60011F15034D5A4A5D4D5A013F4B5A173F4A 4AC7FC17FC017FEC03F84C5A91C7EA1FC04949B45A007F90B548C8FCB712F016803C397C B83F>I<0107B8FCA3903A000FF000034BEB007F183E141F181E5DA2143FA25D181C147F A29238000380A24A130718004A91C7FC5E13015E4A133E167E49B512FEA25EECF8000107 147C163C4A1338A2010F147818E04A13701701011F16C016004A14031880013F15071800 4A5CA2017F151E173E91C8123C177C4915FC4C5A4914070001ED7FF0B8FCA25F38397BB8 38>I<0107B712FEA3903A000FF000074B1300187C021F153CA25DA2143FA25D1838147F A292C8FCEE03804A130718004A91C7FCA201015CA24A131E163E010314FE91B5FC5EA290 3807F800167C4A1378A2130FA24A1370A2011F14F0A24A90C8FCA2133FA25CA2137FA291 CAFCA25BA25B487EB6FCA337397BB836>II<0103B512F8A390390007F8005DA214 0FA25DA2141FA25DA2143FA25DA2147FA292C7FCA25CA25CA21301A25CA21303A25CA213 07A25CA2130FA25CA2131FA25CA2133FA25CA2137FA291C8FC497EB6FCA25C25397CB820 >73 D<0103B500F890387FFFE0A21AC090260007F8C7380FFC004B15E061020F4BC7FC18 3E4B5C18F0021F4A5A4D5A4BEB0F804DC8FC023F143C5F4B5B4C5A027FEB07C04CC9FCED 001E5E4A5BED01FCECFE0315070101497E151FECFC7C4B7E903903FDE07FDAFFC07F1580 ED003F49488014F84A131F83130F160F4A801607011F81A24A130383133F16014A80A201 7F6E7EA291C8FC494A7F007F01FE011F13FCB55CA243397CB840>75 D<0107B512FCA25E9026000FF8C7FC5D5D141FA25DA2143FA25DA2147FA292C8FCA25CA2 5CA21301A25CA21303A25CA21307A25CA2130F170C4A141CA2011F153C17384A1478A201 3F157017F04A14E01601017F140317C091C71207160F49EC1F80163F4914FF0001020713 00B8FCA25E2E397BB834>I<902607FFF8923807FFF0614F13E0D9000FEFF0004F5AA202 1F167FF1EFC0141DDA1CFCEC01CF023C16DF9538039F800238ED071FA20278ED0E3F97C7 FC0270151CA202F04B5AF0707E14E0037E14E0010117FE4D485A02C0EC0380A20103ED07 01610280140EA20107ED1C0305385B14006F137049160705E05B010EEC01C0A2011E9138 03800F61011CEC0700A2013C020E131F4C5C1338ED1FB80178163F04F091C8FC01705CA2 01F04A5B187E00015DD807F816FEB500C09039007FFFFC151E150E4C397AB84A>I<9026 03FFF891B512E0A281D90007923807F8006F6E5A61020F5E81DA0E7F5DA2021E6D130703 3F92C7FC141C82DA3C1F5C70130EEC380FA202786D131E0307141C147082DAF003143C70 133814E0150101016E1378030014705C8201036E13F0604A1480163F010715C1041F5B91 C7FC17E149EC0FE360010E15F31607011E15FF95C8FC011C80A2013C805F133816001378 5F01F8157CEA03FC267FFFE0143CB51538A243397CB83E>II<0107B612F817FF1880903B000F F0003FE04BEB0FF0EF03F8141FEF01FC5DA2023F15FEA25DA2147FEF03FC92C7FCA24A15 F817074A15F0EF0FE01301EF1FC04AEC3F80EFFE0001034A5AEE0FF091B612C04CC7FCD9 07F8C9FCA25CA2130FA25CA2131FA25CA2133FA25CA2137FA291CAFCA25BA25B1201B512 FCA337397BB838>I<0103B612F017FEEFFF80903B0007F8003FC04BEB0FF01707020FEC 03F8EF01FC5DA2021F15FEA25DA2143FEF03FC5DA2027FEC07F818F092C7120F18E04AEC 1FC0EF3F004A14FEEE01F80101EC0FE091B6128004FCC7FC9138FC003F0103EC0F80834A 6D7E8301071403A25C83010F14075F5CA2011F140FA25CA2133F161F4AECE007A2017F16 0F180E91C7FC49020F131C007F01FE153CB5913807F078040313F0CAEAFFE0EF3F80383B 7CB83D>82 D<92383FC00E913901FFF01C020713FC91391FC07E3C91393F001F7C027CEB 0FF84A130749481303495A4948EB01F0A2495AA2011F15E091C7FCA34915C0A36E90C7FC A2806D7E14FCECFF806D13F015FE6D6D7E6D14E0010080023F7F14079138007FFC150F15 031501A21500A2167C120EA3001E15FC5EA3003E4A5AA24B5AA2007F4A5A4B5A6D49C7FC 6D133ED8F9F013FC39F8FC03F839F07FFFE0D8E01F138026C003FCC8FC2F3D7ABA2F>I< 0007B812E0A25AD9F800EB001F01C049EB07C0485AD900011403121E001C5C003C178014 03123800785C00701607140700F01700485CA2140FC792C7FC5DA2141FA25DA2143FA25D A2147FA292C9FCA25CA25CA21301A25CA21303A25CA21307A25CA2130FA25CEB3FF0007F B512F8B6FCA2333971B83B>I86 D<91B712F0A25B9239E0001FE092C7EA3FC0D903FCEC7F8002F015004A14FE1601494849 5A4A495A4C5A49C75B4C5A010E143F011E4A5A011C4AC7FC4B5A5E90C7485A15074B5A4B 5A4B5A5E157F4BC8FC4A5A4A5A4A5A5D140F4A5A4A5A4A5A4AC712E05C13014948130149 485C495A494813034A5C013F1407495A49C7FC48484AC7FC48485C5B0007153E4848147E 4848EB01FE4848EB07FC4848133F90B6FCB7FC5E34397AB833>90 D<14F8EB07FE90381F871C90383E03FE137CEBF801120148486C5A485A120FEBC001001F 5CA2EA3F801403007F5C1300A21407485C5AA2140F5D48ECC1C0A2141F15831680143F15 87007C017F1300ECFF076C485B9038038F8E391F0F079E3907FE03FC3901F000F0222677 A42A>97 D<133FEA1FFFA3C67E137EA313FE5BA312015BA312035BA31207EBE0F8EBE7FE 9038EF0F80390FFC07C013F89038F003E013E0D81FC013F0A21380A2123F1300A214075A 127EA2140F12FE4814E0A2141F15C05AEC3F80A215005C147E5C387801F8007C5B383C03 E0383E07C0381E1F80D80FFEC7FCEA01F01C3B77B926>I<147F903803FFC090380FC1E0 90381F0070017E13784913383901F801F83803F003120713E0120FD81FC013F091C7FC48 5AA2127F90C8FCA35A5AA45AA3153015381578007C14F0007EEB01E0003EEB03C0EC0F80 6CEB3E00380F81F83803FFE0C690C7FC1D2677A426>II<147F903803FFC090380F C1E090383F00F0017E13785B485A485A485A120F4913F8001F14F0383F8001EC07E0EC1F 80397F81FF00EBFFF891C7FC90C8FC5A5AA55AA21530007C14381578007E14F0003EEB01 E0EC03C06CEB0F806CEB3E00380781F83803FFE0C690C7FC1D2677A426>IIIII107 DIII<147F903803FFC090380FC1F090381F00F8017E137C5B4848 137E4848133E0007143F5B120F485AA2485A157F127F90C7FCA215FF5A4814FEA2140115 FC5AEC03F8A2EC07F015E0140F007C14C0007EEB1F80003EEB3F00147E6C13F8380F83F0 3803FFC0C648C7FC202677A42A>I<9039078007C090391FE03FF090393CF0787C903938 F8E03E9038787FC00170497EECFF00D9F0FE148013E05CEA01E113C15CA2D80003143FA2 5CA20107147FA24A1400A2010F5C5E5C4B5A131F5EEC80035E013F495A6E485A5E6E48C7 FC017F133EEC70FC90387E3FF0EC0F8001FEC9FCA25BA21201A25BA21203A25B1207B512 C0A3293580A42A>II<3903C003F0390FF0 1FFC391E783C0F381C7C703A3C3EE03F8038383FC0EB7F800078150000701300151CD8F0 7E90C7FCEAE0FE5BA2120012015BA312035BA312075BA3120F5BA3121F5BA3123F90C9FC 120E212679A423>I<14FE903807FF8090380F83C090383E00E04913F00178137001F813 F00001130313F0A215E00003EB01C06DC7FC7FEBFFC06C13F814FE6C7F6D13807F010F13 C01300143F141F140F123E127E00FE1480A348EB1F0012E06C133E00705B6C5B381E03E0 6CB45AD801FEC7FC1C267AA422>II<13F8D803FEEB01C0D8078FEB03E0390E0F8007121E121C0038140F131F0078 15C01270013F131F00F0130000E015805BD8007E133FA201FE14005B5D120149137EA215 FE120349EBFC0EA20201131E161C15F813E0163CD9F003133814070001ECF07091381EF8 F03A00F83C78E090393FF03FC090390FC00F00272679A42D>I<01F0130ED803FC133FD8 071EEB7F80EA0E1F121C123C0038143F49131F0070140FA25BD8F07E140000E08013FEC6 485B150E12015B151E0003141C5BA2153C000714385B5DA35DA24A5A140300035C6D48C7 FC0001130E3800F83CEB7FF8EB0FC0212679A426>I<01F01507D803FC903903801F80D8 071E903907C03FC0D80E1F130F121C123C0038021F131F49EC800F00701607A249133FD8 F07E168000E0ED000313FEC64849130718000001147E5B03FE5B0003160E495BA2171E00 070101141C01E05B173C1738A217781770020314F05F0003010713016D486C485A000190 391E7C07802800FC3C3E0FC7FC90393FF81FFE90390FE003F0322679A437>I<903907E0 07C090391FF81FF89039787C383C9038F03E703A01E01EE0FE3803C01F018013C0D80700 14FC481480000E1570023F1300001E91C7FC121CA2C75AA2147EA214FEA25CA21301A24A 1370A2010314F016E0001C5B007E1401010714C000FEEC0380010F1307010EEB0F003978 1CF81E9038387C3C393FF03FF03907C00FC027267CA427>I<13F0D803FCEB01C0D8071E EB03E0D80E1F1307121C123C0038140F4914C01270A249131FD8F07E148012E013FEC648 133F160012015B5D0003147E5BA215FE00075C5BA214015DA314035D14070003130FEBF0 1F3901F87FE038007FF7EB1FC7EB000F5DA2141F003F5C48133F92C7FC147E147C007E13 FC387001F8EB03E06C485A383C1F80D80FFEC8FCEA03F0233679A428>I<903903C00380 90380FF007D91FF81300496C5A017F130E9038FFFE1E9038F83FFC3901F007F849C65A49 5B1401C7485A4A5A4AC7FC141E5C5C5C495A495A495A49C8FC131E5B49131C5B4848133C 48481338491378000714F8390FF801F0391FFF07E0383E1FFFD83C0F5B00785CD8700790 C7FC38F003FC38E000F021267BA422>II<001E1338007F13FEEA FF811383A3EB03FC00FE13F8383800F017096AB72A>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmsl12 12 7 /Fd 7 117 df97 D I101 DI111 D<903903F007E0D801FFEB3FF8 48EC787CEDE1FC39000FF1C1903807F383ECE70314EE9138EC01F89138FC00E04A130013 0F5CA35CA2131F5CA5133F5CA5137F91C8FCA55B5BA31201487EB6FCA25C262C7DAB26> 114 D<14C0A313015CA21303A21307A249C7FCA25B5B5B5B485A1203001FB512F0B6FCA2 C648C7FC12015BA512035BA512075BA5120F5BA215C0A3001FEB018013C0A414031500A2 5C1406000F130E6D5A00075B6C6C5AC6B45AEB3F801C3E77BC26>116 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmsy6 6 1 /Fe 1 1 df0 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmmi6 6 3 /Ff 3 117 df<1338137CA2137813701300A7EA0780EA1FC0EA38E01230EA60F0EAC1E0 A3EA03C0A3EA0780A2EA0F0013041306EA1E0CA21318121CEA1E70EA0FE0EA07800F237D A116>105 D<380F01F0381FC7F83831CE1CEA61F8EBF03C00C1137C13E014383803C000 A4485AA448C7FCA4121EA2120C16177D951D>114 D<133013785BA4485AA4485AB51280 A23803C000485AA448C7FCA4121EA25B1480383C03001306A25BEA1C38EA0FF0EA07C011 217D9F18>116 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmr6 6 4 /Fg 4 52 df<1438B2B712FEA3C70038C7FCB227277C9F2F>43 D<13E01201120712FF12 F91201B3A7487EB512C0A212217AA01E>49 D I<13FF000313C0380F03E0381C00F014F8003E13FC147CA2001E13FC120CC712F8A2EB01 F0EB03E0EB0FC03801FF00A2380003E0EB00F01478147C143E143F1230127812FCA2143E 48137E0060137C003813F8381E03F0380FFFC00001130018227DA01E>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh msbm10 12 4 /Fh 4 91 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fi msam10 12 1 /Fi 1 4 df<007FBA1280BB12C0A300F0CB1203B3B3B3A6BBFCA36C198042447BC34D>3 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmex10 12 18 /Fj 18 113 df0 D<12E07E12787E7E7E7F6C7E6C7E7F12016C7E7F137C137E7FA26D7EA26D7EA26D7EA36D 7EA2801301A2801300A280A2147EA2147FA4801580A7EC1FC0B3A5EC3F80A715005CA414 7EA214FEA25CA213015CA213035CA2495AA3495AA2495AA249C7FCA2137E137C13FC5B48 5A12035B485A485A90C8FC121E5A5A5A5A1A777C832E>I8 D<12F012FEEAFFC0EA3FF0EA07FCEA01FE6C6C7E EB3FC06D7E130F801307801303B3B0801301A26D7E806E7E6E7E6E7EEC07F8EC01FC9138 007F80ED1FE01503151FED7F80913801FC00EC07F8EC1FE04A5A4A5A4AC7FC5C495AA213 035CB3B013075C130F5C131F495AEBFF804848C8FCEA07FCEA3FF0EAFFC048C9FC12F023 7775833A>I<12F0B3B3B3AA0440728121>12 D<16F01501ED03E0ED07C0ED0F80ED1F00 5D157E5D5D14014A5A4A5A4A5AA24A5A143F92C7FC147EA25C13015C13035C13075C130F 5C131FA2495AA349C8FCA213FEA312015BA212035BA21207A25BA2120FA25BA2121FA45B A2123FA55B127FA990C9FC5AB3AA7E7FA9123F7FA5121FA27FA4120FA27FA21207A27FA2 1203A27F1201A27F1200A3137FA26D7EA36D7EA2130F801307801303801301801300147E A28081141F6E7EA26E7E6E7E6E7E140081157E8181ED0F80ED07C0ED03E0ED01F0150024 B26E833B>16 D<12F07E127C7E7E6C7E7F6C7E6C7E12017F6C7E137E7FA26D7E80130F6D 7EA26D7E80130180130080147E147F8081A26E7EA36E7EA26E7EA3811403A2811401A281 A21400A281A281A21680A4153FA216C0A5151F16E0A9150F16F0B3AA16E0151FA916C015 3FA51680A2157FA41600A25DA25DA21401A25DA214035DA214075DA34A5AA24A5AA34A5A A292C7FC5C147E14FE5C13015C13035C495AA2495A131F5C49C8FCA2137E5B485A5B1203 485A485A5B48C9FC123E5A5A5A24B27C833B>I<171E173E177C17F8EE01F0EE03E0EE07 C0160FEE1F80EE3F00167E167C16FC4B5A4B5A15075E4B5A4B5A153F93C7FC5D15FE5D14 015D14034A5AA24A5AA24A5AA24A5AA24AC8FCA214FEA213015C13035C1307A25C130F5C 131FA25C133FA3495AA349C9FCA35A5BA312035BA31207A25BA2120FA35BA3121FA35BA3 123FA55BA2127FAB485AB3B06C7EAB123FA27FA5121FA37FA3120FA37FA31207A27FA212 03A37F1201A37F7EA36D7EA36D7EA3131F80A2130F80130780A21303801301801300A214 7FA26E7EA26E7EA26E7EA26E7EA26E7E140181140081157F8182151F6F7E6F7E8215036F 7E6F7E167C167E82EE1F80EE0FC01607EE03E0EE01F0EE00F8177C173E171E2FEE6B8349 >I<12F07E127C7E7E6C7E6C7E7F6C7E6C7E6C7E137C137E7F6D7E80130F6D7E6D7E8013 01806D7E147E147F80816E7EA26E7EA26E7EA26E7EA26E7EA26E7EA2818182153F82A215 1F82150F82A2150782A36F7EA36F7EA38281A31780167FA317C0A2163FA217E0A3161FA3 17F0A3160FA317F8A51607A217FCABEE03FEB3B0EE07FCAB17F8A2160FA517F0A3161FA3 17E0A3163FA317C0A2167FA21780A316FF1700A35D5EA34B5AA34B5AA35E150FA25E151F 5E153FA25E157F93C7FC5D5DA24A5AA24A5AA24A5AA24A5AA24A5AA24A5A92C8FC5C147E 14FE495A5C13035C495A495A131F5C49C9FC137E137C13FC485A485A485A5B485A48CAFC 123E5A5A5A2FEE7C8349>I<170F173F17FF1603EE0FFCEE1FF0EE7FE0EEFF804B13004B 5A4B5A4B5A4B5A4B5A4B5A15FF5E5C93C7FC5C5D14075DA3140F5DB3B3B3AE4A5AA3143F 5DA24A5AA24A5AA24990C8FC495AA2495A495A495A495A495A49C9FC485AEA07FCEA0FF0 EA3FC0B4CAFC12FCA2B4FCEA3FC0EA0FF0EA07FCEA01FE6C7EEB7FC06D7E6D7E6D7E6D7E 6D7EA26D7E6D7FA26E7EA26E7EA281141FA36E7EB3B3B3AE811407A38114038180828082 157F6F7E6F7E6F7E6F7E6F7E6F7E6F1380EE7FE0EE1FF0EE0FFCEE03FF1600173F170F30 EE73834B>26 D56 D58 D<913807FF80B3B3B04A1300A55D141FA35D143F5DA2147F5D14FF5DA249 5B5D5B4990C7FC5C130F5C495A495A495AA2495A485B4890C8FCEA07FC485A485AEA7FE0 EAFF8090C9FC12FCB4FC7FEA7FE0EA1FF06C7E6C7E6CB4FC6C7F6C7F6D7EA26D7E6D7E6D 7E801307806D7F7F816D7FA281147F81143FA281141F81A3140F81A56E1380B3B3B021B5 6F8059>60 D<003EF407C0007FF40FE0486CF31FF0B3B3B3B3B3A56D1B3F007F1DE0A46D 1B7F003F1DC0A26D1BFF001F1D806D62A26C6C501300A26C6C505A6D1A0F6C6D4F5AA26C 6D4F5A6E197F6C6D4F5A6D6C4E5B6D6C4E5B6E606D6C4E5B6D01C0053F90C7FC6D6D4D5A 6D01F84C485A6D01FE04075B6D6D6C031F5B6E01E0037F5B021F01FE0207B512806ED9FF F090B6C8FC020391B712FC6E606E6C17E0031F178003074CC9FC030116F8DB003F15C004 0302FCCAFCDC001F1380648B7B7F6F>91 D<153015FC4A7E913807FF80021F13E0027F13 F89138FFCFFC0103EB03FF90260FFC0013C0D93FF0EB3FF0D97FC0EB0FF84848C7EA03FE D807F89138007F80D81FE0ED1FE0D87F80ED07F800FEC9EA01FC00F8EE007C00E0171C36 1280C937>98 D101 DI<1DC0F401E01C03A2F407C0A2F40F80A2F41F00A21C3EA264A2 64A2641B01A2515AA2515AA2515AA251C7FCA21B3EA263A263A2505AA2505AA2505AA250 5AA250C8FCA21A3EA21A3C1A7CA262A24F5AA24F5AA24F5AA24F5AA24FC9FCA20104173E 130E011E5F137F495F5A486D4B5A120F261C7FC04B5A123826F03FE04B5A124000004D5A 6D7E96CAFC6D6C5DA26D6C153EA2606D7E606D7E4D5A6D7F4D5AA26E6C495AA26E6C495A A26E6C49CBFCA26E6C133EA25F6E7E5F6E7E4C5AEC01FF4C5AA26EEB83C01687ED7FC7EE CF80ED3FEF04FFCCFCA26F5AA26F5AA26F5AA25E15035E6F5A5B78758364>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmsy8 8 10 /Fk 10 70 df0 D<130C131EA50060EB01800078130739FC0C0F C0007FEB3F80393F8C7F003807CCF83801FFE038007F80011EC7FCEB7F803801FFE03807 CCF8383F8C7F397F0C3F8000FCEB0FC039781E078000601301000090C7FCA5130C1A1D7C 9E23>3 D<140381B3A3B812FCA3C7D80380C7FCB3B812FCA32E2F7CAD37>6 D<170EA3170F8384170384170184717E1878187C84180FF007C0BA12F819FC19F8CBEA07 C0F00F00183E601878604D5A60170360170795C7FC5F170EA33E237CA147>33 D<137813FE1201A3120313FCA3EA07F8A313F0A2EA0FE0A313C0121F1380A3EA3F00A312 3E127E127CA35AA35A0F227EA413>48 DI<4A7E1403B3B3 A6007FB712FEB8FC7E2F2E7CAD38>63 D67 D<91383FFFF00107B6FC011F15E0017F812701F83E0313FCD803C09038003FFED80F00EC 07FF001E017E6D1380481500007CEE7FC00078163F48EE1FE048137CC7150FA202FC1407 A25CA3010116C05CA2EF0F8013034A15005F171E49485C177C177849485C4C5A4C5A49C7 485A040EC7FC163C013E14F0ED03E0013CEB0F80017C01FEC8FC90387FFFF848B512E048 49C9FC4813E0332D7EAC37>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmmi8 8 33 /Fl 33 123 df12 D<90B6128012035A481500261E 00E0C7FC5A00705B130112E012C0EA0003A25CA21307A349C8FCA35BA2131E133EA45BA2 1338211E7E9C1F>28 D<1506A3150E150CA3151C1518A315381530A31570D801E0EB6007 D807F8EC1F80EA0E3CD81C3E01E013C0003814C00030150F0070150726607E011480D8E0 7CEB800312C013FC3880F803000002001300120113F04A5B00030106130601E0140E160C 020E131C020C131801C0143801E05C021C5B91381801C0D801F0495A030FC7FC3900FC38 1C90383F30F890380FFFE0010190C8FCEB00701460A314E05CA313015CA42A3C7EAD2E> 32 D34 D39 D<123C127EB4FCA21380A2127F123D1201A312 031300A25A1206120E5A5A5A126009157A8714>59 D<147F903801FFE090380780F89038 0E003C497F497F49148001781307017C14C001FC130316E0A2137090C7FC16F0A314FE90 3807FF8390381F01C390397C00E7E049137748481337D807E0133F49131F484814C0121F 48C7FCA2481580127EA2ED3F0012FE48147EA2157C15FC5D4A5A007C495AA26C495A001E 49C7FC6C133E3807C0F83803FFE038007F8024307DAE25>64 D<013FB512FEEEFFC0903A 00FE0007F0EE01F84AEB007E8301018118804A140F18C00103150718E05CA21307A25CA2 130FA24A140FA2131F18C04A141FA2013F1680173F91C81300A249157EA2017E5D5F01FE 14014C5A494A5A4C5A00014BC7FC163E4914FCED03F00003EC1FC0B7C8FC15F8332D7CAC 3A>68 D77 D<013FB512F816FF903A00FE001FC0EE07E04A6D7E707E01016E7EA24A80A213034C5A5C A201074A5A5F4A495A4C5A010F4A5A047EC7FC9138C003F891B512E04991C8FC9138C007 C04A6C7E6F7E013F80150091C77EA2491301A2017E5CA201FE1303A25BA20001EE038018 005B5F0003913801FC0EB5D8E000133CEE7FF0C9EA0FC0312E7CAC35>82 D<3B7FFFF801FFFEA2D801FCC7EA0FC0178049EC070016060003150E160C5BA20007151C 16185BA2000F153816305BA2001F157016605BA2003F15E05E90C8FCA24814015E127EA2 150300FE92C7FC5A5D1506150E007C5C151815386C5C5D6CEB03C0260F800FC8FC3803E0 3C3801FFF038003FC02F2E7BAC30>85 DI<90260FFFFCEB7FFFA29026007FC0EB0FF06E48148018006E6C 131E1718020F5C6F5B02075C6F485A020349C7FCEDF8065E6E6C5A5E6E6C5A5EED7F8093 C8FC6F7EA26F7E153F156FEDCFE0EC018791380307F0EC0703020E7F141C4A6C7E14704A 6C7E495A4948137F49C7FC010E6E7E5B496E7E5BD801F081D807F8143FD8FFFE0103B5FC A2382D7EAC3A>88 DI<0107B612F8A2903A0FFC0007F002E0EB0FE00280EB1FC049C71380011E 143F011CEC7F004914FE4B5A0130495A0170495A0160495A4B5A4B5A90C748C7FCA215FE 4A5A4A5A4A5A4A5A4A5A4A5A4AC8FC14FE5C130149481306495A4948130E4948130C495A 49C7121C01FE141848481438485A5E484814F048481301484813034848495A48C7127FB7 FC5E2D2D7CAC30>I<151FEC03FFA2EC003FA2153EA2157EA2157CA215FCA215F8A21401 EB07E190381FF9F0EB7C1DEBF80FEA01F03903E007E0EA07C0120FEA1F8015C0EA3F0014 0F5A007E1480A2141F12FE481400A2EC3F021506143E5AEC7E0E007CEBFE0C14FC010113 1C393E07BE18391F0E1E38390FFC0FF03903F003C0202F7DAD24>100 D<131FEA03FFA2EA003FA2133EA2137EA2137CA213FCA25BA21201143F9038F1FFC09038 F3C1F03803FF0001FC7F5BA2485A5BA25B000F13015D1380A2001F13035D1300140748EC C04016C0003E130F1580007E148191381F0180007C1403ED070000FCEB0F06151E48EB07 F80070EB01E0222F7DAD29>104 D<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01 E0EA07F8EA0E3CEA1C3E123812301270EA607EEAE07C12C013FC485A120012015B12035B A21207EBC04014C0120F13801381381F01801303EB0700EA0F06131EEA07F8EA01F0122E 7EAC18>I<15E0EC01F01403A3EC01C091C7FCA9147CEB03FE9038078F80EB0E07131C01 3813C01330EB700F0160138013E013C0EB801F13001500A25CA2143EA2147EA2147CA214 FCA25CA21301A25CA21303A25CA2130700385BEAFC0F5C49C7FCEAF83EEAF0F8EA7FF0EA 1F801C3B81AC1D>I<131FEA03FFA2EA003FA2133EA2137EA2137CA213FCA25BA2120115 F89038F003FCEC0F0E0003EB1C1EEC387EEBE07014E03807E1C09038E3803849C7FC13CE EA0FDC13F8A2EBFF80381F9FE0EB83F0EB01F81300481404150C123EA2007E141C151800 7CEBF038ECF83000FC1470EC78E048EB3FC00070EB0F801F2F7DAD25>I<137CEA0FFCA2 1200A213F8A21201A213F0A21203A213E0A21207A213C0A2120FA21380A2121FA21300A2 5AA2123EA2127EA2127CA2EAFC08131812F8A21338133012F01370EAF860EA78E0EA3FC0 EA0F000E2F7DAD15>I<27078007F0137E3C1FE01FFC03FF803C18F0781F0783E03B3878 E00F1E01263079C001B87F26707F8013B00060010013F001FE14E000E015C0485A491480 0081021F130300015F491400A200034A13076049133E170F0007027EEC8080188149017C 131F1801000F02FCEB3F03053E130049495C180E001F0101EC1E0C183C010049EB0FF000 0E6D48EB03E0391F7E9D3E>I<3907C007E0391FE03FF83918F8783E393879E01E39307B 801F38707F00126013FEEAE0FC12C05B00815C0001143E5BA20003147E157C5B15FC0007 ECF8081618EBC00115F0000F1538913803E0300180147016E0001F010113C015E390C7EA FF00000E143E251F7E9D2B>II<90387C01F89038FE07FE3901CF8E0F3A03879C0780D907B813C0000713F0000690 38E003E0EB0FC0000E1380120CA2D8081F130712001400A249130F16C0133EA2017EEB1F 80A2017C14005D01FC133E5D15FC6D485A3901FF03E09038FB87C0D9F1FFC7FCEBF0FC00 0390C8FCA25BA21207A25BA2120FA2EAFFFCA2232B829D24>I<903807E03090381FF870 90387C1CF0EBF80D3801F00F3903E007E0EA07C0000F1303381F800715C0EA3F00A24813 0F007E1480A300FE131F481400A35C143E5A147E007C13FE5C1301EA3E07EA1F0E380FFC F8EA03F0C7FC13015CA313035CA21307A2EBFFFEA21C2B7D9D20>I<3807C01F390FF07F C0391CF8E0E0383879C138307B8738707F07EA607E13FC00E0EB03804848C7FCA2128112 015BA21203A25BA21207A25BA2120FA25BA2121FA290C8FC120E1B1F7E9D20>II<130E131FA25BA2133EA2137EA2137CA2 13FCA2B512F8A23801F800A25BA21203A25BA21207A25BA2120FA25BA2001F1310143013 001470146014E0381E01C0EB0380381F0700EA0F0EEA07FCEA01F0152B7EA919>I119 D<013F137C9038FFC1FF3A01C1E383803A0380F703C0390700F60F000E13FE4813FC1218 0038EC0700003049C7FCA2EA200100005BA313035CA301075B5D14C000385CD87C0F1306 00FC140E011F130C011B131C39F03BE038D8707113F0393FE0FFC0260F803FC7FC221F7E 9D28>II< 011E1330EB3F809038FFC07048EBE0E0ECF1C03803C0FF9038803F80903800070048130E C75A5C5C5C495A495A49C7FC131E13385B491340484813C0485A38070001000EEB038038 0FE007391FF81F0038387FFF486C5A38601FFC38E00FF038C003C01C1F7D9D21>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmti12 12 53 /Fm 53 124 df<4CB414FC040F9039C003FF80933B3F81F00783C0933B7C00781F01E04C 9038F83F03923C01F001FC3E07F003030103EB7E0F922607E007EB7C1F19FCDB0FC001F8 14E0943A03F0F80FC0DD01E1EB0780031FD9000190C7FC5E180361153F93C7FCA2180761 5D157EA2180F6115FE91B912F0A3DA00FCC7D81F80C7FC1401A25D183F96C8FCA214035D A260187E14075DA218FE60140F5DA2170160141F5DA2170360143F92C7FCA21707605C14 7EA2170F6014FE5CA24D5AA2495A95C9FC5F5C0103153E177E001CEBE038007F02FE137C 26FF07E114FC02C15C4C5AEB0F8100FE903901FC03E0D8F81F9038F007C03B701E00E00F 80D8783CD9F83ECAFCD81FF0EB3FF8D807C0EB0FE04C5A83C53C>11 DI14 D<167016E0ED01C0ED0380ED0700150E153C5D15F85D4A5A4A5A4A5A140F4AC7FC141E14 3E5C147814F8495A5C1303495AA2495AA249C8FCA25B133E137E137CA25BA212015BA212 035BA212075BA2120FA25BA2121FA290C9FCA25AA2123EA3127EA2127CA65AAB1278A612 7C123CA47EA2120E120FA27E6C7EA26C7EA26C7E1360246472CA28>40 D<1560A2157081A281151E150E150FA2811680A3ED03C0A516E0A21501A71503A91507A2 16C0A4150FA21680A2151FA21600A25DA2153EA2157EA2157C15FCA25D1401A25D14035D A214075D140F5DA24AC7FCA2143EA25C147814F8495AA2495A5C1307495A91C8FC131E13 3E5B13785B485A485A485A48C9FC121E5A5A12E05A23647FCA28>I<13F0EA03FC1207A2 EA0FFEA4EA07FCEA03CCEA000C131C1318A2133813301370136013E0EA01C013801203EA 0700120E5A5A5A5A5A0F1D7A891E>44 D<007FB5FCB6FCA214FEA21805789723>I<120F EA3FC0127FA212FFA31380EA7F00123C0A0A76891E>I<16C01501A215031507ED0F8015 1F153F157F913801FF005C140F147F903807FCFEEB0FF0EB0700EB00015DA314035DA314 075DA3140F5DA3141F5DA3143F5DA3147F92C7FCA35C5CA313015CA313035CA313075CA2 130FA2131F133FB612FCA25D224276C132>49 DII<130FEB1FC0133FEB7FE013FFA214 C0EB7F801400131E90C7FCB3A5120FEA3FC0127FA212FFA35B6CC7FC123C132B76AA1E> 58 D65 D<91B712FCF0FF8019E00201903980001FF06E90C7EA07F84A6F7E727E4B 81841A800203167F5DA314075D19FFA2020F17004B5C611803021F5E4B4A5A180F4E5A02 3F4B5A4BEC7F804EC7FCEF03FC027FEC0FF84BEBFFC092B6C8FC18E0913AFF800007F892 C7EA01FC717E187F49834A6F7EA30103835CA313075CA3010F5F4A157FA24E5A131F4A4A 90C7FC601703013F4B5A4A4A5A4D5A017F4B5A4D5A4A4948C8FC01FFEC0FFEB812F817C0 4CC9FC41447AC345>II<91B912C0A30201902680000313806E90C8127F4A163F191F4B150FA30203EE 07005DA314074B5D190EA2140F4B1307A25F021F020E90C7FC5DA2171E023F141C4B133C 177C17FC027FEB03F892B5FCA39139FF8003F0ED00011600A2495D5CA2160101034B1370 5C19F061010791C8FC4A1501611803010F5F4A150796C7FC60131F4A151E183E183C013F 167C4A15FC4D5A017F1503EF0FF04A143F01FF913803FFE0B9FCA26042447AC342>69 D<91B91280A30201902680000713006E90C8FC4A163FA24B81A30203160E5DA314074B15 1E191CA2140F5D17075F021F020E90C7FC5DA2171E023F141C4B133CA2177C027F5CED80 0392B5FCA291B65AED00071601A2496E5A5CA2160101035D5CA2160301075D4A90CAFCA3 130F5CA3131F5CA3133F5CA2137FA313FFB612E0A341447AC340>II<027FB512E091B6FCA202 00EBE000ED7F8015FFA293C7FCA35C5DA314035DA314075DA3140F5DA3141F5DA3143F5D A3147F5DA314FF92C8FCA35B5CA313035CA313075CA3130F5CA3131F5CA2133FA25CEBFF E0B612E0A25D2B447BC326>73 D<91B612F0A25F020101C0C7FC6E5B4A90C8FCA25DA314 035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C9FCA35B5CA301031610 4A1538A21878010716705C18F018E0010F15015C18C01703011F15074A1580170FA2013F ED1F004A5C5F017F15FE16034A130F01FFEC7FFCB8FCA25F35447AC33D>76 D<91B56C93387FFFC08298B5FC02014DEBC0006E614A5FA203DF4C6CC7FC1A0E63912603 CFE05D038F5F1A381A711407030FEEE1FCA2F101C3020FEE0383020E60F107036F6C1507 021E160E021C60191CF1380F143C023804705BA2F1E01F0278ED01C091267003F85EF003 801A3F02F0ED070002E0030E5CA24E137F130102C04B91C8FC606201036D6C5B02805F4D 5A943803800113070200DA07005BA2050E1303495D010E606F6C5A1907011E5D011C4B5C A27048130F133C01384B5C017892C7FC191F01F85C486C027E5DD807FE027C4A7EB500F0 0178013FB512C0A216705A447AC357>I<91B712F018FEF0FF800201903980007FE06E90 C7EA1FF04AED07F818034B15FCF001FE1403A24B15FFA21407A25DA2140FF003FE5DA202 1F16FC18074B15F8180F023F16F0F01FE04B15C0F03F80027FED7F0018FE4BEB03FCEF0F F002FFEC7FC092B6C7FC17F892CAFC5BA25CA21303A25CA21307A25CA2130FA25CA2131F A25CA2133FA25CA2137FA25C497EB67EA340447AC342>80 D<91B77E18F818FE02019039 8001FF806E90C7EA3FC04AED1FE0F00FF04BEC07F8180319FC14034B15FEA314075DA302 0FED07FC5DA2F00FF8141F4B15F0F01FE0F03FC0023F16804BEC7F0018FEEF03F8027F4A 5A4BEB1FC04CB4C7FC92B512F891B612E092380003F8EE00FE177F496F7E4A6E7EA28413 034A140FA2171F13075CA2173F130F5CA24D5A131F5CA3013F170E5CA2017FEE801E191C 4A163C496C1638B66C90383FC070051F13F094380FE1E0CA3803FF80943800FE003F467A C347>82 DI<48B912F85AA291 3B0007FC001FF0D807F84A130701E0010F140349160148485C90C71500A2001E021F15E0 5E121C123C0038143F4C1301007818C0127000F0147F485DA3C800FF91C7FC93C9FCA35C 5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92CAFCA35B5CA213 03A21307497E007FB612C0A25E3D446FC346>I87 D97 DIIIII<15FCEC03FF9139 0F83838091393E01CFC091387C00EF4A13FF4948137F010315804948133F495A131F4A14 00133F91C75A5B167E13FE16FE1201495CA215011203495CA21503A2495CA21507A25EA2 150F151F5E0001143F157F6C6C13FF913801DF8090387C039F90383E0F3FEB0FFCD903F0 90C7FC90C7FC5DA2157EA215FEA25DA2001C495A127F48495A14074A5A485C023FC8FC00 F8137E387C01F8381FFFE0000390C9FC2A407BAB2D>I<14FE137FA3EB01FC13001301A2 5CA21303A25CA21307A25CA2130FA25CA2131FA25C157F90393F83FFC091388F81F09138 1E00F802387F4948137C5C4A137EA2495A91C7FCA25B484814FE5E5BA2000314015E5BA2 000714035E5B1507000F5DA249130F5E001F1678031F1370491480A2003F023F13F0EE00 E090C7FC160148023E13C01603007E1680EE070000FEEC1E0FED1F1E48EC0FF80038EC03 E02D467AC432>I<143C147E14FE1301A3EB00FC14701400AE137C48B4FC3803C7803807 03C0000F13E0120E121C13071238A21278EA700F14C0131F00F0138012E0EA003F1400A2 5B137EA213FE5B12015BA212035B141E0007131C13E0A2000F133CEBC038A21478EB8070 14F014E0EB81C0EA0783EBC7803803FE00EA00F8174378C11E>I<16F0ED03F8A21507A3 16F0ED01C092C7FCAEEC01F0EC07FCEC1E1EEC380F0270138014E0130114C0EB03800107 131F1400A2130E153F131E011C140090C7FC5DA2157EA215FEA25DA21401A25DA21403A2 5DA21407A25DA2140FA25DA2141FA25DA2143FA292C7FCA25C147EA214FE001C5B127F48 485A495AA248485A495AD8F81FC8FCEA707EEA3FF8EA0FC0255683C11E>I<14FE137FA3 EB01FC13001301A25CA21303A25CA21307A25CA2130FA25CA2131FA25C167E013F49B4FC 92380783C09138000E07ED3C1F491370ED603F017E13E0EC01C09026FE03801380913907 000E00D9FC0E90C7FC5C00015B5C495AEBF9C03803FB8001FFC9FCA214F03807F3FCEBF0 7F9038E01FC06E7E000F130781EBC003A2001F150FA20180140EA2003F151E161C010013 E0A2485DA2007E1578167000FE01015B15F1489038007F800038021FC7FC2A467AC42D> IIIIII<91381F800C91387FE01C903901F0703C903907C0387890390F 801CF890381F001D013E130F017E14F05B48481307A2484814E012075B000F140F16C048 5AA2003F141F491480A3007F143F90C71300A35D00FE147EA315FE5DA2007E1301A24A5A 1407003E130FA26C495A143B380F80F33807C3E73901FF87E038007E071300140F5DA314 1F5DA3143F92C7FCA25CA25C017F13FEA25D263F76AB2D>III<1470EB01F8A313035CA313075CA3130F5CA3 131F5CA2007FB512E0B6FC15C0D8003FC7FCA25B137EA313FE5BA312015BA312035BA312 075BA3120F5BA2EC0780001F140013805C140E003F131EEB001C143C14385C6C13F0495A 6C485AEB8780D807FEC7FCEA01F81B3F78BD20>I<137C48B414072603C780EB1F803807 03C0000F7F000E153F121C0107150012385E1278D8700F147E5C011F14FE00F05B00E05D EA003FEC0001A2495C137E150313FE495CA215071201495CA2030F13380003167849ECC0 70A3031F13F0EE80E0153F00011581037F13C06DEBEF8300000101148090397C03C78790 3A3E0F07C70090391FFE01FE903903F000782D2D78AB34>I<017C143848B414FC3A03C7 8001FE380703C0000F13E0120E001C14000107147E1238163E1278D8700F141E5C131F00 F049131C12E0EA003F91C7123C16385B137E167801FE14705BA216F0000115E05B150116 C0A24848EB0380A2ED0700A2150E12015D6D5B000014786D5B90387C01E090383F0780D9 0FFFC7FCEB03F8272D78AB2D>I<017CEE038048B4020EEB0FC02603C780013FEB1FE038 0703C0000E7F5E001C037E130F01071607123804FE130300785DEA700F4A1501011F1301 00F001804914C012E0EA003FDA000314034C14805B137E0307140701FE1700495CA2030F 5C0001170E495CA260A24848495A60A2601201033F5C7F4B6C485A000002F713036D9039 E7E0078090267E01C349C7FC903A1F0781F81E903A0FFF007FF8D901FCEB0FE03B2D78AB 41>I<02F8133FD907FEEBFFE0903A0F0F83C0F0903A1C07C780F890393803CF03017013 EE01E0EBFC07120101C013F8000316F00180EC01C000074AC7FC13001407485C120EC7FC 140F5DA3141F5DA3143F92C8FCA34AEB03C01780147EA202FEEB0700121E003F5D267F81 FC130E6E5BD8FF83143CD903BE5B26FE079E5B3A7C0F1F01E03A3C1E0F83C0271FF803FF C7FC3907E000FC2D2D7CAB2D>I<137C48B414072603C780EB1F80380703C0000F7F000E 153F001C1600130712385E0078157EEA700F5C011F14FE00F0495B12E0EA003FEC00015E 5B137E150301FE5C5BA2150700015D5BA2150F00035D5BA2151F5EA2153F12014BC7FC6D 5B00005BEB7C0390383E0F7EEB1FFEEB03F090C712FE5DA214015D121F397F8003F0A24A 5A4848485A5D48131F00F049C8FC0070137E007813F8383801F0381E07C06CB4C9FCEA01 FC294078AB2F>I<027C130749B4130F49EB800E010F141E49EBC03CEDE03890393F03F0 7890397C00FDF00178EB3FE00170EB03C001F0148049130790C7EA0F00151E5D5D5D4A5A 4A5A4A5A4AC7FC141E5C5C5C495A495A495A49C8FC011E14F04914E05B491301485A4848 EB03C0D807B0130701FEEB0F80390FCF801F3A1F07E07F00393E03FFFED83C015B486C5B 00705C00F0EB7FC048011FC7FC282D7BAB28>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmr8 8 12 /Fn 12 94 df6 D<013FB5FCA29038007F806EC8FCA690 3801FFE0011F13FE90397F3F3F80D801F8EB07E0D807E0EB01F8D80FC06D7ED81F80147E D83F0080481680A2007E151F00FE16C0A5007E1680007F153FA26C1600D81F80147ED80F C05CD807E0495AD801F8EB07E0D8007FEB3F8090261FFFFEC7FC010113E0D9003FC8FCA6 4A7E013FB5FCA22A2D7CAC33>8 D<13031307130E131C1338137013F0EA01E013C01203 EA0780A2EA0F00A2121EA35AA45AA512F8A25AAB7EA21278A57EA47EA37EA2EA0780A2EA 03C0120113E0EA00F013701338131C130E1307130310437AB11B>40 D<12C07E12707E7E7E120FEA0780120313C0EA01E0A2EA00F0A21378A3133CA4131EA513 1FA2130FAB131FA2131EA5133CA41378A313F0A2EA01E0A2EA03C013801207EA0F00120E 5A5A5A5A5A10437CB11B>I43 D48 D<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387F FFFEA2172C7AAB23>III<140EA2141E143EA2 147E14FEA2EB01BE1303143E1306130E130C131813381330136013E013C0EA0180120313 001206120E120C5A123812305A12E0B612FCA2C7EA3E00A9147F90381FFFFCA21E2D7EAC 23>I91 D93 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmmi12 12 62 /Fo 62 123 df11 DI I<1578913807FFE0021F13FC91383C7FFEEC7007EC6003ECE0004A13381600A280A380A2 80147CA2147E143E143F816E7EA26E7E81140781EC3FFC14FF903803E1FEEB07C190381F 00FF133E49EB7F805B0001143F485A484814C049131F120F485AA248C7FC150F5A127EA3 00FEEC1F805AA316005A5DA2153E157E157CA26C5C127C4A5A6C495AA26C495A6C6C485A 6C6C48C7FC3803E07C3800FFF0EB1FC027487CC62B>I<157E913801FF80913807C3E091 381F01F0EC3E004A13F814FC4948137C495A5C0107147E495A131F5C133F49C7127FA213 FEA212015B12034914FF1207A25B000F15FE1501A2485AA21503003F15FC5B90B6FCA248 15F89038800007A2150F00FF15F090C7FCA2ED1FE0A25AED3FC0A21680157F16005A15FE A24A5AA25D14035D4A5A007C495AA24A5A007E49C7FC003E133E5C001E5B6C485A380783 C06CB4C8FCEA00FC28477CC52D>18 D<010FB712E0013F16F05B48B812E04817C02807E0 060030C7FCEB800EEA0F00001E010C13705A0038011C13605A0060011813E000E0133812 40C7FC5C4B5AA214F014E01301150314C01303A3EB078082130FA2EB1F00A34980133E13 7EA24980A2000114015BA26C48EB00E0342C7EAA37>25 DI<0203B612E0021F15F091B7FC4916E0010716C09027 0FF80FF8C7FC90381FC00349486C7E017EC7FC49147E485A4848143E0007153F5B485AA2 485AA2123F90C8FC5E48157E127EA216FE00FE5D5A15015EA24B5A007C5D15074B5A5E6C 4AC8FC153E6C5C5D390F8003F03907C007C02601F03FC9FC38007FFCEB1FE0342C7DAA37 >I<137E48B46C150626078FE0150E260607F0151C260E03F81538000C6D1570D81C0116 E000006D15C0010015016EEC03806EEC0700170E6E6C5B5F5F6E6C136017E04C5A6E6C48 5A4CC7FC0207130E6F5A5E1630913803F8705EEDF9C06EB45A93C8FC5D6E5A81A2157E15 FF5C5C9138073F80140E141C9138181FC014381470ECE00FD901C07FEB038049486C7E13 0E130C011C6D7E5B5B496D7E485A48488048C8FC000681000E6F137048EE806048033F13 E04892381FC0C048ED0FE348923803FF00CA12FC37407DAB3D>31 D<1730A317701760A317E05FA316015FA3160394C8FCA35E1606A3160E160C013E1607D9 FF80ED1F802603C3C0011CEB3FC0260703E01318260601F0157F000E173F001C1538D818 030230131F0038170F0030170700701570D86007026013035CA2D8E00F02E0148000C049 491301EA001F4A150303011500013F5C1400604901031406017E91C7FC180E180C01FE49 141C4901061418183860030E1460030C14E04D5A4D5A031C49C7FC0318130E017E5D5F6D 01385B90261F80305BD90FC0EB03C0D907F0010FC8FC903901FE707C9039003FFFF00203 1380DA0060C9FC15E05DA314015DA3140392CAFCA35C1406A3140E140C3A597DC43F>I< 0110160E0138163F0178EE7F80137001F016FF4848167F5B0003173F49161F120790CA12 0FA2000E1707A248180015060018140FA20038021E14061230A2180E00704A140C126018 1CA203381418183800E05C6015F86C01015D170114030078D907BC495ADA0FBE1307007C D91F3E495A007ED97E3F013FC7FC3B7F83FE1FE0FF263FFFFCEBFFFE4A6C5B6C01F05C6C D9C0075B6CD9000113C0D801FC6D6CC8FC392D7FAB3D>II<177F0130913803FFC00170020F13E0494A13F04848 4A13F84848EC7F0190C838F8007C484A48133C00064A48131C000E4B131E000C4A48130E 001C92C7FC0018140E150C0038021C1406003014185D180E00704A140C1260154003C014 1C00E017184A5A4817386C17304AC8127018E01260007049EC01C0EF0380007801061407 0038EE0F00003C010E141E6C167CD81F805D6C6C48EB03F0D807F0EC0FE0D803FEEC3FC0 2801FFFC03FFC7FC6C6CB55A6D14F8010F14E0010114809026007FF8C8FC02F8C9FCA25C A21301A3495AA31307A25C130FA4131F5C6DCAFC37417BAB40>39 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A78891B>58 D<121EEA7F8012FF13C0A2 13E0A3127FEA1E601200A413E013C0A312011380120313005A1206120E5A5A5A12600B1D 78891B>II<1618163C167CA2167816F8A216F01501 A216E01503A216C01507A21680150FA2ED1F00A2151E153EA2153C157CA2157815F8A25D 1401A24A5AA25D1407A25D140FA292C7FC5CA2141E143EA2143C147CA25CA25C1301A25C 1303A25C1307A25C130FA291C8FC5BA2133EA2133C137CA2137813F8A25B1201A25B1203 A2485AA25B120FA290C9FC5AA2121E123EA2123C127CA2127812F8A25A126026647BCA31 >I<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FE903801FF8090 38007FE0EC1FF8EC03FE913800FF80ED3FE0ED0FF8ED03FF030013C0EE3FF0EE0FFCEE01 FF9338007FC0EF1FF0EF07FCEF01FF9438007FC0F01FE0A2F07FC0943801FF00EF07FCEF 1FF0EF7FC04C48C7FCEE0FFCEE3FF0EEFFC0030390C8FCED0FF8ED3FE0EDFF80DA03FEC9 FCEC1FF8EC7FE0903801FF80D907FECAFCEB1FF0EB7FC04848CBFCEA07FCEA1FF0EA7FC0 48CCFC12FC12703B3878B44C>I64 D<1830187018F0A217011703A24D7EA2170F171FA21737A217 6717E717C793380187FCA2EE0307EE07031606160CA216181638163004607FA216C00301 13011680ED0300A21506150E150C5D845D03707F15605DA24A5A4AB7FCA25C0206C87F5C 021C157F14185CA25C14E05C495A8549C9FC49163F1306130E5B133C137C01FE4C7ED807 FFED01FF007F01F0027FEBFFC0B5FC5C42477DC649>I<91B87E19F019FC02009039C000 03FF6F480100138003FFED3FC01AE093C8121FF10FF0A24A17F84B1507A314035D190FA2 020717F04B151F1AE0193F020F17C04BED7F80F1FF004E5A021F4B5A4B4A5AF01FF0F03F C0023F4AB4C7FC4BEB1FFC92B612F018FEDA7FC0C7EA7F804BEC1FC0F00FF0727E02FF6F 7E92C8FC727EA249835CA313035CA301075F4A1503A24E5A130F4A4B5A4E5AA2011F4C5A 4A4B5A4D485A013F4B48C7FCEF0FFC4AEC3FF801FF913801FFE0B9128005FCC8FC17C045 447CC34A>I<4CB46C1318043F01F013384BB512FC0307D9007E1378DB1FF090380F80F0 DB7F80EB03C1DA01FEC7EA01C34A48EC00E7DA0FF0ED7FE04A48153F4A5A02FFC9121F49 4817C04948160F495A130F4A178049481607495A137F4948170091CAFC5A485A1906485A A2485A96C7FC121F5BA2123F5BA3127F5BA4485AA419C0A2180161127F180396C7FC6018 066C6C160E601818001F17386D5E000F5F6D4B5A6C6C4B5A00034CC8FC6C6C150E6C6C15 3C017F5DD93FC0EB01E0D91FF0EB0FC0D907FE017FC9FC0101B512FCD9003F13E0020790 CAFC45487CC546>I<91B87E19F019FC02009039C00007FF6F489038007FC003FFED1FE0 737E93C86C7E737E19014A707E5D1A7FA20203EF3F805DA21BC014075DA3140F4B17E0A3 141F4B17C0A3143F4B167FA3027F18804B16FFA302FF180092C95A62A24917034A5F1907 6201034D5A5C4F5A620107173F4A5F4FC7FC19FE010F4C5A4A15034E5AF00FE0011F4C5A 4A4B5A06FFC8FC013FED01FCEF0FF84AEC3FE001FF913803FF80B848C9FC17F094CAFC4B 447CC351>I<91B912F8A3020001C0C7123F6F48EC07F003FF1503190193C9FCA21A705C 5DA3020317605DA314075D18C01701020F4B13005DA21703021F92C8FC4B5BA25F023F14 1E4B13FE92B5FCA24A5CED8000173CA202FF141892C7FCA217384915305CA21770010315 604A91C9FCA313075CA3130F5CA3131F5CA2133FA313FFB612F8A345447CC33F>70 D<91B6D8E003B61280A3020001E0C70003EB8000DB7F806E48C7FC03FF1503A293C85BA2 19075C4B5EA2190F14034B5EA2191F14074B5EA2193F140F4B5EA2197F141F4B5EA219FF 143F92B8C8FCA3DA7FC0C712014B5DA2180314FF92C85BA218075B4A5EA2180F13034A5E A2181F13074A5EA2183F130F4A5EA2187F131F4A5EA2013F16FFA24A93C9FCD9FFE00203 7FB6D8E003B67EA351447CC351>72 D<027FB512F8A217F09139007FF000ED3FC0157FA2 5EA315FF93C7FCA35C5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314 FF92C8FCA35B5CA313035CA313075CA3130F5CA2131FA25CEB7FF0007FB512F0B6FCA22D 447DC32B>I<031FB512FC5D18F89239000FFE00705AA35FA2160FA25FA2161FA25FA216 3FA25FA2167FA25FA216FFA294C7FCA25DA25EA21503A25EA21507A25EA2150FA25EA215 1FA25EA2153FA25EEA0F80D83FE0137F5E127FA24BC8FC485A4A5A1300006C495A006049 5A0070495A0030495A0038EB3F806C49C9FC380F81FC3803FFF038007F80364679C336> I<91B600E049B512C0A3020001E0C8383FF800DB7F80ED1FE003FF94C7FC1A3E93C91278 62F101C04A4C5A4B4BC8FC191C6102035E4B5DF003804EC9FC0207150E4B14386060020F 4A5A4B0107CAFC170E5F021F14784B13F84C7E1603023F130F4B487E163BEEE1FF91387F C1C1DB83807FED8700159CDAFFB86D7E5D03C06D7E5D4990C7FC4A6E7EA2717E13034A81 1707A201076F7E5C717EA2130F4A6E7FA2727E131F5C727E133F854A82D9FFE04B7EB600 E0010FB512E05FA252447CC353>I<91B612F8A3020001E0C8FC6F5A4B5AA293C9FCA35C 5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92CAFCA35B4A16C0 A21801010317804A15031900A201075E4A1506180E181E010F161C4A153C18381878011F 16F84A4A5A1703013F150F4D5A4A14FF01FF02075BB9FCA2603A447CC342>I<91B500C0 933803FFFE63630200F1FE00DB6FE0EE1BF803EF171F1B3703CFEF67F0A21BCF0201EF01 8F038F60DB87F0ED030F1B1F020317060307040C5BA2F2183F020717300206616F6C1560 1B7F020E17C0020CDC018090C7FCA24F485A021C16060218606F6C5C1A01023816180230 04305BA2F16003027016C00260606F6CEB01801A0702E0ED03004A03065CA24E130F0101 5E4A60047F5B1A1F01035E91C74A5CA24D48133F494BC7FC010661EE3F861A7F010E158C 010C039892C8FCA205B05C011C15E001186001386E5A190101785D01FC92C75BD803FFEF 07FEB500F8011E0107B512FE161C160C5F447BC35E>I<91B500C0020FB5128082A2DA00 7F9239007FE00070ED1F8074C7FCDBEFF8150E15CF03C7160C70151C1401DB83FE1518A2 DB81FF1538140303001630831A704A6D7E02061760163F7114E0140E020C6D6C5CA2706C 1301141C021801075D83190302386D7E023094C8FC1601715B147002606DEB8006A29438 7FC00E14E04A023F130C18E0191C0101ED1FF04A1618170FF0F838130391C83807FC30A2 943803FE705B01060301136018FF19E0010E81010C5F187FA2131C0118705A1338181F13 7801FC70C9FCEA03FFB512F884180651447CC34E>I<91B712F018FF19E002009039C000 3FF86F48EB07FC03FFEC01FEF0007F93C8EA3F801AC0F11FE05C5D1AF0A214035DA30207 EE3FE05DA2F17FC0020F17804B15FF1A004E5A021F4B5A4B4A5AF00FE04E5A023F037FC7 FC4BEB03FCEF1FF092B612804A4AC8FC923980007F80EF0FC0EF07F002FF6E7E92C77F17 01845B4A1400A2170113035CA2170313075CA24D5A130F5CA3011F18185CA2013F4C1338 1A304A6F1370D9FFE0020314E0B600E0ED01C00501EB0380943900FE0F00CBEA3FFEF007 F045467CC34A>82 D<9339FF8001800307EBF003033F13FC9239FF007E07DA01F8EB0F0F DA07E09038079F004A486DB4FC4AC77E023E804A5D187E5C495A183C495AA213074A1538 A3130F183080A295C7FC806D7E8014FF6D13E015FC6DEBFFC06D14FC6E13FF6E14C0020F 80020314F8EC003F03077F9238007FFE160F1603707E8283A283A21206A4000E163EA212 0C177E001E167CA25F5F003F15014C5A6D4A5A4C5A486C4AC8FC6D143ED87CF85CD8787E 495A3AF01FC00FE0D8E007B51280010149C9FC39C0003FF039487BC53C>I<48BA12C05A A291C7D980001380D807F092C7121F4949150F0180170748C75B1903120E48020316005E 12181238003014074C5C00701806126000E0140F485DA3C8001F92C7FC5EA3153F5EA315 7F5EA315FF93CAFCA35C5DA314035DA314075DA3140F5DA3141F5DA3143F5DA2147FA214 FF01037F001FB612FCA25E42447EC339>I<003FB500F80103B512E0A326003FF8C8381F F800D91FE0ED07E0013F705A615C96C7FC60017F16065CA2180E01FF160C91C9FCA2181C 4817185BA21838000317305BA21870000717605BA218E0120F495EA21701121F495EA217 03123F4993C8FCA25F127F491506A2170E00FF160C90C9FC171CA2171817381730481670 5F6C5E6C15014C5A4CC9FC6C150E6D141E001F5D6D5C6C6CEB01E06C6C495A6C6CEB1F80 C6B401FECAFC90387FFFF8011F13E0010190CBFC43467AC342>I<007FB56C91381FFFF8 B65DA2000101E0C8000313006C0180ED01FCF000F0614E5AA2017F4C5A96C7FC1806A260 6E5DA2013F5E1870186060A24D5A6E4AC8FCA2011F1506170E170C5FA26E5C5FA2010F5D 16015F4CC9FCA26E13065EA201075C5EA25E16E06E5B4B5A13034BCAFC1506A25D151CEC FE185D13015D5DA26E5AA292CBFC5C13005C5CA25CA25C45467BC339>II<023FB5 00E0011FB5FCA39126007FFEC7000313E0DB3FF8913801FE006F486E5A1AF06F6C4A5A62 6F6C4A5A0706C7FC190E6F6C5C616F6C5C6171485A6F5D4EC8FC93387FC00660706C5A60 60706C5A17F193380FFB8005FFC9FC5F705AA2707EA3707E5E04067F5E93381C7FC01638 16704C6C7EED01C04B486C7E160015064B6D7E5D4B6D7E5D5D4A486D7E14034AC76C7E14 0E5C4A6E7F143002E06F7E495A0103707E495A131F496C4B7E2603FFE04A487E007F01FC 021FEBFFF0B5FCA250447EC351>II<020FB812C05C1A809326 800001130003F8C7FCDA3FE04A5A03804A5A92C8485A027E4B5A027C4B5A02784B5A4A4B 5AA24A4A90C7FC4A4A5A01014B5A4D5A4A4A5A01034B5A91C8485A4D5AA290C84890C8FC 4C5A4C5A4C5A4C5A4C5A4C5A4C5AA24B90C9FC4B5A4B5A4B5A4B5A4B5A4B5AA24B5A4A90 CAFC4A5A4A4814064A5A4A5A4A48140E4A48140CA24A48141C4990C8121849481538495A 49485D495A494815F049485D1701494814034890C8485A4848150F4848151F48484B5A48 4815FF48481403043F90C8FC48B8FCB9FC5F42447BC343>I97 D100 DII<157E913803FF8091390FC1E0E091391F0073F0027E1333 4A133F4948131F010315E04948130F495AA2494814C0133F4A131F137F91C713805B163F 5A491500A25E120349147EA216FEA2495CA21501A25EA21503150700015D150F0000141F 6D133F017CEB77E090383E01E790381F078F903807FE0FD901F85B90C7FC151FA25EA215 3FA293C7FCA2001C147E007F14FE485C4A5A140348495AEC0FC000F8495A007C01FEC8FC 381FFFF8000313C02C407EAB2F>I<14FE137FA3EB01FC13001301A25CA21303A25CA213 07A25CA2130FA25CA2131FA25CED3FC090393F81FFF0913887C0FC91380E007E023C133E D97F70133F4A7F4A14805C13FF91C7FC5BA24848143F17005BA200035D167E5BA2000715 FE5E5B1501000F5DA24913035E001F1607030713064914E0150F003FEDC00E170C90C714 1CEE80184816381730007E167017E000FE91380781C0EEC38048913801FF000038EC007C 30467BC438>I<141E143F5C5CA3147E143891C7FCAE133EEBFF803801C3C0380781E038 0601F0120E121CEA180312381230A2EA700700605BA2EAE00F00C05BEA001F5CA2133F91 C7FCA25B137E13FE5BA212015BEC03800003140013F01207495A1406140E140CEBC01C14 1814385C00035BEBE1C0C6B45A013EC7FC19437DC121>I<163C16FEA21501A316FCED00 701600AE15FCEC03FF91380F0780021C13C091383803E0147014E014C01301EC80071303 14005B0106130F130E010C14C090C7FC151FA21680A2153FA21600A25DA2157EA215FEA2 5DA21401A25DA21403A25DA21407A25DA2140FA25DA2141F5DA2143F001C91C7FC127F48 137E5CA248485AEB03E038F807C038781F80D83FFEC8FCEA07F0275681C128>I<14FE13 7FA3EB01FC13001301A25CA21303A25CA21307A25CA2130FA25CA2131FA25C163F013FEC FFC0923803C0E09138000703ED1E0F491338ED701F017E13E0EC01C001FE018013C00203 EB07004948C8FC140E00015B5C495A5C3803FBC001FFC9FC8014F83807F1FE9038F03F80 9038E00FE06E7E000F130381EBC001A2001FED01C017801380A2003F15031700010013F0 5E481506160E007E150C161C00FE01005BED787048EC3FE00038EC0F802B467BC433>I< EB03F8EA01FFA3380007F013031307A214E0A2130FA214C0A2131FA21480A2133FA21400 A25BA2137EA213FEA25BA21201A25BA21203A25BA21207A25BA2120FA25BA2121FA25BA2 123FA290C7FCA248136014E0007E13C0A2130100FE138012FCA21303007C13005B1306EA 3E0EEA1E1CEA0FF8EA03E015467CC41D>I<01F8D903FCEC7F80D803FED91FFF903803FF E0D8071F903B7C0FC00F81F83E0E0F80E007E01C00FC001C9026C3C0030178137C271807 C700D9F0E0137E02CE902601F1C0133E003801DCDAFB80133F003001D892C7FCD90FF814 FF0070495C0060495CA200E04949485CD8C01F187E4A5C1200040715FE013F6091C75BA2 040F14014960017E5D1903041F5D13FE494B130762043F160E0001060F130C4992C713C0 191F4CED801C00031A1849027E1638F2003004FE167000071A60494A16E0F201C0030192 380F0380000FF18700494AEC03FED80380D90070EC00F84F2D7DAB55>I<01F8EB03FCD8 03FEEB1FFFD8071F90387C0FC03B0E0F80E007E03A0C07C3C003001CD9C7007F001801CE 1301003801DC80003013D8EB0FF800705B00605BA200E0491303D8C01F5D5C1200160701 3F5D91C7FCA2160F495D137E161F5F13FE49143F94C7FC187000014B136049147E16FE4C 13E0000317C049150104F81380170300071700495D170EEE781C000FED7C3849EC1FF0D8 0380EC07C0342D7DAB3A>I112 D<91380FC00391383FF0079138F83C0F903903E0 0E1E90390FC0063E90381F800790393F00037E4914FC01FE1301485AA2484814F812075B 000F140316F0485AA2003F14074914E0A3007F140F4914C0A3151F90C713805AA2153F6C 1500A2127E5D007F14FE6C1301A214036C6C485A000F131E3807C0383803E0F13901FFC1 F838003F01130014035DA314075DA3140F5DA2141FA2143F011FB51280A21600283F7DAB 2B>I<01F8EB0FC0D803FEEB7FF0D8070FEBF038000E903883C07C3A0C07C701FC001C13 CE0018EBDC03003813D8003013F8D90FF013F800709038E000E0006015005C12E0EAC01F 5C1200A2133F91C8FCA35B137EA313FE5BA312015BA312035BA312075BA3120F5BEA0380 262D7DAB2C>II<141C147EA314 FE5CA313015CA313035CA313075CA2007FB512FCB6FC15F839000FC000A2131F5CA3133F 91C7FCA35B137EA313FE5BA312015BA312035BA21570000714605B15E015C0000F130101 C013801403EC070000071306140E5C6C6C5A000113F03800FFC0013FC7FC1E3F7EBD23> I<013E140ED9FF80EB3F802603C3C0137F380703E0380601F0120E121CD81803143F0038 151F0030150FA2D87007140700605BA2D8E00F150000C0497FEA001F4A5B1606133F91C7 FC160E49140C137EA2161C01FE14185B1638163016704848146016E05E150100005D1503 6D49C7FC1506017C130E017E5B6D137890380F81E06DB45AD900FEC8FC292D7DAB2F> 118 D<013E1738D9FF80D901C013FC2603C3C0903907E001FE380703E0380601F0000E15 0F001C16C0D8180316000038187E0030031F143E00705ED86007171E5C163FD8E00F92C7 121C00C049160CEA001F4A49141C047E1418133F91C7FC04FE1438491730017E5CA20301 157001FE1760495C19E019C0A24949481301198018031900606D0107140670130E017C01 0F5C017E010C1418013ED91CFC13386DD9387E13F0903B0FC0F01F01C0903B03FFC00FFF 809028007F0001FEC7FC3F2D7DAB46>I<02FCEB07E0903A03FF801FFC903A0F07C0781E 903A1C03E0E01F903A3801F1C07FD9700013804901FB13FF4848EBFF00495B000316FE90 C71438484A130012061401000E5C120CC7FC14035DA314075DA3140F5DA3021F14381730 5D1770023F1460121E003F16E0267F807FEB01C0026F148000FF01EF1303D901CFEB0700 00FE903887C00E267C03835B3A3C0F01E0783A1FFC00FFE0D803F0EB3F80302D7EAB37> I<133ED9FF8014E02603C3C0EB03F0380703E0380601F0000E1507001C16E0EA18031238 0030150F007016C0EA60075C161FD8E00F158000C05BEA001F4A133F1700133F91C7FC5E 49147E137EA216FE01FE5C5BA215015E485AA215035EA200001407150F6D5C017C131F15 3F6D13FF90391F03CFC0903807FF8F903801FC0F90C7121F5EA2153F93C7FCD807C05BD8 1FE0137E5DA24848485A4A5A01805B39380007C00018495A001C49C8FC6C137C380781F8 3803FFE0C66CC9FC2C407DAB30>I<027CEB018049B413034901801300010F6D5A49EBE0 0E6F5A90393F03F838903978007EF80170EB1FF00160EB01E001E05C49495A90C748C7FC 150E5D5D5D5D4A5A4A5A4AC8FC140E5C5C5C5CEB03C049C9FC130E49141C4914185B4914 3848481430491470D8039014F048B4495A3A0FEFC007C0391E03F01FD81C01B55A486C91 C7FC485C00606D5A00E0EB3FF048EB0FC0292D7CAB2D>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmsy10 12 32 /Fp 32 111 df<007FB912E0BA12F0A26C18E03C04789A4D>0 D<121FEA3F80EA7FC0EA FFE0A5EA7FC0EA3F80EA1F000B0B789E1C>I<0060160600F8160F6C161F007E163F6C16 7E6C6C15FC6C6CEC01F86C6CEC03F06C6CEC07E06C6CEC0FC06C6CEC1F80017EEC3F006D 147E6D6C5B6D6C485A6D6C485A6D6C485A6D6C485A6D6C485ADA7E3FC7FCEC3F7E6E5A6E 5A6E5AA24A7E4A7EEC3F7EEC7E3F4A6C7E49486C7E49486C7E49486C7E49486C7E49486C 7E49C7127E017E8049EC1F804848EC0FC04848EC07E04848EC03F04848EC01F84848EC00 FC48C9127E007E163F48161F48160F00601606303072B04D>I<16C04B7EB3AC007FBA12 80BB12C0A26C1980C8D801E0C9FCB3A9007FBA1280BB12C0A26C198042427BC14D>6 D8 D<49B4FC010F13E0013F13F849 7F3901FF01FF3A03F8003F80D807E0EB0FC04848EB07E04848EB03F090C71201003EEC00 F8A248157CA20078153C00F8153EA248151EA56C153EA20078153C007C157CA26C15F8A2 6CEC01F06D13036C6CEB07E06C6CEB0FC0D803F8EB3F803A01FF01FF0039007FFFFC6D5B 010F13E0010190C7FC27277BAB32>14 D<49B4FC010F13E0013F13F8497F48B6FC481580 4815C04815E04815F0A24815F8A24815FCA3B712FEA96C15FCA36C15F8A26C15F0A26C15 E06C15C06C15806C15006C6C13FC6D5B010F13E0010190C7FC27277BAB32>I<007FBA12 80BB12C0A26C1980CEFCB0007FBA1280BB12C0A26C1980CEFCB0007FBA1280BB12C0A26C 1980422C7BAE4D>17 D<19E0F003F0180FF03FE0F0FF80943803FE00EF0FF8EF3FE0EFFF 80DC03FEC7FCEE0FF8EE3FE0EEFF80DB03FEC8FCED1FF8ED7FE0913801FF80DA07FEC9FC EC1FF0EC7FC04948CAFCEB07FCEB1FF0EB7FC04848CBFCEA07FCEA1FF0EA7FC048CCFCA2 EA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FC 913801FF809138007FE0ED1FF8ED07FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF 3FE0EF0FF8EF03FE943800FF80F03FE0F00FF01803F000E01900B0007FB912E0BA12F0A2 6C18E03C4E78BE4D>20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0 EB07FCEB01FF9038007FC0EC1FF0EC07FC913801FF809138007FE0ED1FF8ED07FE923800 FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FE943800FF80F03FE0F00FF0 A2F03FE0F0FF80943803FE00EF0FF8EF3FE0EFFF80DC03FEC7FCEE0FF8EE3FE0EEFF80DB 03FEC8FCED1FF8ED7FE0913801FF80DA07FEC9FCEC1FF0EC7FC04948CAFCEB07FCEB1FF0 EB7FC04848CBFCEA07FCEA1FF0EA7FC048CCFC12FC1270CDFCB0007FB912E0BA12F0A26C 18E03C4E78BE4D>I<037FB612E00207B712F0143F91B812E0010301C0C9FCD907FCCAFC EB0FE0EB3F8049CBFC13FC485A485A485A5B485A121F90CCFC123EA2123C127CA2127812 F8A25AA87EA21278127CA2123C123EA27E7F120F6C7E7F6C7E6C7E6C7E137E6D7EEB1FE0 EB07FC6DB47E010090B712E0023F16F01407020016E03C3A78B54D>26 D<1AF0A3861A78A21A7C1A3CA21A3E1A1E1A1F747EA2747E747E87747E747E1B7E87757E F30FE0F303F8007FBC12FEBE1280A26CF3FE00CEEA03F8F30FE0F31F8051C7FC1B7E6350 5A505A63505A505AA250C8FC1A1E1A3E1A3CA21A7C1A78A21AF862A359347BB264>33 D39 D<49B4EF3FC0010F01E0923803FFF8013F01 FC030F13FE4901FF92383FE01F48B66C91397E0007C02603F80301E0D901F8EB01E02807 E0007FF049486D7E01806D6CD907C0147048C76C6C494880001EDA07FE49C87E001C6E6C 013E150C486E6D48150E71481506486E01E0160793387FF1F0006092263FF3E08193381F FBC000E004FF1780486F4915017090C9FC82707F8482717E844D7E6C4B6D1503006004EF 1700933803E7FE0070922607C7FF5DDC0F837F003004816D140E00384BC6FC0018033E6D 6C5C001C4B6D6C143C6C4BD91FFC5C6C4A486D6C5C6DD907E06D6C13036C6C49486D9038 E00FE0D801F0013FC890B55A27007C03FE6F91C7FC90263FFFF8031F5B010F01E0030313 F8D901FECAEA7FC0592D7BAB64>49 D<92B6FC02071580143F91B7120001030180C8FCD9 07FCC9FCEB1FE0EB3F80017ECAFC5B485A485A485A5B485A121F90CBFC123EA2123C127C A2127812F8A25AA2B9FC1880A2180000F0CBFCA27EA21278127CA2123C123EA27E7F120F 6C7E7F6C7E6C7E6C7E137E6D7EEB1FE0EB07FC6DB47E010090B6FC023F15801407020015 00313A78B542>I<1706170F171FA2173EA2177CA217F8A2EE01F0A2EE03E0A2EE07C0A2 EE0F80A2EE1F00A2163EA25EA25EA24B5AA24B5AA24B5AA24B5AA24BC7FCA2153EA25DA2 5DA24A5AA24A5AA24A5AA24A5AA24AC8FCA2143EA25CA25CA2495AA2495AA2495AA2495A A249C9FCA2133EA25BA25BA2485AA2485AA2485AA2485AA248CAFCA2123EA25AA25AA25A 1260305C72C600>54 D<126012F0B012FC12FEA212FC12F0B0126007267BAB00>I<4B7E 4B7EA21507A25EECFF8F010313EF90260F80FFC7FC90383E003F497F4980484880484849 7E5B0007EC3DF049133C000FEC7CF8A248C7EA787C15F848157E15F0A2140148157F007E 4A7E1403A215C0A200FE01071480A21580140FA21500A25CA2141E143EA2143CA2147CA2 1478A214F8A25C1301A2007E491400A21303A2007F495B1307003F157E5CA2130F001F15 7C018FC712FCD80F9F5CA201DE130100075DD803FE495AA26C48495A00004A5A017C49C7 FC017E133E90387F80F89038FBFFE001F8138049C9FC1201A25BA26C5A29557CCC32>59 D 67 D<031FB512C00203B7FC021F16E091B812F8010317FE010F717E90283FE07FC03F80 D9FE00020080D801F8041F7FD803E04A01077F48481601000F716C7E4848717E003F02FF 151F007F180F90C7707E00FE92C8FC488400F01A80008084C75AA24B81A414035DA21B00 A24A5AA24F5AA24A5A621903624A5A4F5AA24B4B5A023F5F191F4B5E027F4CC7FC197E92 C9127C4A5E4E5A4A4B5A01014C5AF01F804A033EC8FC01035E4A4A5AEF07E00107ED1FC0 4A02FFC9FC010FEC07FC4AEBFFF091B612C0017F4ACAFC90B612F04815804802F8CBFC48 91CCFC49447EC34D>II<0403B712F8043F16FE4BB9FC1507151F157F912601FC0090C7EA07FE912603F001ED 01FCDA07C04915F0DA0F80EE0080021F1800EC3F004A495A5C5C495A4A495A5C495A6DC7 FC90C8485AA35F161FA34C5AA35F167F94B612C0A293B7FC624B93C7FC19FC04FCC71270 030392C8FC5EA24B5AA2150F5E151F5EA24B5AA24BCBFCA215FEA24A5AA24A5AEA018000 0F495AEA1FC0486C485AD87FF05B39FFFC1F80D87FFF90CCFC14FE6C5B6C13F06C5B0003 1380D800FCCDFC50477EC348>I76 D87 D<0060170C00F0171EB3B3A66C173EA20078173C007C177C007E17FC003E17F86CEE01F0 6D15036C6CED07E06C6CED0FC0D803F8ED3F80D801FEEDFF0026007FC0EB07FCD93FFCEB 7FF8010FB612E001031580D9007F01FCC7FC020713C0373D7BBA42>91 D<913807FFC0027F13FC0103B67E010F15E0903A3FFC007FF8D97FC0EB07FCD801FEC8B4 FCD803F8ED3F80D807E0ED0FC04848ED07E04848ED03F090C91201003EEE00F8007E17FC 007C177C0078173C00F8173EA248171EB3B3A60060170C373D7BBA42>I<1538157CA215 FEA24A7EA215EF02037FA2913807C7C0A291380F83E0A291381F01F0A2EC1E00023E7FA2 4A137CA24A7FA249487FA24A7F010381A249486D7EA249486D7EA249C76C7EA2011E1400 013E81A249157CA24981A2484881A24848ED0F80A2491507000717C0A24848ED03E0A248 C9EA01F0A2003EEE00F8A2003C1778007C177CA248173EA248171E0060170C373D7BBA42 >94 D102 D<12FEEAFFE0EA07F8EA00FEEB7F806D7E6D7E130F6D7EA26D7EB3AD6D7EA26D7E806E7E 6E7EEC0FE0EC03FC913800FFE0A2913803FC00EC0FE0EC3FC04A5A4AC7FC5C495AA2495A B3AD495AA2495A131F495A495A01FEC8FCEA07F8EAFFE048C9FC236479CA32>I<126012 F0B3B3B3B3B3A81260046474CA1C>106 D<0070130700F01480B3B3B3B3B3A800701400 196474CA32>I<126012F07EA21278127CA2123C123EA2121E121FA26C7EA212077FA212 037FA212017FA26C7EA21378137CA2133C133EA2131E131FA26D7EA2130780A2130380A2 130180A26D7EA21478147CA2143C143EA280A28081A2140781A2140381A26E7EA2140081 A21578157CA2153C153EA281A2811680A2150716C0A2150316E0A2ED01F0A2150016F8A2 1678167CA2163C163EA2161E160C27647BCA32>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmr9 9 27 /Fq 27 89 df48 D<13075B5B137FEA07FFB5FC13BFEAF83F1200B3B3A2497E007FB51280A319327AB126> IIII<000C14C0380FC00F90B5128015005C5C14F014C0D80C18C7FC90C8FCA9 EB0FC0EB7FF8EBF07C380FC03F9038001F80EC0FC0120E000CEB07E0A2C713F01403A215 F8A41218127E12FEA315F0140712F8006014E01270EC0FC06C131F003C14806CEB7F0038 0F80FE3807FFF8000113E038003F801D347CB126>I<14FE903807FF80011F13E090383F 00F0017C13703901F801F8EBF003EA03E01207EA0FC0EC01F04848C7FCA248C8FCA35A12 7EEB07F0EB1FFC38FE381F9038700F809038E007C039FFC003E0018013F0EC01F8130015 FC1400A24814FEA5127EA4127F6C14FCA26C1301018013F8000F14F0EBC0030007EB07E0 3903E00FC03901F81F806CB51200EB3FFCEB0FE01F347DB126>I<1230123C003FB6FCA3 4814FEA215FC0070C7123800601430157015E04814C01401EC0380C7EA07001406140E5C 141814385CA25CA2495A1303A3495AA2130FA3131F91C7FCA25BA55BA9131C20347CB126 >III<15E0A34A7EA24A7EA34A7EA3EC0DFE140CA2EC187FA34A6C7EA202707FEC601FA202 E07FECC00FA2D901807F1507A249486C7EA301066D7EA2010E80010FB5FCA249800118C7 7EA24981163FA2496E7EA3496E7EA20001821607487ED81FF04A7ED8FFFE49B512E0A333 367DB53A>65 D67 D69 DIIII76 DIII82 D<90381FE00390387FFC0748B5FC3907F01FCF390F8003FF48C7 FC003E80814880A200788000F880A46C80A27E92C7FC127F13C0EA3FF013FF6C13F06C13 FF6C14C06C14F0C680013F7F01037F9038003FFF140302001380157F153FED1FC0150F12 C0A21507A37EA26CEC0F80A26C15006C5C6C143E6C147E01C05B39F1FC03F800E0B512E0 011F138026C003FEC7FC22377CB42B>I<007FB712FEA390398007F001D87C00EC003E00 78161E0070160EA20060160600E01607A3481603A6C71500B3AB4A7E011FB512FCA33033 7DB237>III<267FFFFC90B512C0A3000101E090381FF80026007F80EB0FC0013F6E5A6E91 C7FC6D6C130E010F140C6E5B6D6C133801035C6E13606D6C13E06D6C485A5EDA7F83C8FC EC3FC715C6EC1FECEC0FFC5D14076E7EA26E7E815C6F7E9138063FC0140E4A6C7E913818 0FF0EC380702707F91386003FCECC0010101804A6C7E49C77E4981010E6E7E010C6E7E13 1C496E7E01786E7E13FCD807FEEC1FFEB56C90B512F8A335337EB23A>88 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmr12 12 83 /Fr 83 124 df0 D<1618163CA2167EA216FFA24B7FA24B6C7EA29238063FE0A24B6C7EA24B6C7EA2923838 07FC153092387003FE15609238E001FF15C002016D7F5D02036E7E92C7FC4A6E7E140602 0E6E7E140C021C6E7E141802386E7E143002706E7E146002E06E7E5C01016F7F5C010370 7E91C9FC183F010683181F4983180F49831807498318034983A249707EA24848701380A2 48CBEA7FC0A20006F03FE0A248F01FF0A2001FBA12F8A24819FCA24819FEA2BCFC48477C C651>II6 D<0103B612FCA390C701F0C8FC6F5A6F5AA8913801FFF002 3FEBFF80903A01FF3FDFF0D907F0EBC1FCD91FC0EBC07FD93F00EC1F8001FEED0FE04848 6F7E48486F7E48486F7E48486F7E001F834982003F1880007F18C0A249163F00FF18E0A8 007F18C06D167FA2003F1880001F18006D5E000F5F6C6C4B5A6C6C4B5A6C6C4B5A6C6C4B 5A013FED1F80D91FC0027FC7FCD907F0EBC1FCD901FFEBDFF0D9003FB51280020101F0C8 FC9138003FC0A84B7E4B7E0103B612FCA33B447BC346>8 D<9239FFC001FC020F9038F8 0FFF913B3F803E3F03C0913BFC00077E07E0D903F890390FFC0FF0494890383FF81F4948 EB7FF0495A494814E049C7FCF00FE04991393FC0038049021F90C7FCAFB912F0A3C648C7 D81FC0C7FCB3B2486CEC3FF0007FD9FC0FB512E0A33C467EC539>11 D<4AB4FC020F13E091387F80F8903901FC001C49487FD907E0130F4948137F011FECFF80 495A49C7FCA25B49EC7F00163E93C7FCACEE3F80B8FCA3C648C7FC167F163FB3B0486CEC 7FC0007FD9FC1FB5FCA330467EC536>I<913801FFC0020FEBFB8091387F803F903801FC 00494813FFEB07E0EB1FC0A2495A49C7FC167F49143F5BAFB8FCA3C648C7123FB3B2486C EC7FC0007FD9FC1FB5FCA330467EC536>II<043014 C00478497EA204F81303A24C5CA203011407A24C5CA20303140FA24C91C7FCA203075CA2 4C131EA2030F143EA293C7123CA24B147CA2031E1478A2033E14F8A2033C5CA2037C1301 007FBA12F8BB12FCA26C19F8C72801F00007C0C7FC4B5CA30203140FA24B91C8FCA40207 5CA24B131EA3020F143E007FBA12F8BB12FCA26C19F8C7003EC700F8C8FC023C5CA2027C 1301A202785CA202F81303A24A5CA201011407A24A5CA20103140FA24A91C9FCA201075C A24A131EA2010F143EA291C7123CA249147CA2011E1478A2010C143046587BC451>35 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380120313005A 1206120E5A5A5A12600B1D78C41B>39 D<140C141C1438147014E0EB01C01303EB0780EB 0F00A2131E5BA25B13F85B12015B1203A2485AA3485AA348C7FCA35AA2123EA2127EA412 7CA312FCB3A2127CA3127EA4123EA2123FA27EA36C7EA36C7EA36C7EA212017F12007F13 787FA27F7FA2EB0780EB03C01301EB00E014701438141C140C166476CA26>I<12C07E12 707E7E7E120F6C7E6C7EA26C7E6C7EA21378137C133C133E131E131FA2EB0F80A3EB07C0 A3EB03E0A314F0A21301A214F8A41300A314FCB3A214F8A31301A414F0A21303A214E0A3 EB07C0A3EB0F80A3EB1F00A2131E133E133C137C13785BA2485A485AA2485A48C7FC120E 5A5A5A5A5A16647BCA26>I<16C04B7EB3AB007FBAFCBB1280A26C1900C8D801E0C9FCB3 AB6F5A41407BB84C>43 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0 A312011380120313005A1206120E5A5A5A12600B1D78891B>II< 121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A78891B>I<14FF010713E090381F81F890 383E007C01FC133F4848EB1F8049130F4848EB07C04848EB03E0A2000F15F0491301001F 15F8A2003F15FCA390C8FC4815FEA54815FFB3A46C15FEA56D1301003F15FCA3001F15F8 A26C6CEB03F0A36C6CEB07E0000315C06D130F6C6CEB1F806C6CEB3F00013E137C90381F 81F8903807FFE0010090C7FC28447CC131>48 D<143014F013011303131F13FFB5FC13E7 13071200B3B3B0497E497E007FB6FCA3204278C131>II<49B4FC010F13E0013F13FC9038FE01FE3A01F0007F80D803C0EB3FC0 48C7EA1FE0120EED0FF0EA0FE0486C14F8A215077F5BA26C48130FEA03C0C813F0A3ED1F E0A2ED3FC01680ED7F0015FE4A5AEC03F0EC1FC0D90FFFC7FC15F090380001FCEC007FED 3F80ED1FC0ED0FE016F0ED07F816FC150316FEA2150116FFA3121EEA7F80487EA416FE49 1303A2007EC713FC00701407003015F80038140F6C15F06CEC1FE06C6CEB3FC0D803E0EB 7F803A01FE01FE0039007FFFF8010F13E0010190C7FC28447CC131>II<0006 15C0D807C0130701FCEB7F8090B612005D5D5D15E0158026063FFCC7FC90C9FCAE14FF01 0713C090381F01F090383800FC01F0137ED807C07F49EB1F8016C090C7120F000615E0C8 EA07F0A316F81503A216FCA5123E127F487EA416F890C712075A006015F0A20070140F00 3015E00038EC1FC07E001EEC3F806CEC7F006C6C13FE6C6C485A3901F807F039007FFFE0 011F90C7FCEB07F826447BC131>II<121CA2EA1F80 90B712C0A3481680A217005E0038C8120C0030151C00705D0060153016705E5E4814014B 5A4BC7FCC81206150E5D151815385D156015E04A5AA24A5A140792C8FC5CA25C141E143E A2147E147CA214FCA21301A3495AA41307A6130FAA6D5AEB01C02A457BC231>I<14FF01 0713E0011F13F890387F00FE01FC133FD801F0EB1F804848EB0FC049EB07E00007EC03F0 48481301A290C713F8481400A47FA26D130116F07F6C6CEB03E013FC6C6CEB07C09039FF 800F806C9038C01F006CEBF03EECF87839007FFEF090383FFFC07F01077F6D13F8497F90 381E7FFFD97C1F1380496C13C02601E00313E048486C13F000079038007FF84848EB3FFC 48C7120F003EEC07FE150148140016FF167F48153FA2161FA56C151E007C153EA2007E15 3C003E157C6C15F86DEB01F06C6CEB03E06C6CEB07C0D803F8EB1F80C6B4EBFF0090383F FFFC010F13F00101138028447CC131>I<14FF010713E0011F13F890387F80FC9038FC00 7E48487F4848EB1F804848EB0FC0000FEC07E0485AED03F0485A16F8007F140190C713FC A25AA216FE1500A516FFA46C5CA36C7E5D121F7F000F5C6C6C130E150C6C6C131C6C6C5B D8007C5B90383F01E090390FFF80FE903801FE0090C8FC150116FCA4ED03F8A216F0D80F 801307486C14E0486C130F16C0ED1F80A249EB3F0049137E001EC75A001C495A000F495A 3907E01FE06CB51280C649C7FCEB1FF028447CC131>I<121EEA7F80A2EAFFC0A4EA7F80 A2EA1E00C7FCB3A5121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A2B78AA1B>I<121EEA7F 80A2EAFFC0A4EA7F80A2EA1E00C7FCB3A5121E127FEAFF80A213C0A4127F121E1200A512 011380A3120313005A1206120E120C121C5A5A12600A3E78AA1B>I<007FBAFCBB1280A2 6C1900CEFCB0007FBAFCBB1280A26C190041187BA44C>61 D<16C04B7EA34B7EA34B7EA3 4B7EA3ED19FEA3ED30FFA203707FED607FA203E07FEDC03FA2020180ED801FA2DA03007F 160FA20206801607A24A6D7EA34A6D7EA34A6D7EA20270810260147FA202E08191B7FCA2 49820280C7121FA249C87F170FA20106821707A2496F7EA3496F7EA3496F7EA201788313 F8486C83D80FFF03037FB500E0027FEBFFC0A342477DC649>65 DIIIIIIII<010FB512FEA3D900031380 6E130080B3B3AB123F487E487EA44A5A13801300006C495A00705C6C13076C5C6C495A6C EB1F802603E07FC7FC3800FFFCEB1FE027467BC332>IIIIIII82 D<49B41303010FEBE007013F13F890 39FE00FE0FD801F8131FD807E0EB079F49EB03DF48486DB4FC48C8FC4881003E81127E82 127C00FC81A282A37E82A27EA26C6C91C7FC7F7FEA3FF813FE381FFFE06C13FE6CEBFFE0 6C14FC6C14FF6C15C0013F14F0010F80010180D9001F7F14019138001FFF03031380816F 13C0167F163F161F17E000C0150FA31607A37EA36C16C0160F7E17806C151F6C16006C5D 6D147ED8FBC05CD8F9F0495AD8F07C495A90393FC00FE0D8E00FB51280010149C7FC39C0 003FF02B487BC536>I<003FB912F8A3903BF0001FF8001F01806D481303003EC7150048 187C0078183CA20070181CA30060180CA5481806A5C81600B3B3A54B7EED7FFE49B77EA3 3F447DC346>IIII89 D91 D93 D<121EEA7F80A2EAFFC0A4 EA7F80A2EA1E000A0A78C21B>95 D97 DII<167FED3FFFA315018182B3EC7F 80903803FFF090380FC07C90383F000E017E1307496D5AD803F87F48487F5B000F81485A A2485AA2127FA290C8FC5AAB7E7FA2123FA26C7EA2000F5D7F6C6C5B00035C6C6C903807 7F806C6C010E13C0013F011C13FE90380FC0F8903803FFE09026007F0013002F467DC436 >IIIIII<143C14FFA2491380A46D1300 A2143C91C7FCADEC7F80EB3FFFA31300147F143FB3B3AA123E127F39FF807F00A2147EA2 5C6C485A383C01F06C485A3807FF80D801FEC7FC195785C21E>IIII<3901FC01FE00FF903807FFC091381E 07F091383801F8000701707F0003EBE0002601FDC07F5C01FF147F91C7FCA25BA35BB3A8 486CECFF80B5D8F83F13FEA32F2C7DAB36>II<3901FC03FC00FF90380FFF8091383C07E091387001F83A07FDE000FE00030180137F D801FFEC3F8091C7EA1FC04915E049140F17F0160717F8160317FCA3EE01FEABEE03FCA3 EE07F8A217F0160F6D15E0EE1FC06D143F17806EEB7E00D9FDC05B9039FCF003F891383C 0FE091381FFF80DA03FCC7FC91C9FCAE487EB512F8A32F3F7DAB36>I<91387F80039039 03FFE00790380FE07890393F801C0F90387E000E496D5AD803F8EB039F0007EC01BF4914 FF48487F121F5B003F81A2485AA348C8FCAB6C7EA3123F7F121F6D5C120F6D5B12076C6C 5B6C6C497E6C6C130E013F131C90380FC0F8903803FFE09038007F0091C7FCAEEEFF8003 3F13FEA32F3F7DAB33>I<3903F803F000FFEB1FFCEC3C3EEC707F0007EBE0FF3803F9C0 00015B13FBEC007E153C01FF13005BA45BB3A748B4FCB512FEA3202C7DAB26>I<90383F E0183901FFFC383907E01F78390F0003F8001E1301481300007C1478127800F81438A215 18A27EA27E6C6C13006C7E13FC383FFFE06C13FC6C13FF6C14C06C14E0C614F0011F13F8 1300EC0FFC140300C0EB01FE1400157E7E153EA27EA36C143C6C147C15786C14F86CEB01 F039F38003E039F1F00F8039E07FFE0038C00FF01F2E7DAC26>I<1306A5130EA4131EA3 133E137EA213FE12011207001FB512F0B6FCA2C648C7FCB3A4150CAA017E131C017F1318 A26D133890381F8030ECC070903807E0E0903801FFC09038007F001E3E7EBC26>IIIIII<003FB612E0A29038C0003F90 C713C0003CEC7F800038ECFF00A20030495A0070495AA24A5A0060495AA24A5A4A5AA2C7 485A4AC7FC5B5C495A13075C495A131F4A1360495A495AA249C712C0485AA2485A485A15 01485A48481303A24848EB07804848131F00FF14FF90B6FCA2232B7DAA2B>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmcsc10 12 37 /Fs 37 122 df<121FEA3F80EA7FC0EAFFE0A5EA7FC0EA3F80EA1F000B0B768A20>46 D48 D<1438147814F81303130F13FFB5FC13F713071200B3B3B0497E497EB712C0A3224276C1 37>I<49B4FC010F13F0013F13FC9038FC03FF2601E00013C0D807C0EB3FE048486D7E90 C76C7E001E6E7E4881003814030078811270007C80B416807F7F81A46C485B6CC7FCC8FC 17005DA25E15075E4B5AA24B5A5E4B5A4B5A4BC7FC5D4A5A4A5A4A5AEC0FC04A5A92C8FC 143E5C5C495A4948EB0380EB078049C7FC011EEC07005B5B5B48485C485A49141E48B612 FE5A5A5A5AB75AA329427AC137>II<16E015011503A21507150FA2151F153FA2157F15EF 15CF1401EC038F150F1407140E141CA214381470A214E0EB01C0A2EB0380EB0700A2130E 5B131813385B136013E0485A485AA248C7FC120EA25A5AA25A5AB812F8A3C8381FE000AC 4B7E4B7E027FB512F8A32D437CC237>I<0007151801E014F801FEEB1FF090B65A5E5E93 C7FC15FC15F015C0D90FFEC8FC90CAFCADEC7FC0903803FFF890380F807E90381C003F01 786D7E01E0EB0FE0496D7E5B6F7E90C77FC8120182A28281A21780A5123EEA7F8012FF7F A34915005D48C7FC00705DA200784A5A1238003C4A5A6C5D6C4A5A6D495A6C6C495AD803 F049C7FC3900FE03FE90387FFFF8011F13C0D903FEC8FC29447AC137>II<121C121EEA1FC090B712F0A34816E0A217C017801700003CC8120F48151E5E00 70153816785E4B5A485D15034B5AC848C7FC150E151E5D5D157015F04A5AA24A5AA24A5A 140FA24AC8FCA25C143E147EA35CA21301A31303A25CA21307A6130FAA6D5AEB01C02C45 79C237>III<1638167CA316FEA34B7EA24B7FA34B7F16 7FA2030E7F163FA24B6C7EA2033C7FED380FA203787FED7007A203E07F1603A24A486C7E A20203814B7EA202078192C7127FA2020E81173FA24A6E7EA2023C810238140FA2027FB6 7EA302E0C7EA07FE17030101824A80A20103834A80A249C97F187FA2010E707EA2011E83 181F133E85137E48B483000701C0ED7FFFB500FC021FB512FEA347477CC651>65 D67 D69 DI73 D82 D<157015F8A34A7EA24A7EA34A7E81A291380E3F80A2021E7FEC1C 1FA24A6C7EA34A6C7EA202F07FECE003A249486C7EA349486C7EA201078091C77EA249B6 7EA24981011CC7121FA2013C810138140FA2496E7EA201F081491403120183486C140100 074B7ED81FF84A7EB5027F13F8A335357CB43D>97 DI<4AB4EB0180021FEBF00391B5EAFC0701039038007E0FD907F8EB0F9F D91FE0EB03DF4948EB01FF01FFC8FC4848157F4848153FA24848151F4848150F121F4915 07123F5BA2007F1603A3484892C7FCAB6C7EEF0380A2123FA27F001F16076D1600000F5E 6C6C150E6C6C151E171C6C6C153C6C6C5DD93FC05C6D6CEB03E0D907F8495A902703FF80 7FC7FC0100EBFFFC021F13F00201138031357BB33B>IIIIIII108 DIIII114 D<90390FF0018090387FFE0348B512873907F00FEF390FC001FF48C7FC003E143F151F5A 150F5A1507A36C1403A27E6C91C7FC6C7E7FEA3FF8EBFF806C13FC6CEBFFC06C14F06C80 C614FE011F7F01031480D9001F13C014019138003FE0151F150FED07F0150312E01501A3 7EA216E06C1403A26CEC07C06CEC0F806C6CEB1F0001E0133ED8FBFE13FC00F0B55AD8E0 1F13E0D8C00390C7FC24357BB32E>I<007FB812C0A3903A8007FC003F277E0003F8130F 007C16070078160300701601A200F017E0A2481600A6C71600B3AA4A7E4A7E010FB512FE A333327CB13B>II<267FFFF890383FFFF0A300 0101E0010F13006C49EB07F8017F5D013F15C06D6C5C6D6C49C7FC160E6D6C131E6D6C5B 6D6C133816786D6C5B91387F81E0023F5B15C391381FE7806EB4C8FC5D14076E5A6E7EA2 6E7E4A7FA24A7F9138079FE091380F0FF0140E91381E07F84A6C7E4A6C7E14709138F000 FF49486D7E4948133F4A8001076E7E49C76C7E131E013E6E7E017E6E7E01FE81000182D8 0FFF4A1380B500C0011F13FEA337337DB23D>120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmsy10 10 1 /Ft 1 3 df<0060150600F8150F6C151F007E153F6C157E6C6C14FC6C6CEB01F86C6CEB 03F06C6CEB07E06C6CEB0FC06C6CEB1F80017EEB3F006D137E6D6C5A90380FC1F8903807 E3F0903803F7E06DB45A6D5B6EC7FCA24A7E497F903803F7E0903807E3F090380FC1F890 381F80FC90383F007E017E7F49EB1F804848EB0FC04848EB07E04848EB03F04848EB01F8 4848EB00FC48C8127E007E153F48151F48150F00601506282874A841>2 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmr7 7 2 /Fu 2 51 df<13381378EA01F8121F12FE12E01200B3AB487EB512F8A215267BA521>49 D<13FF000313E0380E03F0381800F848137C48137E00787F12FC6CEB1F80A4127CC7FC15 005C143E147E147C5C495A495A5C495A010EC7FC5B5B903870018013E0EA018039030003 0012065A001FB5FC5A485BB5FCA219267DA521>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmmi10 10 5 /Fv 5 105 df<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A12 06120E5A5A5A12600A19798817>59 D<126012FCB4FCEA7FC0EA1FF0EA07FCEA01FF3800 7FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF9138007FC0ED1FF0ED07FC ED01FF9238007FC0EE1FF0EE07FCEE01FF9338007F80EF1FC0A2EF7F80933801FF00EE07 FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9 FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA3FF0EA7FC048CBFC12FC1270323279AD41> 62 D<902603FFF893383FFF80496081D900079438FF80000206DC01BFC7FCA2020E4C5A 1A7E020C1606190CDA1C7E16FE4F5A02181630A20238166162023016C1F00181DA703F15 8395380303F002601506A202E0ED0C076202C01518183001016D6C140F06605B028015C0 A20103923801801FDD03005B140092380FC00649173F4D91C8FC01065DA2010E4B5B4D13 7E130C6F6C5A011C17FEDCE1805B011802E3C7FCA2013802E6130104EC5C1330ED03F801 7016034C5C01F05CD807FC4C7EB500E0D9C007B512F01680150151397CB851>77 D<92391FE00380DBFFFC130002036D5A91390FE01F8F91393F0007DF027EEB01FE02F813 00495A4948147E177C4948143C495AA2011F153891C8FCA3491530A28094C7FC80806D7E 14FEECFFE06D13FE6DEBFFC06D14F06D806D80021F7F02037FEC003F03037F1500167F16 3F161FA3120C160FA2001C151F94C7FCA3003C153EA25E003E5D127E007F4A5A6D495A6D EB0FC0D8F9F0495AD8F0FE01FEC8FC39E03FFFF8010F13E0D8C00190C9FC313D7CBA33> 83 D 104 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmcsc10 10 8 /Fw 8 117 df<121C127FEAFF80A5EA7F00121C090977881B>46 D<150EA3151FA24B7EA34B7EA3EDDFE0A202017F158FA29138030FF81507A202067F1503 020E7FEC0C01A2021C7FEC1800A24A80167FA24A6D7EA202E0804A131FA2494880160FA2 49B67EA249810106C71203A249811601A2498182A2496F7EA20170820160153F13E06D82 1203D80FFCED7FF8B56C010FB512E0A33B3C7CBB44>65 D<1407A24A7EA34A7EA3EC37E0 A2EC77F01463A2ECC1F8A201017F1480A2903803007EA301067FA2010E80010C131FA249 6D7EA2013FB57EA29038300007496D7EA3496D7EA200018149130012036D801207D81FE0 903801FF80D8FFF8010F13F8A22D2C7DAB33>97 DI<91383FC006903901FFF80E90390FE03E 1E90381F0007017EEB03BE01F8EB01FE484813004848147E0007153E485A001F151E5B00 3F150E90C8FC5A1606A212FE1600AA007F1506A37E6D140E001F150C7F000F151C6C6C14 18000315386C6C14706C6C14E0017EEB01C0011FEB078090390FE03E00903801FFF89038 003FC0272D7BAB31>I114 D<017F13603901FFE0E0380780F9380E001F48130748130312780070130100F01300A315 607EA26C14007E127F13C0EA3FFEEBFFE06C13F8000713FE6C7FC61480010F13C01300EC 0FE01407EC03F01401A212C01400A37E15E06C1301A26CEB03C06CEB0780B4EB0F0038F3 E01E38E0FFF838C01FE01C2D7BAB26>I<007FB712C0A23A7E003FC00F007890381F8003 007015011600126000E016E0A2481660A5C71500B3A8EC7FE0011FB57EA22B2B7DAA31> I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmr10 10 73 /Fx 73 128 df11 DI<030C1303031E497EA2033E130FA2033C91C7FCA2037C5BA20378 131EA303F8133EA24B133CA20201147CA24B1378A2020314F8A24B5BA302071301007FB9 1280BA12C0A26C1880C7271F0007C0C7FC021E5CA3023E130FA2023C91C8FCA2027C5BA2 0278131EA302F8133E007FB91280BA12C0A26C1880280003E000F8C8FC4A5BA301071301 A202805BA2010F1303A202005BA2491307A2011E5CA3013E130FA2013C91C9FCA2017C5B A20178131EA20130130C3A4A7BB945>35 D<141FEC7FC0903801F0E0903803C060010713 7090380F803090381F00381518A25BA2133E133F15381530A215705D5D140190381F8380 92CAFC1487148E02DC49B51280EB0FF85C4A9039003FF8000107ED0FC06E5D71C7FC6E14 0E010F150CD91DFC141C01391518D970FE143801E015302601C07F1470D803805D00076D 6C5BD80F00EBC00148011F5C4890380FE003003E6E48C8FC007E903807F8060203130E00 FE6E5A6E6C5A1400ED7F706C4B13036F5A6F7E6C6C6D6C5B7013066C6C496C130E6DD979 FE5B281FF001F07F133C3C07F80FE03FC0F86CB539800FFFF0C69026FE000313C0D91FF0 D9007FC7FC393E7DBB41>38 D<146014E0EB01C0EB0380EB0700130E131E5B5BA25B485A A2485AA212075B120F90C7FCA25A121EA2123EA35AA65AB2127CA67EA3121EA2121F7EA2 7F12077F1203A26C7EA26C7E1378A27F7F130E7FEB0380EB01C0EB00E01460135278BD20 >40 D<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378A2137C133C133E131EA213 1F7FA21480A3EB07C0A6EB03E0B2EB07C0A6EB0F80A31400A25B131EA2133E133C137C13 78A25BA2485A485AA2485A48C7FC120E5A5A5A5A5A13527CBD20>I<121C127FEAFF80A2 13C0A3127F121C1200A412011380A2120313005A1206120E5A5A5A12600A19798817>44 DI<121C127FEAFF80A5EA7F00121C0909798817>I<150C151E15 3EA2153C157CA2157815F8A215F01401A215E01403A215C01407A21580140FA215005CA2 141E143EA2143C147CA2147814F8A25C1301A25C1303A2495AA25C130FA291C7FC5BA213 1E133EA2133C137CA2137813F8A25B1201A25B1203A25B1207A25B120FA290C8FC5AA212 1E123EA2123C127CA2127812F8A25A12601F537BBD2A>IIIII<1538A2157815F8A2140114031407A2140F141F 141B14331473146314C313011483EB030313071306130C131C131813301370136013C012 01EA038013005A120E120C5A123812305A12E0B712F8A3C73803F800AB4A7E0103B512F8 A325397EB82A>I<0006140CD80780133C9038F003F890B5FC5D5D158092C7FC14FC3806 7FE090C9FCABEB07F8EB3FFE9038780F803907E007E090388003F0496C7E12066E7EC87E A28181A21680A4123E127F487EA490C71300485C12E000605C12700030495A00385C6C13 03001E495A6C6C485A3907E03F800001B5C7FC38007FFCEB1FE0213A7CB72A>II<12301238123E003FB612E0A316C05A168016000070C71206 0060140E5D151800E01438485C5D5DC712014A5A92C7FC5C140E140C141C5CA25CA214F0 495AA21303A25C1307A2130FA3495AA3133FA5137FA96DC8FC131E233B7BB82A>III<121C127FEAFF80A5EA7F00121CC7FC B2121C127FEAFF80A5EA7F00121C092479A317>I<1538A3157CA315FEA34A7EA34A6C7E A202077FEC063FA2020E7FEC0C1FA2021C7FEC180FA202387FEC3007A202707FEC6003A2 02C07F1501A2D901807F81A249C77F167FA20106810107B6FCA24981010CC7121FA2496E 7EA3496E7EA3496E7EA213E0707E1201486C81D80FFC02071380B56C90B512FEA3373C7D BB3E>65 DI<913A01FF800180020FEBE003027F13F8903A01FF807E 07903A03FC000F0FD90FF0EB039F4948EB01DFD93F80EB00FF49C8127F01FE153F120148 48151F4848150FA248481507A2485A1703123F5B007F1601A35B00FF93C7FCAD127F6DED 0180A3123F7F001F160318006C7E5F6C7E17066C6C150E6C6C5D00001618017F15386D6C 5CD91FE05C6D6CEB03C0D903FCEB0F80902701FF803FC7FC9039007FFFFC020F13F00201 1380313D7BBA3C>IIIIIII<013FB512E0A39039001FFC00EC07F8B3B3A3123FEA7F80EAFFC0A44A5A13 80D87F005B0070131F6C5C6C495A6C49C7FC380781FC3801FFF038007F80233B7DB82B> IIIIIII82 DI<003FB812E0A3D9C003EB001F273E0001FE130348EE01F00078160000701770A3 00601730A400E01738481718A4C71600B3B0913807FF80011FB612E0A335397DB83C>I< B6903807FFFEA3000101809038007FE06C90C8EA1F80EF0F001706B3B2170E6D150C8017 1C133F17186D6C14385F6D6C14F06D6C5C6D6C495A6D6CEB07806D6C49C7FC91387F807E 91381FFFF8020713E09138007F80373B7DB83E>III89 D91 D93 D97 DIIII<147E9038 03FF8090380FC1E0EB1F8790383F0FF0137EA213FCA23901F803C091C7FCADB512FCA3D8 01F8C7FCB3AB487E387FFFF8A31C3B7FBA19>IIII107 DI<2703F00FF0EB 1FE000FFD93FFCEB7FF8913AF03F01E07E903BF1C01F83803F3D0FF3800FC7001F802603 F70013CE01FE14DC49D907F8EB0FC0A2495CA3495CB3A3486C496CEB1FE0B500C1B50083 B5FCA340257EA445>I<3903F00FF000FFEB3FFCECF03F9039F1C01F803A0FF3800FC038 03F70013FE496D7EA25BA35BB3A3486C497EB500C1B51280A329257EA42E>II<3903F01FE000FFEB7FF89038F1E07E9039F3801F803A0FF7000FC0D803FEEB07 E049EB03F04914F849130116FC150016FEA3167FAA16FEA3ED01FCA26DEB03F816F06D13 076DEB0FE001F614C09039F7803F009038F1E07E9038F0FFF8EC1FC091C8FCAB487EB512 C0A328357EA42E>II<3807E01F00FFEB7FC09038E1E3E09038E387F0380FE707EA03 E613EE9038EC03E09038FC0080491300A45BB3A2487EB512F0A31C257EA421>II<1318A51338A313 78A313F8120112031207001FB5FCB6FCA2D801F8C7FCB215C0A93800FC011580EB7C0301 7E13006D5AEB0FFEEB01F81A347FB220>III< B53A1FFFE03FFEA3260FF8009038000FF86C48017EEB03E018C00003023EEB0180A26C6C 013FEB0300A36C6CEC8006156FA2017E9038EFC00C15C7A2D93F016D5A15830281EBF038 D91F831430150102C3EBF87090260FC6001360A2D907E66D5A02EC137CA2D903FCEB7F80 4A133FA2010192C7FC4A7FA20100141E4A130E0260130C37257EA33C>III<003FB512FCA2EB8003D83E0013F800 3CEB07F00038EB0FE012300070EB1FC0EC3F800060137F150014FE495AA2C6485A495AA2 495A495A495AA290387F000613FEA2485A485A0007140E5B4848130C4848131CA2484813 3C48C7127C48EB03FC90B5FCA21F247EA325>II<001C131C007F 137F39FF80FF80A5397F007F00001C131C190978B72A>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fy cmbx12 12 55 /Fy 55 124 df12 D46 D48 D III<163FA25E5E5D5DA25D5D5D5D A25D92B5FCEC01F7EC03E7140715C7EC0F87EC1F07143E147E147C14F8EB01F0EB03E013 0714C0EB0F80EB1F00133E5BA25B485A485A485A120F5B48C7FC123E5A12FCB91280A5C8 000F90C7FCAC027FB61280A531417DC038>I<0007150301E0143F01FFEB07FF91B6FC5E 5E5E5E5E16804BC7FC5D15E092C8FC01C0C9FCAAEC3FF001C1B5FC01C714C001DF14F090 39FFE03FFC9138000FFE01FC6D7E01F06D13804915C0497F6C4815E0C8FC6F13F0A317F8 A4EA0F80EA3FE0487E12FF7FA317F05B5D6C4815E05B007EC74813C0123E003F4A1380D8 1FC0491300D80FF0495AD807FEEBFFFC6CB612F0C65D013F1480010F01FCC7FC010113C0 2D427BC038>I<4AB47E021F13F0027F13FC49B6FC01079038807F8090390FFC001FD93F F014C04948137F4948EBFFE048495A5A1400485A120FA248486D13C0EE7F80EE1E00003F 92C7FCA25B127FA2EC07FC91381FFF8000FF017F13E091B512F89039F9F01FFC9039FBC0 07FE9039FF8003FF17804A6C13C05B6F13E0A24915F0A317F85BA4127FA5123FA217F07F 121FA2000F4A13E0A26C6C15C06D4913806C018014006C6D485A6C9038E01FFC6DB55A01 1F5C010714C0010191C7FC9038003FF02D427BC038>I<121E121F13FC90B712FEA45A17 FC17F817F017E017C0A2481680007EC8EA3F00007C157E5E00785D15014B5A00F84A5A48 4A5A5E151FC848C7FC157E5DA24A5A14035D14074A5AA2141F5D143FA2147F5D14FFA25B A35B92C8FCA35BA55BAA6D5A6D5A6D5A2F447AC238>I II65 D IIIIII73 D76 DII<923807FFC092B512 FE0207ECFFC0021F15F091267FFE0013FC902601FFF0EB1FFF01070180010313C04990C7 6C7FD91FFC6E6C7E49486F7E49486F7E01FF8348496F7E48496F1380A248496F13C0A248 90C96C13E0A24819F04982003F19F8A3007F19FC49177FA400FF19FEAD007F19FC6D17FF A3003F19F8A26D5E6C19F0A26E5D6C19E0A26C6D4B13C06C19806E5D6C6D4B13006C6D4B 5A6D6C4B5A6D6C4B5A6D6C4A5B6D01C001075B6D01F0011F5B010101FE90B5C7FC6D90B6 5A023F15F8020715C002004AC8FC030713C047467AC454>II 82 DI<003FBA12E0A59026FE00 0FEB8003D87FE09338003FF049171F90C71607A2007E1803007C1801A300781800A400F8 19F8481978A5C81700B3B3A20107B8FCA545437CC24E>III<007FB6D8C003B61280A5D8000F 01E0C7D801F8C7FC6D4C5A6F14076D6D5D6D6D4A5A4E5A6D6D143F6E6C92C8FC6E157E70 5B6EEBC0016E01E05B4D5A6E6D485A6EEBF80F6E01FC5B4D5A6E6D48C9FC6F6C5A6F137E 5F6F5B815F816F7F81836F7F707E93B5FC844B805D4B8004E77FDB0FC37FED1F83DB3F81 7F04007F037E137F4B8002016E7F4B6D7F4A5A4A486D7F020F6E7F4B7F4A48814AC76C7F 717F147E4A6F7E0101707F4A8149488349486F7F010F707FB600E00103B612FCA54E447D C355>88 D<903801FFE0011F13FE017F6D7E48B612E03A03FE007FF84848EB1FFC6D6D7E 486C6D7EA26F7FA36F7F6C5A6C5AEA00F090C7FCA40203B5FC91B6FC1307013F13F19038 FFFC01000313E0000F1380381FFE00485A5B127F5B12FF5BA35DA26D5B6C6C5B4B13F0D8 3FFE013EEBFFC03A1FFF80FC7F0007EBFFF86CECE01FC66CEB8007D90FFCC9FC322F7DAD 36>97 DIIIII II< 137C48B4FC4813804813C0A24813E0A56C13C0A26C13806C1300EA007C90C7FCAAEB7FC0 EA7FFFA512037EB3AFB6FCA518467CC520>I107 DI<90277F8007FEEC0FFCB590263FFFC090387FFF8092B5D8F001B512E002 816E4880913D87F01FFC0FE03FF8913D8FC00FFE1F801FFC0003D99F009026FF3E007F6C 019E6D013C130F02BC5D02F86D496D7EA24A5D4A5DA34A5DB3A7B60081B60003B512FEA5 572D7CAC5E>I<90397F8007FEB590383FFF8092B512E0028114F8913987F03FFC91388F 801F000390399F000FFE6C139E14BC02F86D7E5CA25CA35CB3A7B60083B512FEA5372D7C AC3E>II<90397FC00FF8B5 90B57E02C314E002CF14F89139DFC03FFC9139FF001FFE000301FCEB07FF6C496D13804A 15C04A6D13E05C7013F0A2EF7FF8A4EF3FFCACEF7FF8A318F017FFA24C13E06E15C06E5B 6E4913806E4913006E495A9139DFC07FFC02CFB512F002C314C002C091C7FCED1FF092C9 FCADB67EA536407DAC3E>I<90387F807FB53881FFE0028313F0028F13F8ED8FFC91389F 1FFE000313BE6C13BC14F8A214F0ED0FFC9138E007F8ED01E092C7FCA35CB3A5B612E0A5 272D7DAC2E>114 D<90391FFC038090B51287000314FF120F381FF003383FC00049133F 48C7121F127E00FE140FA215077EA27F01E090C7FC13FE387FFFF014FF6C14C015F06C14 FC6C800003806C15806C7E010F14C0EB003F020313E0140000F0143FA26C141F150FA27E A26C15C06C141FA26DEB3F8001E0EB7F009038F803FE90B55A00FC5CD8F03F13E026E007 FEC7FC232F7CAD2C>IIIIII< B6903803FFFCA5000101E09038003E006C163C80017F5D8017F8013F5D6E1301011F5D6E 1303010F5D6E13076D5DED800F6D92C7FC15C05E6DEBE01E163E6D143CEDF07C027F1378 EDF8F8023F5B15FD021F5B15FF6E5BA36E5BA26E90C8FCA26E5AA26E5AA21578A215F85D 14015D001F1303D83F805B387FC007D8FFE05B140F92C9FC5C143E495A387FC1F8EB07F0 6CB45A6C5B000790CAFCEA01FC36407EAB3B>I123 D E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 821 724 a Fy(EXISTENCE)47 b(OF)i(ENGEL)f(STR)m(UCTURES)1532 973 y Fx(THOMAS)28 b(V)n(OGEL)685 1215 y Fw(Abstra)n(ct.)41 b Fx(W)-7 b(e)31 b(dev)n(elop)e(a)h(construction)f(of)i(Engel)e (structures)h(on)g(4{)685 1314 y(manifolds)j(based)g(on)g(decomp)r (ositions)g(of)h(manifolds)f(in)n(to)g(round)g(han-)685 1414 y(dles.)67 b(This)38 b(allo)n(ws)e(us)i(to)g(sho)n(w)e(that)i(all) g(parallelizable)e(4{manifolds)685 1513 y(admit)23 b(an)g(Engel)f (structure.)35 b(W)-7 b(e)23 b(also)f(sho)n(w)g(that,)j(giv)n(en)d(t)n (w)n(o)g(Engel)g(ma-)685 1613 y(nifolds)28 b Fv(M)1032 1625 y Fu(1)1069 1613 y Fv(;)14 b(M)1187 1625 y Fu(2)1252 1613 y Fx(satisfying)27 b(a)g(certain)h(condition)g(on)f(the)i(c)n (haracteristic)685 1713 y(foliations,)22 b(there)g(is)g(an)f(Engel)g (structure)h(on)f Fv(M)2222 1725 y Fu(1)2259 1713 y Fx(#)p Fv(M)2409 1725 y Fu(2)2446 1713 y Fx(#\()p Fv(S)2603 1683 y Fu(2)2647 1713 y Ft(\002)7 b Fv(S)2775 1683 y Fu(2)2812 1713 y Fx(\))22 b(whic)n(h)685 1812 y(is)27 b(closely)g(related)g(to)g(the)h(original)e(Engel)h(structures.)1639 2543 y Fs(Contents)486 2717 y Fr(1.)97 b(In)m(tro)s(duction)2133 b(2)486 2833 y(2.)97 b(Engel)32 b(structures)j(and)d(related)g (distributions)885 b(5)486 2949 y(2.1.)97 b(Con)m(tact)33 b(structures)h(and)f(ev)m(en)h(con)m(tact)f(structures)586 b(5)486 3066 y(2.2.)97 b(Engel)32 b(structures)i({)f(De\014nition)e (and)h(\014rst)h(examples)549 b(7)486 3182 y(2.3.)97 b(T)-8 b(ransv)m(erse)35 b(h)m(yp)s(ersurfaces)g(in)d(Engel)g (manifolds)699 b(9)486 3298 y(2.4.)97 b(V)-8 b(ertical)31 b(mo)s(di\014cations)f(of)i(transv)m(erse)j(b)s(oundaries)543 b(11)486 3414 y(3.)97 b(Con)m(tact)33 b(top)s(ology)1877 b(13)486 3531 y(3.1.)97 b(Prop)s(erties)33 b(of)f(Legendrian)g(curv)m (es)1191 b(13)486 3647 y(3.2.)97 b(F)-8 b(acts)32 b(from)g(the)h (theory)g(of)f(con)m(v)m(ex)j(surfaces)817 b(15)486 3763 y(3.3.)97 b(Bypasses)35 b(in)c(o)m(v)m(ert)m(wisted)k(con)m(tact)e (structures)732 b(19)486 3879 y(4.)97 b(Round)33 b(handle)f(decomp)s (ositions)f(and)i(mo)s(del)e(Engel)h(structures)133 b(23)486 3995 y(4.1.)97 b(Round)33 b(handle)f(decomp)s(ositions)1262 b(23)486 4112 y(4.2.)97 b(P)m(erturb)s(ed)34 b(prolongation)1541 b(24)486 4228 y(4.3.)97 b(Mo)s(del)32 b(Engel)g(structures)i(on)f (round)g(handles)g(of)f(index)h(0)322 b(26)486 4344 y(4.4.)97 b(Mo)s(del)32 b(Engel)g(structures)i(on)f(round)g(handles)g(of)f(index) h(1)322 b(27)486 4460 y(4.5.)97 b(Mo)s(del)32 b(Engel)g(structures)i (on)f(round)g(handles)g(of)f(index)h(2)322 b(29)486 4577 y(4.6.)97 b(Mo)s(del)32 b(Engel)g(structures)i(on)f(round)g(handles)g (of)f(index)h(3)322 b(33)486 4693 y(5.)97 b(A)m(ttac)m(hing)32 b(maps)h(for)f(round)h(handles)1159 b(37)486 4809 y(5.1.)97 b(Extending)33 b(Engel)f(structures)1378 b(38)486 4925 y(5.2.)97 b(A)m(ttac)m(hing)32 b(maps)h(for)f(round)g(handles)h(of)f (index)h(1)628 b(38)486 5042 y(5.3.)97 b(A)m(ttac)m(hing)32 b(maps)h(for)f(round)g(handles)h(of)f(index)h(2)628 b(41)486 5158 y(5.4.)97 b(A)m(ttac)m(hing)32 b(maps)h(for)f(round)g(handles)h (of)f(index)h(3)628 b(45)486 5274 y(6.)97 b(Existence)34 b(theorems)1795 b(46)486 5390 y(6.1.)97 b(Manifolds)31 b(with)h(trivial)e(tangen)m(t)j(bundle)938 b(46)486 5506 y(6.2.)97 b(Connected)34 b(sums)f(of)f(Engel)h(manifolds)1019 b(50)486 5623 y(References)2344 b(56)1861 5722 y Fq(1)p eop %%Page: 2 2 2 1 bop 386 259 a Fq(2)1134 b(THOMAS)25 b(V)n(OGEL)1479 459 y Fr(1.)49 b Fs(Intr)n(oduction)486 633 y Fr(Distributions)27 b(are)i(subbundles)i(of)e(the)h(tangen)m(t)f(bundle)h(of)f(a)g (manifold.)40 b(It)386 749 y(is)33 b(natural)f(not)h(to)f(consider)i (general)f(distributions)e(but)j(to)f(mak)m(e)g(geometric)386 866 y(assumptions,)27 b(for)f(example)g(in)m(tegrabilit)m(y)-8 b(.)38 b(In)27 b(this)f(case)h(the)f(distribution)f(de-)386 982 y(\014nes)g(a)e(foliation)d(on)j(the)h(manifold.)37 b(Another)24 b(p)s(ossibilit)m(y)e(is)h(to)g(assume)h(that)f(a)386 1098 y(distribution)h(is)i(no)m(where)h(in)m(tegrable.)41 b(Imp)s(ortan)m(t)25 b(examples)h(of)g(this)g(t)m(yp)s(e)h(are)386 1214 y(con)m(tact)41 b(structures)h(on)e(o)s(dd{dimensional)e (manifolds.)64 b(On)41 b(3{dimensional)386 1330 y(manifolds,)35 b(prop)s(erties)h(of)g(con)m(tact)h(structures)h(re\015ect)f(top)s (ological)c(features)386 1447 y(of)f(the)h(underlying)f(manifold)e(in)i (a)g(surprising)g(w)m(a)m(y)-8 b(.)486 1563 y(Engel)37 b(structures)j(form)c(another)i(class)g(of)f(non{in)m(tegrable)g (distributions)386 1679 y(whic)m(h)e(is)g(closely)f(related)g(to)h(con) m(tact)g(structures.)52 b(By)36 b(de\014nition,)e(an)h(Engel)386 1795 y(structure)45 b(is)f(a)g(smo)s(oth)f(distribution)f Fp(D)47 b Fr(of)c(rank)i(2)e(on)h(a)g(manifold)d Fo(M)55 b Fr(of)386 1912 y(dimension)31 b(4)h(whic)m(h)i(satis\014es)f(the)g (non{in)m(tegrabilit)m(y)d(conditions)1625 2135 y(rank[)p Fp(D)s Fo(;)17 b Fp(D)s Fr(])27 b(=)g(3)1447 2286 y(rank[)p Fp(D)s Fo(;)17 b Fr([)p Fp(D)s Fo(;)g Fp(D)s Fr(]])27 b(=)g(4)33 b Fo(;)386 2515 y Fr(where)38 b([)p Fp(D)s Fo(;)17 b Fp(D)s Fr(])37 b(consists)h(of)e(those)i(tangen)m(t)g(v)m (ectors)h(whic)m(h)e(can)h(b)s(e)f(obtained)386 2632 y(b)m(y)26 b(taking)f(comm)m(utators)f(of)h(lo)s(cal)e(sections)j(of)f Fp(D)s Fr(.)41 b(Examples)25 b(of)g(Engel)g(struc-)386 2748 y(tures)34 b(can)g(b)s(e)f(obtained)g(from)f(con)m(tact)i (structures)h(on)e(3{manifolds.)42 b(In)34 b(this)386 2864 y(article)28 b(w)m(e)j(use)g(results)f(from)e(con)m(tact)j(top)s (ology)d(to)h(in)m(v)m(estigate)h(Engel)f(struc-)386 2980 y(tures.)83 b(This)46 b(sho)m(ws)h(that)f(Engel)f(structures)i (and)f(con)m(tact)g(structures)i(are)386 3096 y(closely)32 b(related.)486 3213 y(Similarly)f(to)36 b(con)m(tact)g(structures)i (and)d(symplectic)h(structures,)i(all)c(Engel)386 3329 y(structures)27 b(are)e(lo)s(cally)d(isomorphic.)39 b(Ev)m(ery)27 b(p)s(oin)m(t)e(of)f(an)h(Engel)g(manifold)d(has)386 3445 y(a)43 b(neigh)m(b)s(orho)s(o)s(d)e(with)i(lo)s(cal)d(co)s (ordinates)j Fo(x;)17 b(y)t(;)g(z)t(;)g(w)45 b Fr(suc)m(h)f(that)f(the) g(Engel)386 3561 y(structure)34 b(is)e(the)h(in)m(tersection)f(of)g (the)h(k)m(ernels)h(of)e(the)h(1{forms)1544 3746 y Fo(\013)c Fr(=)e Fo(dz)g Fp(\000)22 b Fo(x)17 b(dy)1546 3892 y(\014)34 b Fr(=)27 b Fo(dx)22 b Fp(\000)h Fo(w)c(dy)35 b(:)386 3829 y Fr(\(1\))386 4121 y(This)40 b(normal)f(form)g(w)m(as)i(obtained) f(\014rst)g(b)m(y)i(F.)d(Engel)h(in)g([Eng].)67 b(T)-8 b(ogether)386 4237 y(with)43 b(the)h(fact)f(that)g(a)g Fo(C)1392 4201 y Fn(2)1432 4237 y Fr({small)e(p)s(erturbation)h(of)h (an)g(Engel)g(structure)i(is)386 4353 y(again)31 b(an)i(Engel)f (structure,)j(this)d(implies)e(that)j(Engel)g(structures)h(are)f (stable)386 4469 y(in)j(the)i(sense)g(of)f(singularit)m(y)e(theory)-8 b(.)57 b(In)38 b([Mo1])f(R.)g(Mon)m(tgomery)g(classi\014ed)386 4586 y(all)32 b(distribution)f(t)m(yp)s(es)36 b(with)d(this)g(stabilit) m(y)f(prop)s(ert)m(y)-8 b(.)48 b(They)35 b(b)s(elong)e(to)g(one)386 4702 y(of)f(the)h(follo)m(wing)d(classes:)609 4872 y Fp(\017)41 b Fr(foliations)30 b(of)i(rank)h(one)g(on)f(manifolds)f(of)h (arbitrary)f(dimension)609 4988 y Fp(\017)41 b Fr(con)m(tact)34 b(structures)g(on)e(manifolds)f(of)h(o)s(dd)g(dimension)609 5104 y Fp(\017)41 b Fr(ev)m(en)35 b(con)m(tact)e(structures)h(on)f (manifolds)d(of)i(ev)m(en)i(dimension)609 5220 y Fp(\017)41 b Fr(Engel)33 b(structures)h(on)f(manifolds)d(of)i(dimension)f(4.)386 5390 y(Th)m(us)c(Engel)d(structures)j(are)f(sp)s(ecial)e(among)g (general)g(distributions)g(and)h(ev)m(en)386 5506 y(among)32 b(the)h(stable)g(distribution)e(t)m(yp)s(es)k(they)f(seem)f(to)g(b)s(e) h(exceptional)e(since)386 5623 y(they)h(are)g(a)f(p)s(eculiarit)m(y)f (of)h(dimension)g(4.)p eop %%Page: 3 3 3 2 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)723 b(3)486 459 y Fr(On)48 b(the)g(other)g(hand)h(Engel)e(structures)j(app) s(ear)e(quite)g(naturally)-8 b(.)88 b(F)-8 b(or)386 575 y(example,)32 b(a)g(generic)h(plane)f(\014eld)g(on)h(a)f(4{manifold)d (satis\014es)34 b(the)f(Engel)f(con-)386 691 y(ditions)40 b(almost)f(ev)m(erywhere)45 b(and)c(one)h(can)f(construct)h(an)f(Engel) g(structure)386 807 y(from)26 b(a)h(con)m(tact)h(structure)g(on)g(a)f (3{manifold.)38 b(Certain)27 b(non{holonomic)e(con-)386 924 y(strain)m(ts)33 b(studied)g(in)e(classical)g(mec)m(hanics)i(also)f (lead)g(to)g(Engel)g(structures.)486 1040 y(An)46 b(Engel)h(structure)h Fp(D)h Fr(induces)f(a)e(distribution)f(of)h(h)m(yp)s(erplanes)i Fp(E)60 b Fr(=)386 1156 y([)p Fp(D)s Fo(;)17 b Fp(D)s Fr(])22 b(whic)m(h)h(is)f(an)g(ev)m(en)j(con)m(tact)e(structure,)j (i.e.)39 b([)p Fp(E)9 b Fo(;)17 b Fp(E)9 b Fr(])22 b(is)g(the)h(whole)f (tangen)m(t)386 1272 y(bundle.)62 b(Moreo)m(v)m(er,)42 b(to)c(the)h(ev)m(en)i(con)m(tact)e(structure)h Fp(E)47 b Fr(one)39 b(can)g(asso)s(ciate)386 1389 y(its)j(c)m(haracteristic)h (foliation)d Fp(W)52 b Fr(and)43 b(it)f(turns)i(out)e(that)h Fp(W)52 b Fr(is)43 b(tangen)m(t)g(to)386 1505 y Fp(D)s Fr(.)h(This)33 b(is)g(explained)f(in)g(Section)h(2.2.)44 b(Th)m(us)35 b(an)e(Engel)f(structure)j Fp(D)g Fr(on)e Fo(M)386 1621 y Fr(induces)g(a)g(\015ag)f(of)g(distributions)386 1814 y(\(2\))764 b Fp(W)37 b(\032)28 b(D)i(\032)e(E)36 b Fr(=)28 b([)p Fp(D)s Fo(;)17 b Fp(D)s Fr(])27 b Fp(\032)h Fo(T)14 b(M)386 2007 y Fr(suc)m(h)43 b(that)f(eac)m(h)h(distribution)d (has)j(corank)f(one)h(in)e(the)h(next)h(one.)72 b(Hyp)s(er-)386 2123 y(surfaces)49 b(transv)m(erse)h(to)d Fp(W)56 b Fr(carry)48 b(a)g(con)m(tact)g(structure)h(together)f(with)f(a)386 2239 y(Legendrian)31 b(line)g(\014eld.)43 b(W)-8 b(e)32 b(will)d(refer)j(to)f(this)g(line)g(\014eld)g(as)h(the)g(in)m (tersection)386 2355 y(line)f(\014eld.)486 2472 y(The)37 b(presence)i(of)e(the)g(distributions)e(in)i(\(2\))f(leads)h(to)f (strong)h(restrictions)386 2588 y(for)29 b(the)h(top)s(ology)e(of)i (Engel)f(manifolds.)40 b(The)31 b(follo)m(wing)c(result)j(can)g(b)s(e)g (found)386 2704 y(in)i([KMS])h(where)h(it)d(is)h(attributed)g(to)h(V.)f (Gershk)m(ovic)m(h.)386 2859 y Fy(Prop)s(osition)51 b(1.1.)e Fm(If)e(an)g(orientable)g Fr(4)p Fm({manifold)e(admits)i(an)h (orientable)386 2975 y(Engel)34 b(structur)-5 b(e,)36 b(then)f(it)g(has)f(trivial)h(tangent)g(bund)5 b(le.)486 3129 y Fr(W)-8 b(e)45 b(dev)m(elop)g(a)g(construction)g(of)f(Engel)g (manifolds)f(whic)m(h)i(allo)m(ws)f(us)h(to)386 3246 y(pro)m(v)m(e)34 b(the)f(con)m(v)m(erse)i(of)d(Prop)s(osition)f(1.1.) 386 3400 y Fy(Theorem)50 b(1.2.)d Fm(Every)f(p)-5 b(ar)g(al)5 b(lelizable)43 b Fr(4)p Fm({manifold)g(admits)i(an)g(orientable)386 3516 y(Engel)34 b(structur)-5 b(e.)486 3671 y Fr(This)52 b(solv)m(es)i(a)e(problem)g(from)f([ElM])h(and)h([KMS)q(].)103 b(F)-8 b(or)52 b(op)s(en)h(mani-)386 3787 y(folds,)31 b(Theorem)g(1.2)g(can)g(b)s(e)h(pro)m(v)m(ed)g(using)f(the)h Fo(h)p Fr({principle)d(for)i(op)s(en,)g(Di\013)o({)386 3903 y(in)m(v)-5 b(arian)m(t)31 b(relations,)g(cf.)i([ElM].)486 4020 y(In)h(the)h(literature)d(one)j(can)f(\014nd)h(only)e(t)m(w)m(o)i (other)f(constructions)h(of)f(Engel)386 4136 y(structures)47 b(on)e(closed)g(manifolds.)79 b(The)46 b(\014rst)f(one)h(is)e(called)g (prolongation)386 4252 y(and)39 b(w)m(as)g(in)m(tro)s(duced)g(b)m(y)g (E.)g(Cartan.)61 b(With)38 b(this)g(metho)s(d)g(one)h(gets)g(Engel)386 4368 y(structures)25 b(on)e(certain)g Fo(S)1341 4332 y Fn(1)1380 4368 y Fr({bundles)h(o)m(v)m(er)h(3{dimensional)20 b(con)m(tact)k(manifolds.)386 4484 y(The)32 b(second)g(construction)f (is)f(due)h(to)g(H.{J.)g(Geiges,)f(cf.)h([Gei)o(].)43 b(It)31 b(yields)g(En-)386 4601 y(gel)h(structures)i(on)e (parallelizable)d(mapping)i(tori.)42 b(These)35 b(t)m(w)m(o)e (constructions)386 4717 y(co)m(v)m(er)h(only)e(a)g(small)f(p)s(ortion)g (of)h(all)e(parallelizable)f(4{manifolds.)486 4833 y(Our)d (construction)g(is)g(based)h(on)f(decomp)s(ositions)f(of)h(manifolds)e (in)m(to)h(round)386 4949 y(handles.)50 b(A)34 b(round)h(handle)g(of)f (dimension)f Fo(n)i Fr(and)f(index)h Fo(l)f Fp(2)d(f)p Fr(0)p Fo(;)17 b(:)g(:)g(:)f(;)h(n)23 b Fp(\000)h Fr(1)p Fp(g)386 5066 y Fr(is)1357 5218 y Fo(R)1431 5233 y Fl(l)1485 5218 y Fr(=)j Fo(D)1672 5177 y Fl(l)1720 5218 y Fp(\002)c Fo(D)1904 5177 y Fl(n)p Fk(\000)p Fl(l)q Fk(\000)p Fn(1)2139 5218 y Fp(\002)g Fo(S)2305 5177 y Fn(1)2377 5218 y Fo(:)386 5390 y Fr(It)47 b(is)g(attac)m(hed)g(to)g(a)g(manifold)d(with)j(b)s (oundary)g(using)g(an)g(em)m(b)s(edding)f(of)386 5506 y Fo(@)437 5521 y Fk(\000)496 5506 y Fo(R)570 5521 y Fl(l)631 5506 y Fr(=)35 b Fo(S)808 5470 y Fl(l)q Fk(\000)p Fn(1)949 5506 y Fp(\002)26 b Fo(D)1136 5470 y Fl(n)p Fk(\000)p Fl(l)q Fk(\000)p Fn(1)1374 5506 y Fp(\002)g Fo(S)1543 5470 y Fn(1)1619 5506 y Fr(in)m(to)36 b(the)h(b)s(oundary)g (of)f Fo(M)10 b Fr(.)57 b(y)37 b(a)g(theorem)f(of)386 5623 y(D.)27 b(Asimo)m(v)f(in)h([As)q(])g(ev)m(ery)i(4{manifold)24 b(with)j(v)-5 b(anishing)26 b(Euler)h(c)m(haracteristic)p eop %%Page: 4 4 4 3 bop 386 259 a Fq(4)1134 b(THOMAS)25 b(V)n(OGEL)386 459 y Fr(can)38 b(b)s(e)g(obtained)g(from)e Fo(R)1424 474 y Fn(0)1502 459 y Fr(b)m(y)j(attac)m(hing)e(round)h(handles)g(of)g (higher)f(index)386 575 y(successiv)m(ely)-8 b(.)486 691 y(W)g(e)37 b(\014x)g(a)g(set)g(of)g(mo)s(del)e(Engel)h(structures)j (on)e(round)g(handles)g(suc)m(h)h(that)386 807 y(the)24 b(c)m(haracteristic)g(foliation)d(is)i(transv)m(erse)j(to)e Fo(@)2214 822 y Fk(\000)2273 807 y Fo(R)2347 822 y Fl(l)2398 807 y Fr(and)g Fo(@)2630 822 y Fn(+)2689 807 y Fo(R)2763 822 y Fl(l)2817 807 y Fr(=)k Fo(@)5 b(R)3051 822 y Fl(l)3082 807 y Fp(n)g Fo(@)3188 822 y Fk(\000)3247 807 y Fo(R)3321 822 y Fl(l)3348 807 y Fr(.)386 924 y(These)24 b(mo)s(del)d(Engel)h (structures)i(are)f(constructed)h(using)e(a)g(p)s(erturb)s(ed)h(v)m (ersion)386 1040 y(of)33 b(the)i(prolongation)c(construction.)48 b(The)35 b(con)m(tact)g(structure)g(on)f(eac)m(h)h Fo(@)3215 1055 y Fk(\000)3274 1040 y Fo(R)3348 1055 y Fl(l)386 1156 y Fr(dep)s(ends)40 b(essen)m(tially)e(only)g(on)g Fo(l)j Fr(while)c(the)i(in)m(tersection)f(line)g(\014eld)g(and)g(the) 386 1272 y(orien)m(tation)43 b(of)i(the)h(con)m(tact)f(structure)i(are) e(di\013eren)m(t)g(for)g(di\013eren)m(t)g(mo)s(del)386 1389 y(Engel)32 b(structures)j(on)d Fo(R)1324 1404 y Fl(l)1350 1389 y Fr(.)486 1505 y(When)42 b(w)m(e)h(attac)m(h)e(a)h (round)g(handle)f Fo(R)2020 1520 y Fl(l)2088 1505 y Fr(with)g(a)g(mo)s (del)f(Engel)h(structure)386 1621 y(to)34 b(an)f(Engel)h(manifold)d Fo(M)1426 1585 y Fk(0)1484 1621 y Fr(with)j(transv)m(erse)i(b)s (oundary)e(w)m(e)h(ha)m(v)m(e)h(to)d(ensure)386 1737 y(that)k(the)g(attac)m(hing)g(map)f(preserv)m(es)k(the)e(orien)m(ted)f (con)m(tact)h(structures)h(and)386 1853 y(the)28 b(in)m(tersection)f (line)f(\014eld)h(if)g(w)m(e)h(w)m(an)m(t)g(to)f(extend)i(the)f(Engel)f (structure)i(from)386 1970 y Fo(M)490 1934 y Fk(0)546 1970 y Fr(to)k Fo(M)770 1934 y Fk(0)815 1970 y Fp([)23 b Fo(R)978 1985 y Fl(l)1036 1970 y Fr(b)m(y)34 b(the)f(mo)s(del)e (Engel)h(structure.)44 b(In)33 b(order)g(to)f(satisfy)g(these)386 2086 y(conditions)d(w)m(e)j(isotop)s(e)d(the)i(attac)m(hing)f(map)f (and)i(w)m(e)g(c)m(ho)s(ose)g(a)f(mo)s(del)f(Engel)386 2202 y(structure)48 b(suitably)-8 b(.)84 b(F)-8 b(or)45 b(this)h(w)m(e)i(use)f(sev)m(eral)g(constructions)g(of)f(con)m(tact)386 2318 y(top)s(ology)-8 b(.)46 b(It)34 b(turns)h(out)f(that)f(it)g(is)h (con)m(v)m(enien)m(t)i(to)d(ensure)j(that)e(the)g(con)m(tact)386 2435 y(structures)39 b(on)f(b)s(oundaries)f(transv)m(erse)j(to)d(the)h (c)m(haracteristic)g(foliation)c(are)386 2551 y(o)m(v)m(ert)m(wisted)g (throughout)f(the)g(construction.)486 2667 y(Since)22 b Fo(M)834 2631 y Fk(0)860 2667 y Fp([)r Fo(R)1002 2682 y Fl(l)1051 2667 y Fr(is)g(again)f(an)i(Engel)f(manifold)e(with)i (transv)m(erse)j(b)s(oundary)e(w)m(e)386 2783 y(can)g(iterate)f(this)g (construction.)41 b(F)-8 b(or)22 b(the)h(construction)g(of)f(an)h (Engel)f(structure)386 2900 y(on)42 b(a)g(manifold)e Fo(M)53 b Fr(with)42 b(trivial)e(tangen)m(t)i(bundle)h(w)m(e)g(\014x)g (a)f(round)h(handle)386 3016 y(decomp)s(osition)31 b(of)i Fo(M)44 b Fr(and)33 b(w)m(e)h(use)g(the)f(mo)s(del)f(Engel)h (structures)h(to)f(extend)386 3132 y(the)f(Engel)g(structure)i(when)f (the)g(round)f(handles)g(are)g(attac)m(hed)h(successiv)m(ely)-8 b(.)386 3248 y(The)34 b(condition)d(that)i Fo(M)44 b Fr(has)33 b(trivial)d(tangen)m(t)k(bundle)f(will)e(b)s(e)i(used)h(to)f (sho)m(w)386 3364 y(that)27 b(there)h(is)e(a)h(mo)s(del)e(Engel)i (structure)i(with)d(the)i(desired)f(prop)s(erties)g(in)g(our)386 3481 y(collection)j(of)j(mo)s(del)e(Engel)h(structures)i(on)f Fo(R)2159 3496 y Fl(l)2217 3481 y Fr(for)f Fo(l)e Fr(=)e(1)p Fo(;)17 b Fr(2)p Fo(;)g Fr(3.)486 3597 y(Another)45 b(result)f(whic)m (h)h(can)g(b)s(e)g(obtained)f(using)g(round)h(handles)g(is)f(the)386 3713 y(follo)m(wing)30 b(theorem.)386 3912 y Fy(Theorem)39 b(1.3.)j Fm(L)-5 b(et)37 b Fo(M)1343 3927 y Fn(1)1383 3912 y Fo(;)17 b(M)1521 3927 y Fn(2)1596 3912 y Fm(b)-5 b(e)36 b(manifolds)f(with)h(Engel)f(structur)-5 b(es)37 b Fp(D)3175 3927 y Fn(1)3215 3912 y Fo(;)17 b Fp(D)3336 3927 y Fn(2)386 4029 y Fm(such)31 b(that)h(the)g(char)-5 b(acteristic)30 b(foliations)h(admit)g(close)-5 b(d)30 b(tr)-5 b(ansversals)31 b Fo(N)3144 4044 y Fn(1)3184 4029 y Fo(;)17 b(N)3306 4044 y Fn(2)3345 4029 y Fm(.)386 4145 y(Then)42 b Fo(M)742 4160 y Fn(1)782 4145 y Fr(#)p Fo(M)957 4160 y Fn(2)997 4145 y Fr(#\()p Fo(S)1182 4109 y Fn(2)1249 4145 y Fp(\002)29 b Fo(S)1421 4109 y Fn(2)1460 4145 y Fr(\))43 b Fm(admits)f(an)g(Engel)g(structur)-5 b(e)44 b(which)e(c)-5 b(oincides)386 4261 y(with)45 b Fp(D)685 4276 y Fn(1)769 4261 y Fm(and)g Fp(D)1046 4276 y Fn(2)1130 4261 y Fm(away)g(fr)-5 b(om)44 b(a)h(neighb)-5 b(orho)g(o)g(d)44 b(of)g Fo(N)2514 4276 y Fn(1)2554 4261 y Fo(;)17 b(N)2676 4276 y Fn(2)2760 4261 y Fm(wher)-5 b(e)45 b(al)5 b(l)44 b(c)-5 b(on-)386 4377 y(ne)g(cte)g(d)48 b(sums)h(ar)-5 b(e)48 b(p)-5 b(erforme)g(d.)86 b(The)48 b(char)-5 b(acteristic)48 b(foliation)g(of)h(the)f(new)386 4494 y(Engel)34 b(structur)-5 b(e)36 b(again)e(admits)h(a)f(close)-5 b(d)34 b(tr)-5 b(ansversal.)486 4693 y Fr(This)43 b(theorem)f(can)h(b)s (e)h(used)g(to)e(construct)i(Engel)f(manifolds)d({)j(lik)m(e)f(for)386 4809 y(example)48 b(\()p Fo(n)33 b Fr(+)g(1\)\()p Fo(S)1213 4773 y Fn(3)1285 4809 y Fp(\002)g Fo(S)1461 4773 y Fn(1)1500 4809 y Fr(\)#)p Fo(n)p Fr(\()p Fo(S)1781 4773 y Fn(2)1854 4809 y Fp(\002)g Fo(S)2030 4773 y Fn(2)2070 4809 y Fr(\))48 b(for)g Fo(n)55 b Fp(\025)g Fr(1)48 b({)g(whic)m(h)h(are)g(not)386 4925 y(co)m(v)m(ered)34 b(b)m(y)g(the)f(construction)g(of)f(Geiges)g (or)g(b)m(y)i(prolongation.)486 5042 y(This)k(article)f(is)g(organized) h(as)g(follo)m(ws.)60 b(Section)38 b(2)g(con)m(tains)g(sev)m(eral)h (def-)386 5158 y(initions)h(and)h(prop)s(erties)h(of)g(Engel)f (structures)j(and)e(related)f(distributions.)386 5274 y(In)d(Section)f(2.4)g(w)m(e)h(in)m(tro)s(duce)g(v)m(ertical)e(mo)s (di\014cations)g(of)h(the)g(b)s(oundary)h(of)386 5390 y(Engel)45 b(manifolds.)81 b(This)46 b(construction)g(will)e(b)s(e)i (used)h(frequen)m(tly)g(in)e(later)386 5506 y(sections.)62 b(Our)38 b(construction)h(relies)e(on)i(sev)m(eral)g(facts)f(from)g (3{dimensional)386 5623 y(con)m(tact)28 b(top)s(ology)-8 b(.)40 b(W)-8 b(e)28 b(summarize)f(them)g(in)g(Section)g(3)g(for)g(the) h(con)m(v)m(enience)p eop %%Page: 5 5 5 4 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)723 b(5)386 459 y Fr(of)33 b(the)h(reader.)48 b(The)35 b(theorems)e (discussed)j(here)f(will)c(b)s(e)j(used)h(later)e(in)g(order)386 575 y(to)f(bring)g(the)h(attac)m(hing)f(maps)g(of)g(round)h(handles)g (in)e(a)i(suitable)e(form.)486 691 y(In)41 b(Section)g(4)f(w)m(e)j (discuss)f(round)f(handle)g(decomp)s(ositions)e(of)i(manifolds)386 807 y(and)e(mo)s(del)f(Engel)h(structures)i(on)e(round)h(handles.)64 b(The)40 b(v)-5 b(arious)39 b(isotopies)386 924 y(of)31 b(attac)m(hing)g(maps)g(for)g(round)h(handles)g(and)f(the)h(c)m(hoice)g (of)g(the)g(righ)m(t)e(mo)s(del)386 1040 y(Engel)h(structure)h(are)f (explained)g(in)g(Section)g(5.)42 b(W)-8 b(e)32 b(giv)m(e)f(the)h(pro)s (ofs)f(of)f(The-)386 1156 y(orem)i(1.2)g(and)h(Theorem)g(1.3)f(in)f (Section)i(6.)486 1272 y Fm(A)-5 b(cknow)5 b(le)-5 b(dgements:)126 b Fr(This)43 b(article)e(con)m(tains)i(the)g(main)e(results)j(of)e(the) 386 1389 y(authors)k(thesis.)85 b(It)46 b(is)g(a)g(pleasure)h(for)e(me) h(to)g(thank)g(m)m(y)h(advisor)f(Dieter)386 1505 y(Kotsc)m(hic)m(k)38 b(for)e(his)g(con)m(tin)m(uous)i(supp)s(ort)f(and)g(help.)56 b(I)37 b(w)m(ould)g(lik)m(e)f(to)g(thank)386 1621 y(Kai)27 b(Cieliebak,)i(Y)-8 b(ak)m(o)m(v)29 b(Eliash)m(b)s(erg)f(and)h(P)m (aolo)e(Ghiggini)f(for)i(helpful)f(discus-)386 1737 y(sions.)53 b(I)36 b(am)f(also)f(grateful)h(to)g(the)i(Studienstiftung)e(des)i (Deutsc)m(hen)g(V)-8 b(olk)m(es)386 1853 y(for)32 b(their)g (\014nancial)f(supp)s(ort.)685 2205 y(2.)48 b Fs(Engel)38 b(str)n(uctures)e(and)i(rela)-7 b(ted)37 b(distributions)486 2380 y Fr(W)-8 b(e)44 b(\014rst)g(de\014ne)h(all)d(the)i(distributions) e(w)m(e)j(will)d(encoun)m(ter)j(in)e(our)h(con-)386 2496 y(structions)30 b(and)h(the)f(relations)f(b)s(et)m(w)m(een)k(them.)42 b(This)30 b(will)e(lead)i(to)g(a)g(pro)s(of)f(of)386 2612 y(Prop)s(osition)g(1.1.)42 b(T)-8 b(o)30 b(ev)m(ery)i(Engel)e (structure)i(one)f(can)f(asso)s(ciate)g(a)g(foliation)386 2728 y(of)d(rank)h(1)f(and)h(h)m(yp)s(ersurfaces)i(transv)m(erse)g(to)d (the)h(this)f(foliation)e(carry)j(a)f(con-)386 2844 y(tact)33 b(structure)i(together)f(with)f(a)g(Legendrian)g(line)f(\014eld.)46 b(In)34 b(Section)g(2.4)f(w)m(e)386 2961 y(explain)41 b(v)m(ertical)h(mo)s(di\014cations)e(of)i(the)h(b)s(oundary)-8 b(,)45 b(a)d(construction)g(whic)m(h)386 3077 y(will)30 b(b)s(e)j(used)h(frequen)m(tly)-8 b(.)386 3359 y(2.1.)48 b Fy(Con)m(tact)g(structures)g(and)h(ev)m(en)g(con)m(tact)e (structures.)i Fr(Con)m(tact)386 3475 y(structures)36 b(and)f(ev)m(en)i(con)m(tact)e(structures)i(arise)d(naturally)g(on)g (Engel)h(mani-)386 3592 y(folds.)49 b(They)36 b(will)c(pla)m(y)j(an)f (imp)s(ortan)m(t)f(role)h(in)g(all)e(our)j(constructions.)50 b(Here)386 3708 y(w)m(e)34 b(summarize)d(the)i(de\014nitions)f(and)h (elemen)m(tary)f(prop)s(erties.)386 3871 y Fy(De\014nition)c(2.1.)36 b Fr(A)25 b Fm(c)-5 b(ontact)28 b(structur)-5 b(e)27 b Fp(C)k Fr(on)25 b(a)h(\(2)p Fo(n)7 b Fp(\000)g Fr(1\){dimensional)23 b(mani-)386 3988 y(fold)28 b Fo(N)40 b Fr(is)29 b(a)g(smo)s(oth)g (subbundle)h(of)f Fo(T)14 b(N)40 b Fr(with)29 b(corank)h(1)f(suc)m(h)i (that)e(for)g(ev)m(ery)386 4104 y(lo)s(cal)38 b(de\014ning)i(1{form)f Fo(\013)i Fr(the)f(rank)h(of)f Fo(d\013)2080 4019 y Fj(\014)2080 4079 y(\014)2112 4143 y Fk(C)2198 4104 y Fr(is)f(maximal,)g(or)h(equiv) -5 b(alen)m(tly)386 4230 y Fo(\013)23 b Fp(^)f Fo(d\013)673 4194 y Fl(n)p Fk(\000)p Fn(1)837 4230 y Fp(6)p Fr(=)28 b(0.)486 4394 y(If)34 b Fo(n)g Fr(is)f(ev)m(en)j(the)f(sign)e(of)h Fo(\013)24 b Fp(^)f Fo(d\013)1774 4358 y Fl(n)p Fk(\000)p Fn(1)1945 4394 y Fr(is)33 b(indep)s(enden)m(t)j(of)d(the)i(c)m(hoice)f (of)g Fo(\013)q Fr(.)386 4510 y(Th)m(us)25 b(a)e(con)m(tact)h(manifold) c(of)j(dimension)f(4)p Fo(n)s Fp(\000)s Fr(1)i(has)g(a)f(preferred)h (orien)m(tation.)386 4626 y(In)42 b(dimension)e(3)h(the)h(orien)m (tabilit)m(y)d(of)i Fo(M)53 b Fr(is)41 b(the)h(only)f(obstruction)g(to) g(the)386 4742 y(existence)34 b(of)e(a)g(con)m(tact)i(structure)f(b)m (y)h(a)e(result)h(of)f(J.)h(Martinet)f(in)f([Mar)q(].)486 4859 y(The)g(follo)m(wing)c(t)m(w)m(o)k(theorems)g(will)d(b)s(e)j(used) g(later.)42 b(W)-8 b(e)31 b(sk)m(etc)m(h)h(the)f(pro)s(ofs)386 4975 y(b)s(ecause)43 b(w)m(e)g(will)d(apply)i(them)f(in)h(concrete)h (situations.)70 b(First)41 b(w)m(e)j(explain)386 5091 y(Gra)m(y's)29 b(theorem)e(whic)m(h)i(states)g(a)f(remark)-5 b(able)26 b(stabilit)m(y)h(prop)s(ert)m(y)i(of)e(con)m(tact)386 5207 y(structures.)386 5371 y Fy(Theorem)49 b(2.2)g Fr(\(Gra)m(y)-8 b(,)46 b([Gr)o(]\))p Fy(.)h Fm(L)-5 b(et)45 b Fp(C)1894 5386 y Fl(s)1931 5371 y Fo(;)17 b(s)45 b Fp(2)h Fr([0)p Fo(;)17 b Fr(1])44 b Fm(b)-5 b(e)44 b(a)g(family)g(of)g(c)-5 b(ontact)386 5487 y(structur)g(es)29 b(on)f Fo(N)39 b Fm(which)28 b(is)g(c)-5 b(onstant)28 b(outside)h(a)f(c)-5 b(omp)g(act)28 b(subset)g(of)g Fo(N)10 b Fm(.)44 b(Then)386 5603 y(ther)-5 b(e)35 b(is)f(an)h(isotopy)g Fo( )1270 5618 y Fl(s)1342 5603 y Fm(such)f(that)i Fo( )1826 5618 y Fl(s)p Fk(\003)1898 5603 y Fp(C)1950 5618 y Fn(0)2018 5603 y Fr(=)27 b Fp(C)2173 5618 y Fl(s)2210 5603 y Fm(.)p eop %%Page: 6 6 6 5 bop 386 259 a Fq(6)1134 b(THOMAS)25 b(V)n(OGEL)386 459 y Fm(Pr)-5 b(o)g(of.)41 b Fr(W)-8 b(e)43 b(assume)h(that)f Fp(C)1484 474 y Fl(s)1564 459 y Fr(is)f(globally)f(de\014ned)j(b)m(y)g (a)f(smo)s(oth)f(family)e(of)386 575 y(1{forms)26 b Fo(\013)q Fr(\()p Fo(s)p Fr(\).)42 b(The)28 b(pro)s(of)f(without)g(this)g (assumption)g(is)g(sligh)m(tly)f(more)h(com-)386 691 y(plicated,)h(cf.)g([Mar].)42 b(The)29 b(desired)f(isotop)m(y)g(is)f (the)i(\015o)m(w)f(of)g(a)f(time{dep)s(enden)m(t)386 807 y(v)m(ector)40 b(\014eld)f Fo(Z)7 b Fr(\()p Fo(s)p Fr(\).)62 b(This)39 b(is)g(the)g(unique)h(v)m(ector)g(\014eld)f(whic)m (h)g(is)g(tangen)m(t)g(to)386 924 y Fp(C)438 939 y Fl(s)503 924 y Fr(=)27 b(k)m(er)q(\()p Fo(\013)q Fr(\()p Fo(s)p Fr(\)\))32 b(and)h(satis\014es)386 1089 y(\(3\))808 b Fo(i)1352 1104 y Fl(Z)5 b Fn(\()p Fl(s)p Fn(\))1496 1089 y Fo(d\013)q Fr(\()p Fo(s)p Fr(\))27 b(=)h Fp(\000)21 b Fr(_)-48 b Fo(\013)q Fr(\()p Fo(s)p Fr(\))32 b(on)h Fp(C)2345 1104 y Fl(s)2415 1089 y Fo(:)386 1254 y Fr(Because)42 b Fo(d\013)q Fr(\()p Fo(s)p Fr(\))e(is)g(a)g(non{degenerate)i(2{form)d (on)h Fp(C)2471 1269 y Fl(s)2508 1254 y Fr(,)j(suc)m(h)f(a)e(v)m(ector) i(\014eld)386 1370 y(exists)24 b(and)g(is)f(uniquely)h(determined.)41 b(One)24 b(can)g(sho)m(w)g(b)m(y)h(a)f(direct)f(calculation)386 1486 y(that)32 b(the)h(\015o)m(w)h(of)e Fo(Z)7 b Fr(\()p Fo(s)p Fr(\))32 b(has)h(the)g(desired)g(prop)s(erties.)863 b Fi(\003)486 1662 y Fr(An)37 b(immediate)d(consequence)40 b(of)d(Gra)m(y's)g(theorem)g(is)g(a)f(normal)f(form)h(for)386 1778 y(the)24 b(con)m(tact)g(structure)h(on)e(neigh)m(b)s(orho)s(o)s (ds)g(of)g(curv)m(es)i(tangen)m(t)f(to)f(the)h(con)m(tact)386 1894 y(structure,)34 b(i.e.)e(Legendrian)g(curv)m(es.)386 2035 y Fy(Corollary)h(2.3.)39 b Fm(If)31 b Fo(\015)37 b Fm(is)31 b(a)h(close)-5 b(d)30 b(L)-5 b(e)g(gendrian)31 b(curve)g(in)h(a)f Fr(3)p Fm({manifold)f(with)386 2151 y(c)-5 b(ontact)44 b(structur)-5 b(e)46 b(then)f(we)f(c)-5 b(an)44 b(cho)-5 b(ose)44 b(c)-5 b(o)g(or)g(dinates)44 b Fo(x;)17 b(z)t(;)g(t)45 b Fm(on)g(a)f(tubular)386 2267 y(neighb)-5 b(orho)g(o)g(d)31 b(of)h Fo(\015)38 b Fm(such)32 b(that)h Fo(\015)g Fr(=)27 b Fp(f)p Fo(x)h Fr(=)g Fo(z)k Fr(=)27 b(0)p Fp(g)33 b Fm(and)f(the)g(c)-5 b(ontact)33 b(structur)-5 b(e)386 2384 y(is)35 b(de\014ne)-5 b(d)33 b(by)i Fo(dz)27 b Fp(\000)c Fo(x)17 b(dt)p Fm(.)486 2524 y Fr(In)36 b(particular,)f(ev)m(ery)j(p)s(oin)m(t)d(of)h(a)f(con)m (tact)i(manifold)c(admits)i(a)h(neigh)m(b)s(or-)386 2640 y(ho)s(o)s(d)28 b(with)h(co)s(ordinates)g Fo(x;)17 b(y)t(;)g(z)33 b Fr(suc)m(h)d(that)f(the)h(con)m(tact)f(structure)i(is)d(de\014ned)386 2757 y(b)m(y)33 b Fo(\013)c Fr(=)e Fo(dz)g Fp(\000)c Fo(x)17 b(dy)t Fr(.)486 2873 y(Next)33 b(w)m(e)h(consider)f(v)m(ector)h (\014elds)f(whic)m(h)g(preserv)m(e)i(a)d(giv)m(en)h(con)m(tact)g (struc-)386 2989 y(ture,)40 b(so)f(called)e(con)m(tact)h(v)m(ector)i (\014elds.)61 b(First)37 b(recall)g(that)h(to)g(eac)m(h)h(con)m(tact) 386 3105 y(form)44 b Fo(\013)h Fr(one)g(can)g(asso)s(ciate)g(a)f (unique)i(con)m(tact)f(v)m(ector)h(\014eld)f Fo(R)q Fr(,)j(the)d Fm(R)-5 b(e)g(eb)386 3222 y(ve)g(ctor)35 b(\014eld)p Fr(,)d(suc)m(h)i(that)e Fo(\013)q Fr(\()p Fo(R)q Fr(\))27 b Fp(\021)h Fr(1)33 b(and)f Fo(i)1989 3237 y Fl(R)2047 3222 y Fo(d\013)c Fp(\021)h Fr(1.)386 3362 y Fy(Prop)s(osition)37 b(2.4.)42 b Fm(The)35 b(map)g(which)g(assigns)g(to)h(e)-5 b(ach)35 b(c)-5 b(ontact)36 b(ve)-5 b(ctor)35 b(\014eld)386 3478 y Fo(X)47 b Fm(the)39 b(function)g Fo(\013)q Fr(\()p Fo(X)8 b Fr(\))39 b Fm(is)g(an)g(isomorphism)e(b)-5 b(etwe)g(en)39 b(the)g(sp)-5 b(ac)g(e)39 b(of)g(c)-5 b(ontact)386 3595 y(ve)g(ctor)35 b(\014elds)f(and)g Fo(C)1189 3559 y Fk(1)1264 3595 y Fr(\()p Fo(M)10 b Fr(\))p Fm(.)386 3771 y(Pr)-5 b(o)g(of.)41 b Fr(Injectivit)m(y)c(follo)m(ws)f(immediately)e(from)i (De\014nition)f(2.1.)57 b(No)m(w)38 b(let)e Fo(f)386 3887 y Fr(b)s(e)i(a)f(smo)s(oth)g(function)g(on)h Fo(M)48 b Fr(and)38 b Fo(R)h Fr(the)f(Reeb)g(v)m(ector)h(\014eld.)58 b(Since)38 b Fo(d\013)3297 3802 y Fj(\014)3297 3862 y(\014)3330 3926 y Fk(C)386 4003 y Fr(is)h(non{degenerate)i(there)f(is)f(a)g (unique)h(v)m(ector)h(\014eld)e Fo(Y)61 b Fr(tangen)m(t)40 b(to)f Fp(C)46 b Fr(suc)m(h)386 4120 y(that)29 b(\()p Fo(i)665 4135 y Fl(Y)726 4120 y Fo(d\013)q Fr(\))894 4035 y Fj(\014)894 4095 y(\014)928 4159 y Fk(C)1000 4120 y Fr(=)f Fp(\000)p Fo(d)-16 b(f)1275 4035 y Fj(\014)1275 4095 y(\014)1308 4159 y Fk(C)1353 4120 y Fr(.)42 b(Then)31 b Fo(f)11 b(R)17 b Fr(+)f Fo(Y)51 b Fr(has)31 b(the)f(desired)g(prop)s (erties.)42 b(This)386 4236 y(pro)m(v)m(es)34 b(surjectivit)m(y)-8 b(.)2114 b Fi(\003)486 4412 y Fr(W)-8 b(e)34 b(will)e(discuss)j(sev)m (eral)f(theorems)h(from)d(3{dimensional)f(con)m(tact)k(top)s(ol-)386 4528 y(ogy)42 b(in)g(Section)g(3.)72 b(No)m(w)43 b(w)m(e)g(state)g(the) g(de\014nition)e(and)i(some)f(elemen)m(tary)386 4644 y(prop)s(erties)32 b(of)h(ev)m(en)h(con)m(tact)f(structures.)386 4785 y Fy(De\014nition)48 b(2.5.)f Fr(Let)c Fo(M)54 b Fr(b)s(e)43 b(a)f(2)p Fo(n)p Fr({dimensional)e(manifold)h(and)i Fp(E)51 b Fr(a)42 b(dis-)386 4901 y(tribution)34 b(on)j Fo(M)47 b Fr(of)35 b(corank)i(one.)55 b(If)36 b(for)g(ev)m(ery)i(lo)s (cal)c(de\014ning)i(1{form)e Fo(\013)j Fr(of)386 5017 y Fp(E)49 b Fr(the)41 b(2{form)e Fo(d\013)i Fr(has)g(maximal)d(rank)j (on)f Fp(E)9 b Fr(,)43 b(then)e Fp(E)49 b Fr(is)40 b(an)h(ev)m(en)h (con)m(tact)386 5133 y(structure.)486 5274 y(Equiv)-5 b(alen)m(tly)d(,)26 b Fp(E)35 b Fr(is)26 b(an)g(ev)m(en)i(con)m(tact)e (structure)i(if)d(for)h(ev)m(ery)i(lo)s(cal)c(de\014ning)386 5390 y(form)j Fo(\013)q Fr(,)i(the)f(\(2)p Fo(n)13 b Fp(\000)g Fr(1\){form)27 b Fo(\013)14 b Fp(^)f Fo(d\013)1773 5354 y Fl(n)p Fk(\000)p Fn(1)1938 5390 y Fr(has)29 b(no)f(zero)s(es.)43 b(In)28 b(dimension)f(4)h(this)386 5506 y(condition)e(ma)m(y)h(b)s(e)h (rephrased)h(as)f([)p Fp(E)9 b Fo(;)17 b Fp(E)9 b Fr(])27 b(=)g Fo(T)14 b(M)c Fr(.)42 b(The)29 b(de\014nitions)e(of)g(con)m(tact) 386 5623 y(structures)34 b(and)f(ev)m(en)h(con)m(tact)f(structures)i (are)d(v)m(ery)i(similar.)40 b(Still)30 b(there)k(are)p eop %%Page: 7 7 7 6 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)723 b(7)386 459 y Fr(signi\014can)m(t)35 b(di\013erences.)54 b(One)36 b(of)f(them)h(is)f(the)h(presence)i(of)d(a)g(distinguished)386 575 y(line)c(\014eld)i(on)f(a)g(manifold)e(with)i(an)h(ev)m(en)h(con)m (tact)f(structure.)486 691 y(Since)e Fp(E)39 b Fr(has)32 b(rank)f(2)p Fo(n)19 b Fp(\000)h Fr(1,)31 b(the)h(rank)f(of)g Fo(d\013)2171 607 y Fj(\014)2171 666 y(\014)2204 730 y Fk(E)2282 691 y Fr(is)f(2)p Fo(n)20 b Fp(\000)f Fr(2.)43 b(Hence)32 b Fo(d\013)3122 607 y Fj(\014)3122 666 y(\014)3155 730 y Fk(E)3233 691 y Fr(has)386 808 y(a)40 b(k)m(ernel)h Fp(W)49 b(\032)42 b(E)48 b Fr(of)40 b(dimension)f(1)h(and)g(it)g(is)g (easy)h(to)f(sho)m(w)h(that)f Fp(W)49 b Fr(do)s(es)386 924 y(not)36 b(dep)s(end)i(on)f(the)g(c)m(hoice)g(of)f(the)h (de\014ning)g(form)e Fo(\013)q Fr(.)56 b(All)35 b(\015o)m(ws)i(whic)m (h)h(are)386 1040 y(tangen)m(t)33 b(to)f Fp(W)41 b Fr(preserv)m(e)35 b Fp(E)9 b Fr(.)386 1184 y Fy(De\014nition)41 b(2.6.)j Fr(The)37 b(line)f(\014eld)g Fp(W)45 b Fr(is)37 b(the)g Fm(char)-5 b(acteristic)38 b(line)g(\014eld)e Fr(of)g Fp(E)9 b Fr(.)386 1300 y(The)29 b(foliation)c(induced)k(b)m(y)h(this)e (line)f(\014eld)h(is)g(called)f(the)i Fm(char)-5 b(acteristic)31 b(folia-)386 1416 y(tion)p Fr(.)41 b(A)24 b(h)m(yp)s(ersurface)j(is)d Fm(tr)-5 b(ansverse)23 b Fr(if)h(it)f(is)h(transv)m(erse)j(to)d(the)h (c)m(haracteristic)386 1532 y(foliation.)486 1676 y(The)33 b(follo)m(wing)d(lemma)h(is)h(a)g(simple)g(consequence)j(of)e(the)g (de\014nition)e(of)i(an)386 1792 y(ev)m(en)h(con)m(tact)f(structures)i (and)d(its)g(c)m(haracteristic)h(foliation.)386 1936 y Fy(Lemma)39 b(2.7.)k Fm(L)-5 b(et)37 b Fp(E)45 b Fm(b)-5 b(e)36 b(an)g(even)f(c)-5 b(ontact)37 b(structur)-5 b(e)37 b(on)f Fo(M)48 b Fm(and)35 b Fp(W)46 b Fm(b)-5 b(e)36 b(the)386 2052 y(char)-5 b(acteristic)44 b(line)h(foliation)f(of)h Fp(E)9 b Fm(.)75 b(If)45 b Fo(N)56 b Fm(is)45 b(a)g(tr)-5 b(ansverse)44 b(hyp)-5 b(ersurfac)g(e,)386 2168 y(then)35 b Fo(T)14 b(N)32 b Fp(\\)23 b(E)43 b Fm(is)35 b(a)f(c)-5 b(ontact)35 b(structur)-5 b(e)36 b(on)e Fo(N)10 b Fm(.)486 2284 y(If)33 b Fo(N)675 2248 y Fk(0)732 2284 y Fm(is)h(another)f(tr)-5 b(ansversal)34 b(such)f(that)h(two)g(interior)g(p)-5 b(oints)33 b Fo(p)28 b Fp(2)g Fo(N)44 b Fm(and)386 2401 y Fo(q)31 b Fp(2)e Fo(N)643 2364 y Fk(0)701 2401 y Fm(lie)34 b(on)g(the)g(same)g(le)-5 b(af)34 b Fp(W)1661 2416 y Fl(p)1735 2401 y Fm(of)h(the)f(char)-5 b(acteristic)34 b(foliation,)f(then)i(the)386 2517 y(map)h(obtaine)-5 b(d)36 b(by)h(fol)5 b(lowing)36 b(ne)-5 b(arby)37 b(le)-5 b(aves)36 b(pr)-5 b(eserves)36 b(the)h(induc)-5 b(e)g(d)36 b(c)-5 b(ontact)386 2633 y(structur)g(es)36 b(on)e(neighb)-5 b(orho)g(o)g(ds)33 b(of)i Fo(p)g Fm(and)f Fo(p)2032 2597 y Fk(0)486 2777 y Fr(As)44 b(w)m(e)g(ha)m(v)m(e)h(explained)f(ab)s(o)m (v)m(e)g(a)f(con)m(tact)i(structure)f(on)g(a)f(manifold)e(of)386 2893 y(dimension)34 b(2)p Fo(n)25 b Fp(\000)f Fr(1)36 b(induces)g(an)g(orien)m(tation)e(of)h(this)g(manifold)e(if)i Fo(n)h Fr(is)f(ev)m(en.)386 3009 y(In)h(this)f(situation)f(an)h(orien)m (tation)f(of)g(the)i(c)m(haracteristic)g(foliation)31 b(together)386 3125 y(with)38 b(the)g(con)m(tact)h(orien)m(tation)d(on) j(a)e(transv)m(ersal)i(de\014ne)h(an)e(orien)m(tation)e(of)386 3241 y Fo(M)10 b Fr(.)386 3385 y Fy(Prop)s(osition)26 b(2.8.)36 b Fm(L)-5 b(et)28 b Fp(E)37 b Fm(b)-5 b(e)27 b(an)g(even)g(c)-5 b(ontact)28 b(structur)-5 b(e)28 b(on)g(a)f Fr(4)p Fo(n)p Fm({manifold)386 3501 y Fo(M)10 b Fm(.)44 b(Then)31 b(an)f(orientation)h(of)g Fo(M)42 b Fm(induc)-5 b(es)31 b(an)f(orientation)h(of)g(the)g(char)-5 b(acter-)386 3617 y(istic)35 b(line)f(\014eld)g Fp(W)44 b Fm(and)34 b(vic)-5 b(e)34 b(versa.)386 3820 y Fr(2.2.)48 b Fy(Engel)37 b(structures)f({)h(De\014nition)f(and)i(\014rst)f(examples.)49 b Fr(If)32 b Fp(D)j Fr(is)d(a)386 3936 y(distribution)j(w)m(e)j(denote) f(the)h(sheaf)f(of)f(all)f(v)m(ectors)k(whic)m(h)e(can)g(b)s(e)g (obtained)386 4052 y(as)c(comm)m(utators)e(of)h(pairs)g(of)g(lo)s(cal)f (sections)i(of)f Fp(D)j Fr(b)m(y)f([)p Fp(D)s Fo(;)17 b Fp(D)s Fr(].)386 4195 y Fy(De\014nition)32 b(2.9.)38 b Fr(An)29 b(Engel)g(structure)h(is)e(a)g(distribution)f Fp(D)k Fr(of)d(rank)h(t)m(w)m(o)h(on)386 4312 y(a)i(manifold)e Fo(M)43 b Fr(of)32 b(dimension)g(four)g(with)g(the)h(follo)m(wing)d (prop)s(erties.)556 4455 y(\(i\))40 b Fp(E)d Fr(=)27 b([)p Fp(D)s Fo(;)17 b Fp(D)s Fr(])27 b Fp(\032)h Fo(T)14 b(M)43 b Fr(is)32 b(a)h(subbundle)g(of)f(rank)h(3.)529 4571 y(\(ii\))39 b Fo(T)14 b(M)39 b Fr(=)27 b([)p Fp(E)9 b Fo(;)17 b Fp(E)9 b Fr(].)486 4715 y(The)33 b(second)g(condition)e(in) g(De\014nition)f(2.9)i(implies)d(that)j Fp(E)k Fr(=)28 b([)p Fp(D)s Fo(;)17 b Fp(D)s Fr(])31 b(is)h(an)386 4831 y(ev)m(en)39 b(con)m(tact)g(structure.)61 b(T)-8 b(o)37 b Fp(E)47 b Fr(w)m(e)39 b(asso)s(ciate)e(the)h(c)m(haracteristic)g (foliation)386 4947 y Fp(W)j Fr(of)32 b Fp(E)9 b Fr(.)43 b(By)33 b(the)g(de\014nition)f(of)g Fp(W)41 b Fr(w)m(e)34 b(ha)m(v)m(e)f Fp(W)k(\032)28 b(E)9 b Fr(.)386 5091 y Fy(Lemma)44 b(2.10.)g Fm(If)c Fp(E)46 b Fr(=)37 b([)p Fp(D)s Fo(;)17 b Fp(D)s Fr(])40 b Fm(is)g(an)f(even)h(c)-5 b(ontact)40 b(structur)-5 b(e)41 b(induc)-5 b(e)g(d)40 b(by)386 5207 y(Engel)34 b(structur)-5 b(e)36 b Fp(D)s Fm(,)f(then)f Fp(W)j(\032)28 b(D)s Fm(.)386 5390 y(Pr)-5 b(o)g(of.)41 b Fr(Assume)36 b(that)g Fp(W)1361 5405 y Fl(p)1436 5390 y Fr(is)f(not)h(con)m(tained)f(in)g Fp(D)2352 5405 y Fl(p)2427 5390 y Fr(at)g Fo(p)e Fp(2)g Fo(M)10 b Fr(.)53 b(Let)36 b Fo(\013)g Fr(b)s(e)g(a)386 5506 y(lo)s(cal)20 b(de\014ning)h(form)g(for)g Fp(E)31 b Fr(on)22 b(a)f(neigh)m(b)s(orho)s(o)s(d)g(of)g Fo(p)p Fr(.)40 b(If)22 b Fo(X)r(;)17 b(Y)43 b Fr(are)22 b(t)m(w)m(o)h(linearly)386 5623 y(indep)s(enden)m(t)35 b(lo)s(cal)d(sections)j(of)f Fp(D)j Fr(around)d Fo(p)p Fr(,)h(then)g Fo(d\013)q Fr(\()p Fo(X)2665 5638 y Fl(p)2704 5623 y Fo(;)17 b(Y)2805 5638 y Fl(p)2844 5623 y Fr(\))31 b Fp(6)p Fr(=)f(0)k(b)m(y)h(the)p eop %%Page: 8 8 8 7 bop 386 259 a Fq(8)1134 b(THOMAS)25 b(V)n(OGEL)386 459 y Fr(assumption)41 b(on)h Fp(W)8 b Fr(.)73 b(On)42 b(the)g(other)g(hand)h Fo(d\013)q Fr(\()p Fo(X)r(;)17 b(Y)j Fr(\))44 b(=)g Fp(\000)p Fo(\013)q Fr(\([)p Fo(X)r(;)17 b(Y)k Fr(]\))44 b(=)g(0)386 575 y(since)h([)p Fo(X)r(;)17 b(Y)22 b Fr(])45 b(is)g(a)f(lo)s(cal)f(section)i(of)g Fp(E)57 b Fr(=)49 b([)p Fp(D)s Fo(;)17 b Fp(D)s Fr(].)80 b(Th)m(us)47 b(the)e(assumption)386 691 y Fp(W)484 706 y Fl(p)552 691 y Fp(6\032)28 b(D)734 706 y Fl(p)806 691 y Fr(leads)k(to)h(a)f(con)m(tradiction.)1457 b Fi(\003)486 905 y Fr(T)-8 b(ogether)34 b(with)g(Lemma)f(2.7)g(this)h(allo)m(ws)f (us)i(to)e(giv)m(e)h(the)h(follo)m(wing)c(in)m(ter-)386 1021 y(pretation)k(of)g(the)h(condition)f Fp(E)41 b Fr(=)34 b([)p Fp(D)s Fo(;)17 b Fp(D)s Fr(].)52 b(As)36 b(one)g(mo)m(v)m(es)h (along)d(the)j(lea)m(v)m(es)386 1137 y(of)30 b Fp(W)8 b Fr(,)32 b(the)f(plane)g(\014eld)f Fp(D)j Fr(rotates)e(in)f Fp(E)40 b Fr(around)30 b(the)i(c)m(haracteristic)e(foliation)386 1254 y(without)i(stopping.)486 1370 y(W)-8 b(e)39 b(ha)m(v)m(e)i(sho)m (wn)f(that)f(an)h(Engel)f(structure)h Fp(D)i Fr(on)d Fo(M)50 b Fr(induces)40 b(a)f(\015ag)g(of)386 1486 y(distributions)386 1682 y(\(4\))929 b Fp(W)36 b(\032)28 b(D)i(\032)f(E)36 b(\032)28 b Fo(T)14 b(M)43 b(:)386 1878 y Fr(Eac)m(h)32 b(of)f(these)i(distributions)d(has)h(corank)h(one)g(in)e(the)i (distribution)e(con)m(tain-)386 1994 y(ing)e(it.)41 b(By)30 b(Prop)s(osition)e(2.8,)h(an)g(orien)m(tation)f(of)h(the)g(c)m (haracteristic)g(foliation)386 2110 y(of)39 b(an)g(Engel)f(structure)j (induces)f(an)f(orien)m(tation)e(of)i(the)g(underlying)g(mani-)386 2227 y(fold)26 b(and)i(vice)f(v)m(ersa.)43 b(In)28 b(addition,)f Fp(E)36 b Fr(=)27 b([)p Fp(D)s Fo(;)17 b Fp(D)s Fr(])27 b(is)g(orien)m(ted)g(b)m(y)i(the)e(follo)m(wing)386 2343 y(prop)s(osition.)386 2499 y Fy(Prop)s(osition)39 b(2.11.)44 b Fm(If)37 b Fp(D)j Fm(is)e(an)f(Engel)h(structur)-5 b(e,)39 b(then)f(the)g(even)f(c)-5 b(ontact)386 2615 y(structur)g(e)36 b Fp(E)g Fr(=)28 b([)p Fp(D)s Fo(;)17 b Fp(D)s Fr(])34 b Fm(has)g(a)h(distinguishe)-5 b(d)33 b(orientation.)386 2829 y(Pr)-5 b(o)g(of.)41 b Fr(Let)36 b Fo(X)r(;)17 b(Y)57 b Fr(b)s(e)36 b(lo)s(cal)d(sections)j(of)f Fp(D)k Fr(around)c Fo(p)e Fp(2)g Fo(M)47 b Fr(suc)m(h)37 b(that)e Fo(X)8 b Fr(\()p Fo(p)p Fr(\))386 2945 y(and)41 b Fo(Y)21 b Fr(\()p Fo(p)p Fr(\))41 b(are)g(linearly)e(indep)s(enden)m (t.)70 b(Then)42 b Fo(X)8 b Fr(\()p Fo(p)p Fr(\))p Fo(;)17 b(Y)j Fr(\()p Fo(p)p Fr(\))p Fo(;)d Fr([)p Fo(X)r(;)g(Y)k Fr(]\()p Fo(p)p Fr(\))41 b(is)g(an)386 3062 y(orien)m(tation)31 b(of)h Fp(E)9 b Fr(\()p Fo(p)p Fr(\))32 b(whic)m(h)h(is)f(indep)s (enden)m(t)i(of)e(the)h(c)m(hoice)g(of)f Fo(X)r(;)17 b(Y)k Fr(.)243 b Fi(\003)486 3275 y Fr(This)32 b(leads)h(to)f(the)h (follo)m(wing)d(orien)m(tation)h(con)m(v)m(en)m(tions.)556 3431 y(\(i\))40 b(If)33 b Fp(W)41 b Fr(is)32 b(orien)m(ted)h(w)m(e)g (orien)m(t)g Fo(M)43 b Fr(b)m(y)33 b(Prop)s(osition)e(2.8.)529 3548 y(\(ii\))39 b Fp(E)e Fr(=)27 b([)p Fp(D)s Fo(;)17 b Fp(D)s Fr(])32 b(is)g(orien)m(ted)h(according)f(to)g(Prop)s(osition)f (2.11)h(.)502 3664 y(\(iii\))38 b(T)-8 b(ransv)m(erse)45 b(h)m(yp)s(ersurfaces)f(are)d(orien)m(ted)h(b)m(y)g(the)g(induced)g (con)m(tact)700 3780 y(structure.)61 b(In)38 b(particular)e(transv)m (erse)k(b)s(oundaries)d(are)h(orien)m(ted)g(b)m(y)700 3896 y(this)33 b(con)m(v)m(en)m(tion.)505 4013 y(\(iv\))40 b(If)49 b Fp(W)57 b Fr(is)47 b(orien)m(ted,)53 b(then)c(w)m(e)g(orien)m (t)f(the)g(con)m(tact)h(structure)h(on)e(a)700 4129 y(transv)m(erse)30 b(h)m(yp)s(ersurface)f(suc)m(h)g(that)e(the)g(orien)m(tation)f(of)g (the)i(con)m(tact)700 4245 y(structure)33 b(follo)m(w)m(ed)e(b)m(y)h (the)g(orien)m(tation)e(of)h Fp(W)40 b Fr(is)31 b(the)h(orien)m(tation) e(of)700 4361 y(the)j(ev)m(en)i(con)m(tact)e(structure.)386 4517 y(Let)f Fo(M)43 b Fr(b)s(e)32 b(an)g(Engel)g(manifold)d(with)j (orien)m(ted)g(c)m(haracteristic)g(foliation)d(and)386 4634 y(transv)m(erse)c(b)s(oundary)-8 b(.)40 b(W)-8 b(e)23 b(write)g Fo(@)1752 4649 y Fn(+)1811 4634 y Fo(M)34 b Fr(for)22 b(those)i(connected)g(comp)s(onen)m(ts)f(of)386 4750 y Fo(@)5 b(M)43 b Fr(where)32 b(the)g(c)m(haracteristic)f (foliation)d(p)s(oin)m(ts)i(out)h(of)g Fo(M)10 b Fr(.)44 b(The)32 b(remaining)386 4866 y(b)s(oundary)h(comp)s(onen)m(ts)g(are)g Fo(@)1576 4881 y Fk(\000)1635 4866 y Fo(M)39 b Fr(=)27 b Fo(@)5 b(M)34 b Fp(n)22 b Fo(@)2178 4881 y Fn(+)2237 4866 y Fo(M)10 b Fr(.)486 4982 y(It)24 b(is)g(clear)g(from)f(\(4\))h (that)g(the)h(existence)h(of)e(an)g(Engel)g(structure)i(has)f(strong) 386 5099 y(implications)35 b(for)k(the)g(top)s(ology)f(of)h(the)g (underlying)g(manifold.)60 b(A)39 b(pro)s(of)f(of)386 5215 y(the)h(follo)m(wing)e(result)i(can)g(b)s(e)g(found)h(in)e([KMS].) 64 b(It)39 b(w)m(as)h(kno)m(wn)g(earlier)e(b)m(y)386 5331 y(V.)33 b(Gershk)m(o)m(vic.)386 5487 y Fy(Prop)s(osition)44 b(2.12.)i Fm(L)-5 b(et)42 b Fp(D)j Fm(b)-5 b(e)42 b(an)f(oriente)-5 b(d)42 b(Engel)f(structur)-5 b(e)43 b(on)e(an)h(ori-)386 5603 y(ente)-5 b(d)34 b Fr(4)p Fm({manifold)g Fo(M)10 b Fm(.)45 b(Then)34 b(the)h(tangent)g(bund)5 b(le)34 b(of)h Fo(M)45 b Fm(is)35 b(trivial.)p eop %%Page: 9 9 9 8 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)723 b(9)386 459 y Fm(Pr)-5 b(o)g(of.)41 b Fr(Consider)h(the)f(\015ag)g Fp(W)51 b(\032)43 b(D)h(\032)f(E)51 b(\032)43 b Fo(T)14 b(M)51 b Fr(of)41 b(subbundles)h(of)f Fo(T)14 b(M)c Fr(.)386 575 y(According)32 b(to)h(our)f(orien)m(tation)g(con)m(v)m(en)m(tions)i (eac)m(h)g(of)e(the)h(quotien)m(t)g(bundles)386 691 y Fp(D)s Fo(=)p Fp(W)8 b Fr(,)33 b Fp(E)9 b Fo(=)p Fp(D)34 b Fr(and)f Fo(T)14 b(M)5 b(=)p Fp(E)40 b Fr(is)33 b(orien)m(ted.)43 b(Th)m(us)386 911 y(\(5\))769 b Fo(T)14 b(M)38 b Fr(=)28 b Fp(W)j(\010)1839 843 y(D)p 1825 888 107 4 v 1825 979 a(W)1964 911 y(\010)2083 843 y(E)p 2074 888 80 4 v 2074 979 a(D)2185 911 y(\010)2295 843 y Fo(T)14 b(M)p 2295 888 176 4 v 2352 979 a Fp(E)386 1108 y Fr(is)32 b(isomorphic)f(to)h (the)h(sum)f(of)h(four)f(trivial)e(line)h(bundles.)685 b Fi(\003)486 1286 y Fr(Under)39 b(the)g(assumptions)f(of)g(Prop)s (osition)f(2.12)h(w)m(e)h(can)g(c)m(ho)s(ose)h(orien)m(ted)386 1403 y(orthogonal)21 b(complemen)m(ts)h(in)g(\(4\))g(after)h(ha)m(ving) f(\014xed)i(an)f(auxiliary)d(Riemann-)386 1519 y(ian)34 b(metric)g(and)h(w)m(e)h(obtain)d(a)i(trivialization)c(of)j Fo(T)14 b(M)c Fr(.)51 b(Since)35 b(w)m(e)h(only)e(c)m(hose)386 1635 y(the)39 b(Riemannian)d(metric,)k(this)e(framing)e(is)j(w)m(ell)f (de\014ned)i(up)f(to)f(homotop)m(y)-8 b(.)386 1751 y(W)g(e)33 b(will)d(refer)j(to)f(suc)m(h)i(framings)d(as)i Fm(Engel)h(fr)-5 b(amings)p Fr(.)486 1867 y(A)35 b(v)m(ery)i(simple)d(example)i(of)f(an) g(Engel)h(structure)h(is)e(the)h(standard)g(Engel)386 1984 y(structure)e(in)e(the)h(follo)m(wing)c(Darb)s(oux)k(theorem)f (for)g(Engel)g(structures.)386 2125 y Fy(Theorem)39 b(2.13)g Fr(\(Engel,)34 b([Eng]\))p Fy(.)42 b Fm(Every)37 b(p)-5 b(oint)35 b(of)h(an)g(Engel)g(manifold)e(has)386 2242 y(a)f(neighb)-5 b(orho)g(o)g(d)31 b(with)i(c)-5 b(o)g(or)g(dinates)31 b Fo(w)s(;)17 b(x;)g(y)t(;)g(z)36 b Fm(such)d(that)g(the)g(Engel)g (structur)-5 b(e)386 2358 y(is)35 b(given)f(by)386 2525 y Fr(\(6\))670 b(k)m(er)q(\()p Fo(dz)26 b Fp(\000)d Fo(x)17 b(dy)t Fr(\))k Fp(\\)h Fr(k)m(er)q(\()p Fo(dx)h Fp(\000)f Fo(w)d(dy)t Fr(\))34 b Fo(:)486 2692 y Fr(Here,)d(the)f(c)m (haracteristic)f(foliation)e(is)i(spanned)i(b)m(y)g Fo(@)2538 2707 y Fl(w)2624 2692 y Fr(and)f(the)g(ev)m(en)i(con-)386 2809 y(tact)c(structure)i(is)e(de\014ned)i(b)m(y)f Fo(dz)18 b Fp(\000)c Fo(x)j(dy)t Fr(.)42 b(W)-8 b(e)29 b(no)m(w)g(discuss)g (more)f(in)m(teresting)386 2925 y(examples.)82 b(Apart)45 b(from)g(the)g(construction)h(w)m(e)g(presen)m(t)i(later,)f(t)m(w)m(o)g (other)386 3041 y(constructions)32 b(of)f(closed)h(Engel)f(manifolds)e (can)i(b)s(e)h(found)g(in)e(the)i(literature.)386 3183 y Fy(Example)h(2.14.)39 b Fr(The)31 b(starting)e(p)s(oin)m(t)f(of)h (the)h Fm(pr)-5 b(olongation)28 b Fr(construction)i(is)386 3299 y(a)e(con)m(tact)i(structure)g Fp(C)35 b Fr(on)28 b(a)h(3{manifold)d Fo(N)10 b Fr(.)42 b(The)30 b(pro)5 b(jectivization)27 b Fh(P)p Fp(C)37 b Fr(of)28 b Fp(C)386 3415 y Fr(is)h(a)g(circle)g(bundle)h(o)m(v)m(er)h Fo(N)10 b Fr(.)43 b(The)30 b(pro)5 b(jection)29 b(pr)f(:)g Fh(P)p Fp(C)36 b(!)27 b Fo(N)40 b Fr(of)29 b(this)h(\014bration)386 3531 y(maps)i(eac)m(h)i(Legendrian)e(line)f(to)h(its)h(base)g(p)s(oin)m (t)f(in)f Fo(N)10 b Fr(.)44 b(One)33 b(can)g(sho)m(w)h(that)1209 3699 y Fp(D)1286 3714 y Fk(C)1358 3699 y Fr(=)1462 3618 y Fj(\010)1520 3699 y Fo(v)e Fp(2)c Fo(T)1750 3714 y Fn([)p Fl(l)q Fn(])1815 3699 y Fh(P)p Fp(C)1956 3614 y Fj(\014)1956 3674 y(\014)2028 3699 y Fr(pr)2120 3722 y Fk(\003)2160 3699 y Fr(\()p Fo(v)t Fr(\))f Fp(2)h Fr([)p Fo(l)r Fr(])2493 3618 y Fj(\011)386 3866 y Fr(is)41 b(an)g(Engel)g (structure)i(whose)f(c)m(haracteristic)g(foliation)c(is)i(tangen)m(t)i (to)f(the)386 3982 y(\014b)s(ers)j(of)f(pr.)76 b(The)44 b(asso)s(ciated)f(ev)m(en)i(con)m(tact)f(structure)h(is)d(pr)2881 3946 y Fk(\000)p Fn(1)2881 4007 y Fk(\003)2992 3982 y Fp(C)6 b Fr(.)76 b(Simi-)386 4098 y(larly)-8 b(,)34 b(one)i(obtains)e Fm(orientable)h Fr(Engel)g(structures)i(on)e(the)h(space)g(of)f Fm(oriente)-5 b(d)386 4214 y Fr(Legendrian)32 b(lines.)486 4356 y(The)c(second)h(previously)e(kno)m(wn)i(construction)e(of)g (closed)h(Engel)f(manifolds)386 4472 y(is)47 b(due)h(to)f(H.{J.)h (Geiges.)87 b(It)48 b(yields)f(Engel)g(structures)i(on)f (parallelizable)386 4589 y(mapping)31 b(tori)g(of)h(compact)h (3{manifolds,)c(cf.)k([Gei)o(].)386 4784 y(2.3.)48 b Fy(T)-9 b(ransv)m(erse)35 b(h)m(yp)s(ersurfaces)g(in)e(Engel)g (manifolds.)48 b Fr(According)29 b(to)386 4900 y(Lemma)g(2.7)h(transv)m (erse)i(h)m(yp)s(ersurfaces)h(carry)e(a)f(natural)f(con)m(tact)i (structure.)386 5016 y(In)h(this)g(section)g(w)m(e)h(discuss)g(a)e (Legendrian)h(line)f(\014eld)g(induced)i(b)m(y)g(the)f(Engel)386 5132 y(structure)37 b(on)f(a)g(transv)m(erse)i(h)m(yp)s(ersurface.)56 b(This)37 b(line)d(\014eld)i(can)g(b)s(e)h(used)g(to)386 5249 y(obtain)32 b(a)g(normal)f(form)g(for)i(the)g(Engel)g(structure)h (near)f(a)f(transv)m(erse)j(h)m(yp)s(er-)386 5365 y(surface)e Fo(N)10 b Fr(.)386 5506 y Fy(De\014nition)40 b(2.15.)k Fr(The)36 b Fm(interse)-5 b(ction)38 b(line)f(\014eld)e Fr(of)h Fp(D)i Fr(on)e Fo(N)47 b Fr(is)35 b(the)i(Legen-)386 5623 y(drian)32 b(line)f(\014eld)h Fo(T)14 b(N)33 b Fp(\\)22 b(D)s Fr(.)p eop %%Page: 10 10 10 9 bop 386 259 a Fq(10)1096 b(THOMAS)25 b(V)n(OGEL)486 459 y Fr(If)33 b(the)i(Engel)f(structure)h Fp(D)h Fr(is)e(orien)m(ted,) g(then)h(w)m(e)g(orien)m(t)e(the)i(in)m(tersection)386 575 y(line)c(\014eld)g(suc)m(h)j(that)e(the)g(orien)m(tation)e(of)i Fp(W)40 b Fr(follo)m(w)m(ed)31 b(b)m(y)i(the)g(orien)m(tation)d(of)386 691 y(the)j(in)m(tersection)f(line)g(\014eld)g(is)g(the)h(orien)m (tation)e(of)h Fp(D)s Fr(.)486 807 y(The)i(follo)m(wing)c(theorem,)k (as)f(w)m(ell)g(as)g(Prop)s(osition)f(2.17,)h(can)g(b)s(e)h(found)f(in) 386 924 y([Mo2])d(but)f(according)g(to)h(that)f(article)f(they)i(w)m (ere)h(kno)m(wn)g(b)s(efore.)43 b(Both)29 b(deal)386 1040 y(with)j(Engel)g(structures)j(obtained)d(b)m(y)h(prolongation.)486 1156 y(If)c(one)h(applies)f(the)h(prolongation)e(construction)h(to)h (the)g(con)m(tact)g(structure)386 1272 y Fp(C)42 b Fr(on)36 b(a)f(transv)m(erse)j(h)m(yp)s(ersurface)g Fo(N)46 b Fr(in)35 b(an)h(Engel)g(manifold,)e(then)i(one)g(ob-)386 1389 y(tains)30 b(the)h(manifold)d Fh(P)p Fp(C)38 b Fr(with)31 b(its)f(canonical)f(Engel)h(structure.)44 b(Let)31 b Fo(\033)j Fr(b)s(e)d(the)386 1505 y(section)37 b(of)g Fh(P)p Fp(C)46 b Fr(whic)m(h)38 b(assigns)f(to)g(eac)m(h)i(p)s(oin)m(t) d Fo(p)h Fr(of)g Fo(N)48 b Fr(the)38 b(Legendrian)f(line)386 1621 y Fo(T)14 b(N)32 b Fp(\\)23 b(D)s Fr(\()p Fo(p)p Fr(\).)386 1773 y Fy(Theorem)j(2.16.)34 b Fm(A)n(ny)26 b(su\016ciently)g(smal)5 b(l)25 b(tubular)i(neighb)-5 b(orho)g(o)g(d)24 b(of)i Fo(N)36 b Fm(in)26 b Fo(M)386 1889 y Fm(is)38 b(c)-5 b(anonic)g(al)5 b(ly)38 b(di\013e)-5 b(omorphic)37 b(as)h(an)g(Engel)h(manifold)e(to)i(a)f(tubular)i(neigh-) 386 2005 y(b)-5 b(orho)g(o)g(d)34 b(of)g Fo(\033)39 b Fm(in)c Fh(P)p Fp(C)6 b Fm(.)486 2157 y Fr(This)31 b(means)f(that)h (the)g(in)m(tersection)g(line)e(\014eld)i(and)g(the)g(con)m(tact)g (structure)386 2273 y Fp(C)k Fr(on)28 b(a)h(transv)m(erse)i(h)m(yp)s (ersurface)f Fo(N)40 b Fr(su\016ce)30 b(to)e(reco)m(v)m(er)j(the)e (Engel)f(structure)386 2389 y(on)k(a)h(small)d(neigh)m(b)s(orho)s(o)s (d)h(of)h Fo(N)10 b Fr(.)486 2505 y(Another)37 b(prop)s(ert)m(y)h(of)f (prolonged)f(Engel)h(structures)i(is)d(that)h(they)h(admit)386 2621 y(man)m(y)32 b(di\013eomorphisms)f(whic)m(h)i(preserv)m(e)i(the)e (Engel)f(structure,)i(cf.)f([Mo2].)386 2773 y Fy(Prop)s(osition)j (2.17.)42 b Fm(L)-5 b(et)35 b Fo(N)1510 2788 y Fn(1)1550 2773 y Fo(;)17 b(N)1672 2788 y Fn(2)1746 2773 y Fm(b)-5 b(e)35 b Fr(3)p Fm({manifolds)e(with)i(c)-5 b(ontact)35 b(structur)-5 b(es)386 2889 y Fp(C)438 2904 y Fn(1)478 2889 y Fo(;)17 b Fp(C)574 2904 y Fn(2)654 2889 y Fm(and)39 b(let)i Fo(')c Fr(:)h Fo(N)1235 2904 y Fn(1)1312 2889 y Fp(!)f Fo(N)1527 2904 y Fn(2)1607 2889 y Fm(b)-5 b(e)40 b(a)g(c)-5 b(ontact)40 b(di\013e)-5 b(omorphism.)59 b(The)40 b(di\013e)-5 b(o-)386 3005 y(morphism)32 b(b)-5 b(etwe)g(en)33 b Fh(P)p Fp(C)1305 3020 y Fn(1)1380 3005 y Fm(and)g Fh(P)p Fp(C)1679 3020 y Fn(2)1754 3005 y Fm(which)f(is)h(induc)-5 b(e)g(d)33 b(by)g(the)h(action)f(of)g Fo(')g Fm(on)386 3122 y Fp(C)438 3137 y Fn(1)513 3122 y Fm(pr)-5 b(eserves)34 b(the)h(Engel)f(structur)-5 b(es.)486 3238 y(Every)35 b(di\013e)-5 b(omorphism)32 b Fh(P)p Fp(C)1559 3253 y Fn(1)1629 3238 y Fp(!)27 b Fh(P)p Fp(C)1867 3253 y Fn(2)1944 3238 y Fm(with)34 b(this)h(pr)-5 b(op)g(erty)35 b(is)g(of)f(this)h (form.)486 3389 y Fr(W)-8 b(e)33 b(will)d(mainly)h(b)s(e)i(concerned)h (with)e(the)i(case)f(that)g Fo(N)43 b Fr(is)32 b(the)h(transv)m(erse) 386 3506 y(b)s(oundary)42 b(of)g(a)g(compact)g(Engel)g(manifold.)69 b(In)42 b(the)h(follo)m(wing)c(section)k(w)m(e)386 3622 y(will)e(explain)h(a)h(mo)s(di\014cation)d(of)i(the)i(Engel)f (structure)h(on)f(a)g(collar)e(of)h(the)386 3738 y(b)s(oundary)35 b(whic)m(h)h(allo)m(ws)d(us)j(to)e(v)-5 b(ary)35 b(the)h(in)m (tersection)e(line)g(\014eld)g(within)g(its)386 3854 y(homotop)m(y)g(class.)48 b(Therefore)35 b(it)e(su\016ces)j(to)e(con)m (trol)g(the)g(homotop)m(y)g(class)g(of)386 3971 y(the)f(in)m (tersection)f(line)g(\014eld)g(and)h(not)f(the)h(induced)g(foliation.) 486 4087 y(In)22 b(the)g(remaining)d(part)j(of)f(this)h(section)g(w)m (e)h(de\014ne)g(simple)d(in)m(v)-5 b(arian)m(ts)21 b(whic)m(h)386 4203 y(determine)32 b(the)i(in)m(tersection)e(line)g(\014eld)g(on)h (transv)m(erse)i(b)s(oundaries)e(of)f(Engel)386 4319 y(manifolds.)40 b(F)-8 b(or)30 b(this)f(w)m(e)i(assume)g(that)f(the)g (c)m(haracteristic)g(foliation)d(and)j(the)386 4435 y(Engel)35 b(structure)h(are)f(orien)m(ted.)51 b(This)35 b(induces)g(orien)m (tations)f(of)h(the)g(con)m(tact)386 4552 y(structure)f(and)e(the)h(in) m(tersection)g(line)e(\014eld)i(on)f Fo(N)10 b Fr(.)486 4668 y(If)47 b(w)m(e)h(\014x)f(an)g(orien)m(ted)h(framing)d Fo(C)1895 4683 y Fn(1)1934 4668 y Fo(;)17 b(C)2048 4683 y Fn(2)2134 4668 y Fr(of)46 b(the)i(con)m(tact)g(structure,)k(w)m(e)386 4784 y(can)32 b(assign)g(an)f(elemen)m(t)h(of)f Fo(H)1545 4748 y Fn(1)1584 4784 y Fr(\()p Fo(N)10 b Fr(;)17 b Fh(Z)p Fr(\))30 b(to)h(a)h(no)m(where)h(v)-5 b(anishing)31 b(Legendrian)386 4900 y(line)44 b(\014eld)g Fo(X)53 b Fr(as)45 b(follo)m(ws.)80 b(There)46 b(are)f(unique)h(functions)f Fo(f)2748 4915 y Fn(1)2787 4900 y Fo(;)17 b(f)2879 4915 y Fn(2)2963 4900 y Fr(suc)m(h)47 b(that)386 5017 y Fo(X)35 b Fr(=)28 b Fo(f)654 5032 y Fn(1)693 5017 y Fo(C)763 5032 y Fn(1)810 5017 y Fr(+)7 b Fo(f)941 5032 y Fn(2)980 5017 y Fo(C)1050 5032 y Fn(2)1090 5017 y Fr(.)41 b(The)26 b(elemen)m(t)f(of)g Fo(H)1897 4980 y Fn(1)1936 5017 y Fr(\()p Fo(N)10 b Fr(;)17 b Fh(Z)p Fr(\))22 b(corresp)s(onding)k(to)e Fo(X)33 b Fr(assigns)386 5133 y(the)g(winding)e(n)m(um)m(b)s(er)i(of)1204 5320 y(\()p Fo(f)1290 5335 y Fn(1)1352 5320 y Fp(\016)22 b Fo(\015)5 b(;)17 b(f)1572 5335 y Fn(2)1633 5320 y Fp(\016)22 b Fo(\015)5 b Fr(\))28 b(:)f Fo(S)1947 5279 y Fn(1)2014 5320 y Fp(\000)-16 b(!)28 b Fh(R)2269 5279 y Fn(2)2336 5320 y Fp(n)22 b(f)p Fr(0)p Fp(g)386 5506 y Fr(around)37 b(0)g(to)f(the)i(homology)d(class)i(represen)m(ted)j(b)m(y)e Fo(\015)5 b Fr(.)56 b(This)38 b(dep)s(ends)g(only)386 5623 y(on)27 b(the)h(homotop)m(y)e(class)i(of)e Fo(X)35 b Fr(as)27 b(no)m(where)i(v)-5 b(anishing)26 b(Legendrian)h(line)f (\014eld.)p eop %%Page: 11 11 11 10 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(11)486 459 y Fr(In)30 b(general)f(w)m(e)i(will)c(not)j(use)h(a)e (distinguished)g(global)e(framing)h(but)i(w)m(e)h(will)386 575 y(compare)e(the)h(homotop)m(y)f(classes)h(of)f(no)m(where)i(v)-5 b(anishing)28 b(Legendrian)i(v)m(ector)386 691 y(\014elds)44 b(using)g(rotation)e(n)m(um)m(b)s(ers)j(along)d(Legendrian)i(curv)m (es.)80 b(Let)44 b Fo(X)52 b Fr(b)s(e)44 b(a)386 807 y(no)m(where)g(v)-5 b(anishing)41 b(Legendrian)g(v)m(ector)j(\014eld)e (on)g Fo(N)52 b Fr(and)42 b(let)g Fo(X)r(;)17 b(Y)63 b Fr(b)s(e)43 b(an)386 924 y(orien)m(ted)37 b(framing)e(of)h(the)i(con) m(tact)g(structure)g(on)f Fo(N)10 b Fr(.)57 b(If)37 b Fo(\015)42 b Fr(is)37 b(an)g(immersed,)386 1040 y(closed)k(Legendrian)g (curv)m(e,)j(then)56 b(_)-41 b Fo(\015)45 b Fr(is)c(a)f(Legendrian)h(v) m(ector)h(\014eld)e(along)g Fo(\015)386 1156 y Fr(and)33 b(there)g(are)f(unique)i(functions)e Fo(f)1773 1171 y Fn(1)1813 1156 y Fo(;)17 b(f)1905 1171 y Fn(2)1976 1156 y Fr(suc)m(h)34 b(that)47 b(_)-41 b Fo(\015)32 b Fr(=)c Fo(f)2643 1171 y Fn(1)2682 1156 y Fo(X)i Fr(+)22 b Fo(f)2939 1171 y Fn(2)2979 1156 y Fo(Y)f Fr(.)386 1304 y Fy(De\014nition)48 b(2.18.)f Fr(The)d(winding)e(n)m(um)m(b)s(er)h(of)f(the)i(map)e(\()p Fo(f)2766 1319 y Fn(1)2834 1304 y Fp(\016)29 b Fo(\015)5 b(;)17 b(f)3061 1319 y Fn(2)3130 1304 y Fp(\016)28 b Fo(\015)5 b Fr(\))46 b(:)386 1420 y Fo(S)452 1384 y Fn(1)519 1420 y Fp(\000)-16 b(!)27 b Fh(R)773 1384 y Fn(2)841 1420 y Fp(n)21 b(f)p Fr(0)p Fp(g)32 b Fr(around)h(zero)g(is)f(the)h Fm(r)-5 b(otation)35 b(numb)-5 b(er)32 b Fr(of)g Fo(X)40 b Fr(along)32 b Fo(\015)5 b Fr(.)486 1568 y(The)37 b(sign)g(of)f(the)h (rotation)e(n)m(um)m(b)s(er)j(c)m(hanges)g(if)d(the)j(orien)m(tation)d (of)h Fo(\015)5 b Fr(,)38 b(or)386 1684 y(if)29 b(the)i(orien)m(tation) d(of)i(the)h(con)m(tact)f(structure)i(is)e(rev)m(ersed.)45 b(In)30 b(particular,)f(its)386 1800 y(sign)h(c)m(hanges)i(if)d(w)m(e)j (c)m(hange)f(the)g(orien)m(tation)e(of)h(the)h(c)m(haracteristic)f (foliation)386 1917 y(of)i Fp(D)j Fr(but)e(it)f(is)g(indep)s(enden)m(t) h(of)g(the)g(orien)m(tation)d(of)i Fp(D)s Fr(.)486 2033 y(In)46 b(order)g(to)f(determine)g(the)i(homotop)m(y)e(class)h(of)f (the)h(in)m(tersection)g(line)386 2149 y(\014eld,)41 b(it)d(su\016ces)j(to)e(kno)m(w)i(the)e(rotation)f(n)m(um)m(b)s(er)i (along)e(su\016cien)m(tly)i(man)m(y)386 2265 y(Legendrian)32 b(curv)m(es.)386 2484 y(2.4.)48 b Fy(V)-9 b(ertical)24 b(mo)s(di\014cations)g(of)i(transv)m(erse)g(b)s(oundaries.)50 b Fr(In)23 b(this)f(sec-)386 2601 y(tion)36 b(w)m(e)i(consider)g(a)f (manifold)d Fo(M)48 b Fr(with)37 b(orien)m(ted)h(Engel)f(structure)h Fp(D)i Fr(with)386 2717 y(orien)m(ted)e(c)m(haracteristic)g(foliation)d (suc)m(h)k(that)f Fo(@)5 b(M)50 b Fr(is)37 b(transv)m(erse)k(and)d (com-)386 2833 y(pact.)68 b(V)-8 b(ertical)40 b(mo)s(di\014cations)e (of)j(the)g(b)s(oundary)g(allo)m(w)f(us)h(to)g(c)m(hange)g(the)386 2949 y(in)m(tersection)35 b(line)g(\014eld)g(on)g Fo(@)5 b(M)10 b Fr(.)54 b(In)36 b(this)g(construction)f(w)m(e)i(mo)s(dify)d (the)i(En-)386 3066 y(gel)g(structure)i(near)g(the)f(b)s(oundary)g(suc) m(h)i(that)e(the)g(ev)m(en)i(con)m(tact)e(structure)386 3182 y(remains)32 b(unc)m(hanged.)45 b(Assume)34 b(that)f Fp(L)f Fr(is)h(a)f(Legendrian)h(line)e(\014eld)i(whic)m(h)g(is)386 3298 y(homotopic)f(to)g(the)i(in)m(tersection)f(line)f(\014eld)h Fp(L)2142 3313 y Fk(D)2236 3298 y Fr(of)g Fp(D)s Fr(.)45 b(Then)34 b(w)m(e)h(can)e(mo)s(dify)386 3414 y Fp(D)i Fr(suc)m(h)e(that)f(the)h(in)m(tersection)f(line)f(\014eld)h(of)f(the)i (new)g(Engel)f(structure)h(is)f Fp(L)p Fr(.)486 3531 y(This)i(can)g(b)s(e)h(generalized)e(to)h(the)h(case)g(of)f(a)g(transv) m(erse)i(h)m(yp)s(ersurface)g Fo(N)10 b Fr(,)386 3647 y(but)34 b(then)g(the)h(p)s(ossible)e(c)m(hanges)i(of)e(the)h(in)m (tersection)g(line)f(\014elds)h(dep)s(end)h(on)386 3763 y(the)24 b(b)s(eha)m(vior)f(of)g(the)h(Engel)f(structure)i(along)d(the) i(lea)m(v)m(es)g(of)f(the)h(c)m(haracteristic)386 3879 y(foliation)35 b(through)k Fo(N)10 b Fr(.)63 b(This)39 b(problem)e(will)g(arise)h(only)g(in)g(Theorem)i(6.5)e(at)386 3995 y(the)33 b(end)g(of)f(this)h(article.)486 4112 y(Let)46 b Fo(X)755 4127 y Fn(0)840 4112 y Fr(b)s(e)h(a)f(Legendrian)g(v)m (ector)h(\014eld)f(whic)m(h)h(spans)g(and)f(orien)m(ts)g(the)386 4228 y(in)m(tersection)31 b(line)e(\014eld)i(on)g Fo(@)5 b(M)10 b Fr(.)44 b(Assume)32 b(that)f(the)g(Legendrian)g(v)m(ector)h (\014eld)386 4344 y Fo(X)467 4359 y Fn(1)530 4344 y Fr(on)24 b Fo(@)5 b(M)35 b Fr(is)23 b(homotopic)f(to)h Fo(X)1590 4359 y Fn(0)1653 4344 y Fr(through)h(a)g(family)d Fo(X)2456 4359 y Fl(s)2492 4344 y Fo(;)c(s)28 b Fp(2)g Fr([0)p Fo(;)17 b Fr(1])23 b(of)g(no)m(where)386 4460 y(v)-5 b(anishing)36 b(Legendrian)i(v)m(ector)g(\014elds.)59 b(Our)38 b(aim)d(is)i(to)h(construct)g(an)g(Engel)386 4577 y(structure)c(on)f Fo(M)44 b Fr(suc)m(h)34 b(that)f(the)g(in)m (tersection)g(line)f(\014eld)g(on)h Fo(@)5 b(M)45 b Fr(is)32 b(spanned)386 4693 y(and)k(orien)m(ted)f(b)m(y)i Fo(X)1178 4708 y Fn(1)1253 4693 y Fr(without)e(c)m(hanging)g(the)h(ev)m(en)h(con) m(tact)f(structure.)54 b(W)-8 b(e)386 4809 y(treat)34 b(the)g(b)s(oundary)g(comp)s(onen)m(ts)h Fo(@)1825 4824 y Fn(+)1884 4809 y Fo(M)45 b Fr(where)35 b Fp(W)42 b Fr(p)s(oin)m(ts)34 b(out)f(of)h Fo(M)10 b Fr(.)48 b(The)386 4925 y(comp)s(onen)m(ts)33 b Fo(@)974 4940 y Fk(\000)1034 4925 y Fo(M)38 b Fr(=)28 b Fo(@)5 b(M)33 b Fp(n)22 b Fo(@)1576 4940 y Fn(+)1636 4925 y Fo(M)43 b Fr(can)33 b(b)s(e)g(treated)g(similarly)-8 b(.)486 5042 y(Cho)s(ose)40 b(a)g(collar)d Fo(U)51 b Fp(')40 b Fo(@)1478 5057 y Fn(+)1538 5042 y Fo(M)e Fp(\002)27 b Fr(\()p Fp(\000)p Fr(1)p Fo(;)17 b Fr(0])40 b(of)f Fo(@)2267 5057 y Fn(+)2327 5042 y Fo(M)50 b Fr(suc)m(h)42 b(that)d(the)h(second)386 5158 y(factor)c(with)g(its)g (standard)h(orien)m(tation)e(corresp)s(onds)j(to)e(the)h(orien)m(ted)g (c)m(har-)386 5274 y(acteristic)32 b(foliation)d(of)j Fp(D)s Fr(.)43 b(W)-8 b(e)33 b(write)f Fo(w)j Fr(for)d(the)h(co)s (ordinate)f(on)g(\()p Fp(\000)p Fr(1)p Fo(;)17 b Fr(0].)486 5390 y(F)-8 b(urthermore)34 b(let)h Fo(Y)1254 5405 y Fn(0)1329 5390 y Fr(b)s(e)g(a)h(v)m(ector)g(\014eld)f(suc)m(h)i(that)f Fo(X)2577 5405 y Fn(0)2616 5390 y Fo(;)17 b(Y)2717 5405 y Fn(0)2791 5390 y Fr(is)35 b(an)h(orien)m(ted)386 5506 y(trivialization)21 b(of)26 b(the)g(con)m(tact)h(structure)g(on)f Fo(@)2151 5521 y Fn(+)2210 5506 y Fo(M)10 b Fr(.)42 b(The)27 b(ev)m(en)h(con)m(tact)e(struc-)386 5623 y(ture)42 b(on)g Fo(U)52 b Fr(is)42 b(spanned)h(and)f(orien)m(ted)g(b)m(y)h Fo(X)2172 5638 y Fn(0)2211 5623 y Fo(;)17 b(Y)2312 5638 y Fn(0)2351 5623 y Fo(;)g(@)2446 5638 y Fl(w)2503 5623 y Fr(.)71 b(There)43 b(is)f(a)f(unique)p eop %%Page: 12 12 12 11 bop 386 259 a Fq(12)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fr(smo)s(oth)36 b(function)h Fo(f)47 b Fr(:)36 b Fo(U)47 b Fp(')36 b Fo(@)1557 474 y Fn(+)1617 459 y Fo(M)g Fp(\002)26 b Fr(\()p Fp(\000)p Fr(1)p Fo(;)17 b Fr(0])35 b Fp(\000)-16 b(!)36 b Fh(R)48 b Fr(whic)m(h)38 b(v)-5 b(anishes)38 b(along)386 575 y Fo(@)437 590 y Fn(+)496 575 y Fo(M)44 b Fr(suc)m(h)34 b(that)e Fo(@)1116 590 y Fl(w)1206 575 y Fr(and)386 769 y(\(7\))345 b Fo(X)8 b Fr(\()p Fo(p;)17 b(w)s Fr(\))26 b(=)i(cos)q(\()p Fo(f)11 b Fr(\()p Fo(p;)17 b(w)s Fr(\)\))p Fo(X)1906 784 y Fn(0)1943 769 y Fr(\()p Fo(p)p Fr(\))22 b(+)g(sin\()p Fo(f)11 b Fr(\()p Fo(p;)17 b(w)s Fr(\)\))p Fo(Y)2742 784 y Fn(0)2780 769 y Fr(\()p Fo(p)p Fr(\))386 962 y(span)39 b(and)f(orien)m(t)g Fp(D)s Fr(\()p Fo(p;)17 b(w)s Fr(\).)60 b(Because)40 b Fp(D)h Fr(is)d(an)g(Engel)g(structure)i Fo(X)r(;)17 b Fr([)p Fo(@)3158 977 y Fl(w)3215 962 y Fo(;)g(X)8 b Fr(])386 1079 y(and)43 b Fo(@)637 1094 y Fl(w)737 1079 y Fr(are)h(linearly)d(indep)s(enden)m(t)j(ev)m(erywhere.)78 b(This)44 b(implies)c(that)j Fo(f)54 b Fr(is)386 1195 y(strictly)37 b(monotone)g(along)g(the)h(lea)m(v)m(es)h(of)f Fp(W)8 b Fr(.)60 b(By)38 b(our)g(orien)m(tation)e(con)m(v)m(en-)386 1311 y(tion)41 b(for)h(the)g(con)m(tact)h(structure)g(on)f(transv)m (erse)j(b)s(oundaries,)f Fo(f)53 b Fr(is)41 b(strictly)386 1427 y(increasing.)486 1544 y(There)27 b(is)e(a)g(unique)h(family)e(of) h(real)g(functions)h Fo(g)2276 1559 y Fl(s)2338 1544 y Fr(suc)m(h)h(that)e(cos)q(\()p Fo(g)2971 1559 y Fl(s)3007 1544 y Fr(\()p Fo(p)p Fr(\)\))p Fo(X)3251 1559 y Fn(0)3299 1544 y Fr(+)386 1660 y(sin\()p Fo(g)591 1675 y Fl(s)627 1660 y Fr(\()p Fo(p)p Fr(\)\))p Fo(Y)847 1675 y Fn(0)929 1660 y Fr(is)43 b(a)g(p)s(ositiv)m(e)g(m)m(ultiple)e(of)i Fo(X)2098 1675 y Fl(s)2178 1660 y Fr(and)h Fo(g)2426 1675 y Fn(0)2511 1660 y Fr(=)i(0.)76 b(Since)43 b Fo(@)3101 1675 y Fn(+)3161 1660 y Fo(M)54 b Fr(is)386 1776 y(compact)35 b(w)m(e)h(can)f(c)m(ho)s(ose)h Fo(k)f Fp(2)d Fh(Z)h Fr(suc)m(h)j(that)f Fo(g)2187 1791 y Fn(1)2250 1776 y Fr(+)24 b(2)p Fo(\031)t(k)38 b Fr(is)c(at)h(least)f(2)p Fo(\031)t Fr(.)51 b(No)m(w)386 1902 y(extend)30 b Fo(f)39 b Fr(from)27 b Fo(@)1063 1917 y Fn(+)1122 1902 y Fo(M)d Fp(\002)13 b Fr(\()p Fp(\000)p Fr(1)p Fo(;)k Fr(0])29 b(to)f(a)g(smo)s(oth)f(function)2571 1876 y Fj(b)2553 1902 y Fo(f)39 b Fr(on)28 b Fo(@)2822 1917 y Fn(+)2882 1902 y Fo(M)c Fp(\002)13 b Fr(\()p Fp(\000)p Fr(1)p Fo(;)k Fr(1])386 2033 y(suc)m(h)45 b(that)857 2007 y Fj(b)839 2033 y Fo(f)55 b Fr(is)43 b(strictly)g(increasing)g (and)h(coincides)f(with)g Fo(g)2767 2048 y Fn(1)2836 2033 y Fr(+)30 b(2)p Fo(\031)t(k)46 b Fr(along)386 2150 y Fo(@)437 2165 y Fn(+)496 2150 y Fo(M)14 b Fp(\002)t(f)p Fr(1)p Fp(g)p Fr(.)41 b(W)-8 b(e)24 b(extend)i Fp(D)g Fr(from)c Fo(M)35 b Fr(to)23 b(a)g(distribution)f(on)h Fo(M)14 b Fp([)2828 2165 y Fl(@)2879 2150 y Fo(@)2930 2165 y Fn(+)2989 2150 y Fo(M)g Fp(\002)t Fr([0)p Fo(;)j Fr(1])386 2266 y(b)m(y)33 b(the)g(span)h(of)e(the)h(v)m(ector)g (\014eld)g Fo(@)1752 2281 y Fl(w)1841 2266 y Fr(and)826 2469 y Fo(X)8 b Fr(\()p Fo(p;)17 b(w)s Fr(\))27 b(=)g(cos)q(\()1474 2443 y Fj(b)1456 2469 y Fo(f)10 b Fr(\()p Fo(p;)17 b(w)s Fr(\)\))p Fo(X)1875 2484 y Fn(0)1914 2469 y Fr(\()p Fo(p)p Fr(\))22 b(+)g(sin)o(\()2334 2443 y Fj(b)2316 2469 y Fo(f)11 b Fr(\()p Fo(p;)17 b(w)s Fr(\)\))p Fo(Y)2712 2484 y Fn(0)2750 2469 y Fr(\()p Fo(p)p Fr(\))32 b Fo(:)386 2678 y Fr(The)47 b(fact)f(that)h Fo(@)1083 2693 y Fl(w)1158 2652 y Fj(b)1140 2678 y Fo(f)61 b(>)51 b Fr(0)c(implies)c(that)k Fo(X)54 b Fr(and)46 b([)p Fo(@)2458 2693 y Fl(w)2515 2678 y Fo(;)17 b(X)8 b Fr(])46 b(are)h(linearly)d(in-)386 2794 y(dep)s(enden)m(t.)86 b(Moreo)m(v)m(er,)51 b Fo(@)1459 2809 y Fl(w)1562 2794 y Fr(together)46 b(with)g Fo(X)r(;)17 b Fr([)p Fo(@)2401 2809 y Fl(w)2458 2794 y Fo(;)g(X)8 b Fr(],)49 b(resp)s(ectiv)m(ely)e(the)386 2911 y(horizon)m(tal)33 b(lifts)h(of)g Fo(X)1236 2926 y Fn(0)1276 2911 y Fo(;)17 b(Y)1377 2926 y Fn(0)1415 2911 y Fr(,)36 b(span)g(the)f(same)g(ev)m(en) h(con)m(tact)g(structure.)52 b(Th)m(us)386 3027 y(the)32 b(extended)h(distribution)c(is)i(an)g(Engel)g(structure)h(whose)h(in)m (tersection)e(line)386 3143 y(\014eld)42 b(on)g Fo(@)803 3158 y Fn(+)863 3143 y Fo(M)e Fp(\002)29 b(f)p Fr(1)p Fp(g)42 b Fr(is)g(spanned)h(b)m(y)h Fo(X)2018 3158 y Fn(1)2057 3143 y Fr(.)73 b(Using)42 b(a)g(\015o)m(w)h(tangen)m(t)g(to)f (the)386 3259 y(c)m(haracteristic)29 b(foliation)e(one)j(can)g(iden)m (tify)f Fo(M)40 b Fr(and)30 b Fo(M)d Fp([)2574 3274 y Fl(@)2636 3259 y Fo(@)2687 3274 y Fn(+)2747 3259 y Fo(M)g Fp(\002)16 b Fr([0)p Fo(;)h Fr(1])30 b(suc)m(h)386 3376 y(that)i(the)h(ev)m(en)h(con)m(tact)g(structures)g(are)f(preserv)m(ed.) 386 3531 y Fy(De\014nition)c(2.19.)38 b Fr(The)28 b(mo)s(di\014cations) c(of)i(an)h(Engel)g(manifold)d(with)i(b)s(ound-)386 3647 y(ary)h(describ)s(ed)i(ab)s(o)m(v)m(e)f(will)d(b)s(e)j(called)e(a)h Fm(vertic)-5 b(al)30 b(mo)-5 b(di\014c)g(ations)28 b(of)i(the)g(b)-5 b(ound-)386 3763 y(ary)p Fr(.)486 3918 y(Let)32 b(us)h(summarize)f(the) h(discussion)g(in)e(the)i(follo)m(wing)d(theorem.)386 4073 y Fy(Theorem)43 b(2.20.)h Fm(L)-5 b(et)40 b Fr(\()p Fo(M)5 b(;)17 b Fp(D)s Fr(\))39 b Fm(b)-5 b(e)39 b(an)g(oriente)-5 b(d)38 b(Engel)h(manifold)f(such)h(that)386 4189 y Fo(@)5 b(M)49 b Fm(is)37 b(tr)-5 b(ansverse)37 b(and)g(c)-5 b(omp)g(act.)51 b(Assume)37 b(that)h(the)f(L)-5 b(e)g(gendrian)37 b(line)f(\014eld)386 4305 y Fp(L)f Fm(on)f Fo(@)680 4320 y Fn(+)740 4305 y Fo(M)45 b Fm(is)35 b(homotopic)f(to)h(the)f(interse) -5 b(ction)35 b(line)f(\014eld)g Fp(L)2730 4320 y Fk(D)2826 4305 y Fm(of)g Fp(D)s Fm(.)486 4422 y(Then)27 b Fp(L)h Fm(is)g(the)g(interse)-5 b(ction)27 b(line)h(\014eld)g(of)f(a)h (suitable)g(vertic)-5 b(al)28 b(mo)-5 b(di\014c)g(ation)386 4538 y(of)34 b Fo(@)551 4553 y Fn(+)611 4538 y Fo(M)10 b Fm(.)486 4693 y Fr(There)37 b(is)f(an)g(analogous)f(construction)h (in)g(the)g(con)m(text)i(of)e(con)m(tact)g(struc-)386 4809 y(tures.)49 b(Let)35 b Fo(N)44 b Fr(b)s(e)35 b(a)f(compact)g (3{manifold)d(carrying)i(a)h(con)m(tact)h(structure)g Fp(C)386 4925 y Fr(and)26 b(a)g(con)m(tact)g(v)m(ector)h(\014eld)f Fo(V)47 b Fr(transv)m(erse)28 b(to)e(the)g(b)s(oundary)-8 b(.)42 b(F)-8 b(or)25 b(this,)i(let)e Fo(X)386 5042 y Fr(b)s(e)k(a)g(no)m(where)h(v)-5 b(anishing)28 b(Legendrian)h(v)m (ector)h(\014eld)f(suc)m(h)h(that)f([)p Fo(V)5 b(;)17 b(X)8 b Fr(])29 b(and)g Fo(X)386 5158 y Fr(are)k(linearly)e(indep)s (enden)m(t.)45 b(Assume)34 b(that)f Fo(X)2145 5173 y Fn(1)2217 5158 y Fr(is)g(a)f(Legendrian)h(v)m(ector)h(\014eld)386 5274 y(along)j Fo(@)5 b(N)51 b Fr(whic)m(h)39 b(is)g(homotopic)e(to)i (the)g(restriction)f(of)h Fo(X)47 b Fr(to)38 b Fo(@)5 b(N)51 b Fr(through)386 5390 y(no)m(where)43 b(v)-5 b(anishing)40 b(Legendrian)h(v)m(ector)h(\014elds.)70 b(Then)42 b(w)m(e)h(can)e(mo)s (dify)f Fo(X)386 5506 y Fr(near)29 b(the)g(b)s(oundary)f(of)g Fo(N)39 b Fr(suc)m(h)30 b(that)f(the)g(resulting)e(Legendrian)h(v)m (ector)i(\014eld)386 5623 y Fo(X)475 5587 y Fk(0)521 5623 y Fr(coincides)24 b(with)f Fo(X)1219 5638 y Fn(1)1282 5623 y Fr(along)f Fo(@)5 b(N)34 b Fr(and)24 b Fo(X)1971 5587 y Fk(0)1994 5623 y Fo(;)17 b Fr([)p Fo(V)5 b(;)17 b(X)2260 5587 y Fk(0)2283 5623 y Fr(])23 b(are)h(linearly)d(indep)s (enden)m(t.)p eop %%Page: 13 13 13 12 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(13)386 459 y Fy(Remark)44 b(2.21.)h Fr(In)39 b(our)f(applications)f (the)i(Legendrian)f(v)m(ector)i(\014eld)e Fo(X)3231 474 y Fn(1)3309 459 y Fr(is)386 575 y(giv)m(en)e(only)f(on)h(a)g(compact)g (em)m(b)s(edded)h(submanifold)d Fo(U)44 b Fp(\032)34 b Fo(@)2769 590 y Fn(+)2828 575 y Fo(M)10 b Fr(.)55 b(W)-8 b(e)36 b(then)386 691 y(pic)m(k)f(a)f(p)s(ositiv)m(e)h(function)f Fo(g)1472 706 y Fn(1)1534 691 y Fr(+)24 b(2)p Fo(\031)t(k)37 b Fr(as)e(ab)s(o)m(v)m(e)h(and)f(extend)h(it)d(to)i(a)f(p)s(ositiv)m(e) 386 807 y(function)d Fj(e)-56 b Fo(g)35 b Fr(whic)m(h)c(equals)g(2)p Fo(\031)k Fr(outside)c(of)f(a)h(small)d(neigh)m(b)s(orho)s(o)s(d)i(of)h Fo(U)10 b Fr(.)43 b(If)31 b Fo(U)386 924 y Fr(is)c(not)h(connected)h(w) m(e)g(can)f(c)m(ho)s(ose)g(di\013eren)m(t)g(v)-5 b(alues)28 b(for)f Fo(k)k Fr(on)c(eac)m(h)i(connected)386 1040 y(comp)s(onen)m(t)k (of)f Fo(U)10 b Fr(.)1343 1283 y(3.)49 b Fs(Cont)-7 b(a)n(ct)38 b(topology)486 1457 y Fr(In)31 b(this)g(section)g(w)m(e)h(summarize)e (facts)i(from)e(the)h(theory)h(of)f(con)m(tact)g(struc-)386 1574 y(tures.)56 b(Section)36 b(3.1)g(deals)g(with)h(Legendrian)f(curv) m(es)i(in)e(con)m(tact)h(manifolds.)386 1690 y(W)-8 b(e)37 b(discuss)i(in)d(particular)f(stabilization)f(of)j(Legendrian)g(curv)m (es.)59 b(This)37 b(will)386 1806 y(pla)m(y)28 b(an)h(imp)s(ortan)m(t)d (role)i(in)g(the)h(construction)f(of)g(attac)m(hing)g(maps)g(for)g (round)386 1922 y(handles)33 b(of)f(index)h(1)f(carrying)g(mo)s(del)f (Engel)h(structures.)486 2039 y(In)39 b(Section)h(3.2)f(w)m(e)i (collect)d(those)i(facts)g(ab)s(out)f(con)m(v)m(ex)j(surfaces)f(in)e (con-)386 2155 y(tact)33 b(manifolds)d(whic)m(h)k(w)m(e)g(will)c(use)k (in)e(Section)h(3.3)f(for)h(the)g(construction)g(of)386 2271 y(b)m(ypasses)f(in)d(o)m(v)m(ert)m(wisted)j(con)m(tact)e (manifolds)e(as)i(w)m(ell)f(as)h(in)f(Section)g(4.5)h(and)386 2387 y(Section)42 b(4.6)h(for)f(the)h(construction)f(of)g(mo)s(del)f (Engel)i(structures)h(on)e(round)386 2504 y(handles)33 b(of)f(index)h(2)f(and)h(3.)486 2620 y(Most)41 b(of)f(the)h(material)c (presen)m(ted)43 b(in)d(this)g(section)h(\(with)f(the)h(exception)386 2736 y(of)48 b(the)h(construction)g(of)g(b)m(ypasses)i(in)d(o)m(v)m (ert)m(wisted)j(con)m(tact)e(manifolds)d(in)386 2852 y(Section)32 b(3.3\))g(can)h(b)s(e)g(found)g(in)f([Aeb,)h(EH1,)g(EH2,)g (Gi1)n(,)g(Ho].)386 3059 y(3.1.)48 b Fy(Prop)s(erties)32 b(of)h(Legendrian)g(curv)m(es.)49 b Fr(Throughout)29 b(this)f(section)h(w)m(e)386 3175 y(consider)35 b(an)g(orien)m(ted)h (con)m(tact)f(structure)i Fp(C)k Fr(on)35 b(a)g(3{manifold)d Fo(N)10 b Fr(.)52 b(A)35 b(curv)m(e)386 3291 y(is)d Fm(L)-5 b(e)g(gendrian)31 b Fr(if)h(it)f(is)h(tangen)m(t)h(to)g(the)g(con)m (tact)g(structure.)386 3436 y Fy(De\014nition)40 b(3.1.)k Fr(A)37 b(framing)d(\()p Fo(S;)17 b(T)d Fr(\))36 b(of)g(an)g(em)m(b)s (edded)h(closed)g(Legendrian)386 3552 y(curv)m(e)d Fo(\015)k Fr(is)32 b(an)g Fm(oriente)-5 b(d)35 b(c)-5 b(ontact)34 b(fr)-5 b(aming)32 b Fr(if)556 3696 y(\(i\))40 b Fo(S)48 b Fr(is)41 b(tangen)m(t)h(to)g(the)g(con)m(tact)g(structure)h(and)56 b(_)-41 b Fo(\015)5 b(;)17 b(S)47 b Fr(represen)m(ts)d(the)700 3813 y(orien)m(tation)31 b(of)h(the)h(con)m(tact)h(structure)f(and)529 3929 y(\(ii\))39 b Fo(T)51 b Fr(is)36 b(transv)m(erse)j(to)e(the)g(con) m(tact)h(structure)g(and)51 b(_)-41 b Fo(\015)5 b(;)17 b(S;)g(T)50 b Fr(represen)m(ts)700 4045 y(the)33 b(orien)m(tation)e (induced)i(b)m(y)h(the)f(con)m(tact)g(structure.)486 4190 y(Ev)m(ery)h(Legendrian)f(curv)m(e)i(has)e(a)g(con)m(tact)h (framing)d(whic)m(h)i(is)g(w)m(ell)f(de\014ned)386 4306 y(up)j(to)f(homotop)m(y)-8 b(.)50 b(The)36 b(homotop)m(y)e(class)h(of)f (framings)f(of)i Fo(\015)k Fr(represen)m(ted)f(b)m(y)386 4422 y(con)m(tact)30 b(framings)e(is)h(denoted)i(b)m(y)g(fr)o(\()p Fo(\015)5 b Fr(\).)43 b(On)29 b(the)h(set)h(of)e(framings)f(of)h Fo(\015)35 b Fr(there)386 4538 y(is)d(a)g Fh(Z)p Fr({action)d (de\014ned)34 b(b)m(y)667 4627 y Fj(\000)713 4707 y Fo(m)23 b Fp(\001)e Fr(\()p Fo(S;)c(T)d Fr(\))1121 4627 y Fj(\001)1167 4707 y Fr(\()p Fo(\015)5 b Fr(\()p Fo(t)p Fr(\)\))27 b(=)1541 4627 y Fj(\000)1603 4707 y Fr(cos)q(\()p Fo(mt)p Fr(\))p Fo(S)6 b Fr(\()p Fo(\015)f Fr(\()p Fo(t)p Fr(\)\))22 b(+)g(sin\()p Fo(mt)p Fr(\))p Fo(T)14 b Fr(\()p Fo(\015)5 b Fr(\()p Fo(t)p Fr(\)\))p Fo(;)1627 4868 y Fp(\000)23 b Fr(sin)o(\()p Fo(mt)p Fr(\))p Fo(S)6 b Fr(\()p Fo(\015)f Fr(\()p Fo(t)p Fr(\)\))23 b(+)f(cos\()p Fo(mt)p Fr(\))p Fo(T)14 b Fr(\()p Fo(\015)5 b Fr(\()p Fo(t)p Fr(\)\))3112 4788 y Fj(\001)3191 4868 y Fo(:)386 4790 y Fr(\(8\))386 5042 y(This)25 b(action)f(is)h(transitiv)m(e)g(on)g(the)h(set)f(of)g (homotop)m(y)g(classes)h(of)f(framings)e(suc)m(h)386 5158 y(that)47 b(_)-41 b Fo(\015)37 b Fr(follo)m(w)m(ed)32 b(b)m(y)i(the)f(framing)e(induces)j(the)f(con)m(tact)g(orien)m(tation.) 43 b(When)386 5274 y(w)m(e)23 b(rev)m(erse)h(the)e(orien)m(tation)e(of) i Fo(\015)27 b Fr(the)22 b(co)s(orien)m(tation)e(of)h Fo(\015)27 b Fr(c)m(hanges.)42 b(Therefore)386 5390 y(the)32 b Fh(Z)p Fr({action)27 b(on)k(the)h(framings)e(do)s(es)i(not)f(dep)s (end)h(on)f(the)h(orien)m(tation)e(of)h Fo(\015)5 b Fr(.)386 5506 y(Con)m(tact)37 b(framings)e(are)h(related)g(to)g(the)h(Th)m (urston{Bennequin)h(in)m(v)-5 b(arian)m(t)35 b(of)386 5623 y(n)m(ull{homologous)29 b(Legendrian)k(knots.)p eop %%Page: 14 14 14 13 bop 386 259 a Fq(14)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fy(De\014nition)40 b(3.2.)k Fr(Let)37 b Fo(\015)k Fr(b)s(e)c(a)f(n)m(ull{homologous)d(Legendrian)j(curv)m(e)i(in)e Fo(N)10 b Fr(.)386 575 y(Fix)39 b(a)h(relativ)m(e)f(homology)f(class)i ([\006])h Fp(2)g Fo(H)2049 590 y Fn(2)2088 575 y Fr(\()p Fo(N)5 b(;)17 b(\015)5 b Fr(;)17 b Fh(Z)p Fr(\))37 b(whic)m(h)k(is)e (represen)m(ted)386 691 y(b)m(y)i(an)f(orien)m(ted)g(surface)h(\006)f (suc)m(h)i(that)e Fo(@)5 b Fr(\006)42 b(=)e Fo(\015)45 b Fr(and)40 b Fo(\015)45 b Fr(is)40 b(orien)m(ted)g(as)g(the)386 807 y(b)s(oundary)49 b(of)f(\006.)92 b(A)48 b(new)i(curv)m(e)g Fo(\015)1827 771 y Fk(0)1899 807 y Fr(is)e(obtained)g(b)m(y)h(pushing)g Fo(\015)k Fr(sligh)m(tly)386 924 y(along)28 b(a)h(v)m(ector)i(\014eld)e (whic)m(h)h(is)f(transv)m(erse)j(to)d Fp(C)6 b Fr(.)43 b(The)30 b Fm(Thurston{Benne)-5 b(quin)386 1040 y(invariant)37 b Fr(tb\()p Fo(\015)5 b(;)17 b Fr([\006]\))39 b(is)f(the)g(homological) c(in)m(tersection)k(n)m(um)m(b)s(er)h(of)f Fo(\015)3124 1004 y Fk(0)3185 1040 y Fr(with)386 1156 y(\006.)486 1309 y(In)44 b(De\014nition)f(3.2)h(the)g(surface)h(\006)g(singles)f (out)g(those)h(framings)d(of)i Fo(\015)53 b Fr(=)386 1425 y Fo(@)5 b(N)53 b Fr(whose)42 b(\014rst)g(comp)s(onen)m(t)f(is)g (tangen)m(t)g(to)g(\006.)70 b(W)-8 b(e)42 b(write)f(fr)2853 1440 y Fn(\006)2908 1425 y Fr(\()p Fo(\015)5 b Fr(\))42 b(for)e(the)386 1541 y(corresp)s(onding)d(homotop)m(y)g(class)g(of)g (framings.)55 b(The)38 b(Th)m(urston{Bennequin)386 1657 y(in)m(v)-5 b(arian)m(t)31 b(measures)i(the)g(di\013erence)h(b)s(et)m (w)m(een)g(fr)2262 1672 y Fn(\006)2317 1657 y Fr(\()p Fo(\015)5 b Fr(\))33 b(and)f(fr\()p Fo(\015)5 b Fr(\))32 b(of)h Fo(\015)386 1847 y Fr(\(9\))815 b(tb\()p Fo(\015)5 b(;)17 b Fr([\006]\))22 b Fp(\001)g Fr(fr)1858 1862 y Fn(\006)1913 1847 y Fr(\()p Fo(\015)5 b Fr(\))27 b(=)h(fr\()p Fo(\015)5 b Fr(\))32 b Fo(:)386 2036 y Fr(A)h(simple)e(application)f (of)j(Gra)m(y's)g(theorem)g(\(Theorem)g(2.2\))f(sho)m(ws)j(that)d(the) 386 2152 y(con)m(tact)k(framing)d(of)j(a)f(Legendrian)g(curv)m(e)i (determines)f(the)g(isotop)m(y)g(t)m(yp)s(e)g(of)386 2268 y(the)d(con)m(tact)g(structure)h(on)e(a)h(neigh)m(b)s(orho)s(o)s (d)e(of)h(the)h(Legendrian)g(curv)m(e.)386 2421 y Fy(Lemma)42 b(3.3.)j Fm(L)-5 b(et)39 b Fp(C)1231 2436 y Fn(0)1271 2421 y Fo(;)17 b Fp(C)1367 2436 y Fn(1)1446 2421 y Fm(b)-5 b(e)38 b(two)i(c)-5 b(ontact)39 b(structur)-5 b(es)40 b(on)e Fo(N)50 b Fm(which)38 b(induc)-5 b(e)386 2537 y(the)33 b(same)f(orientation)g(of)g Fo(M)44 b Fm(and)32 b Fo(\015)38 b Fm(a)32 b(curve)h(tangent)f(to)h(b)-5 b(oth)33 b Fp(C)2882 2552 y Fn(0)2954 2537 y Fm(and)f Fp(C)3193 2552 y Fn(1)3233 2537 y Fm(.)44 b(If)386 2653 y(the)31 b(c)-5 b(ontact)30 b(fr)-5 b(amings)30 b(of)g Fo(\015)36 b Fm(with)30 b(r)-5 b(esp)g(e)g(ct)30 b(to)h Fp(C)2156 2668 y Fn(0)2227 2653 y Fm(and)f Fp(C)2464 2668 y Fn(1)2534 2653 y Fm(ar)-5 b(e)31 b(homotopic,)f(then)386 2770 y(ther)-5 b(e)37 b(is)g(an)g(isotopy)g Fo( )1279 2785 y Fl(s)1316 2770 y Fo(;)17 b(s)32 b Fr(=)g([0)p Fo(;)17 b Fr(1])p Fo(;)37 b Fm(of)g Fo(N)48 b Fm(r)-5 b(elative)37 b Fo(\015)42 b Fm(such)37 b(that)h Fo( )2976 2785 y Fn(1)p Fk(\003)3051 2770 y Fp(C)3103 2785 y Fn(0)3175 2770 y Fr(=)32 b Fp(C)3335 2785 y Fn(1)386 2886 y Fm(on)i(a)h(tubular)h (neighb)-5 b(orho)g(o)g(d)33 b(of)h Fo(\015)5 b Fm(.)486 3002 y(If)35 b Fp(C)641 3017 y Fn(0)718 3002 y Fm(and)h Fp(C)961 3017 y Fn(1)1037 3002 y Fm(ar)-5 b(e)36 b(oriente)-5 b(d,)36 b(then)h(one)f(c)-5 b(an)36 b(cho)-5 b(ose)35 b Fo( )2572 3017 y Fl(s)2646 3002 y Fm(such)h(that)h Fo( )3133 3017 y Fn(1)3209 3002 y Fm(pr)-5 b(e-)386 3118 y(serves)34 b(also)g(the)h(orientations)f(of)h(the)g(c)-5 b(ontact)35 b(structur)-5 b(es.)486 3271 y Fr(No)m(w)27 b(w)m(e)i(discuss)f(the)g(other)g(classical)d(in)m(v)-5 b(arian)m(t)26 b(of)h(n)m(ull{homologous)d(Leg-)386 3387 y(endrian)32 b(curv)m(es.)386 3540 y Fy(De\014nition)f(3.4.)39 b Fr(Let)28 b(\006)h(b)s(e)g(a)f(connected)i(orien)m(table)d(surface)i (with)f Fo(@)5 b Fr(\006)29 b(=)f Fo(\015)5 b Fr(.)386 3656 y(Fix)24 b(an)g(orien)m(ted)h(trivialization)20 b Fo(X)r(;)d(Y)45 b Fr(of)24 b Fp(C)31 b Fr(along)23 b(\006.)41 b(There)26 b(are)f(unique)g(func-)386 3772 y(tions)37 b Fo(f)678 3787 y Fl(x)722 3772 y Fo(;)17 b(f)814 3787 y Fl(y)892 3772 y Fr(suc)m(h)39 b(that)52 b(_)-42 b Fo(\015)5 b Fr(\()p Fo(t)p Fr(\))36 b(=)g Fo(f)1696 3787 y Fl(X)1764 3772 y Fr(\()p Fo(t)p Fr(\))p Fo(X)d Fr(+)25 b Fo(f)2138 3787 y Fl(Y)2199 3772 y Fr(\()p Fo(t)p Fr(\))p Fo(Y)d Fr(.)58 b(The)38 b(winding)e(n)m(um)m(b)s(er)386 3889 y(of)1440 4078 y Fo(S)1506 4037 y Fn(1)1573 4078 y Fp(\000)-16 b(!)27 b Fh(R)1827 4037 y Fn(2)1895 4078 y Fp(n)22 b(f)p Fr(0)p Fp(g)1510 4229 y Fo(t)28 b Fp(7\000)-16 b(!)27 b Fr(\()p Fo(f)1847 4244 y Fl(Y)1908 4229 y Fr(\()p Fo(t)p Fr(\))p Fo(;)17 b(f)2111 4244 y Fl(Y)2172 4229 y Fr(\()p Fo(t)p Fr(\)\))386 4424 y(around)33 b(0)f(is)g(the)h Fm(r)-5 b(otation)35 b(numb)-5 b(er)32 b Fr(rot\()p Fo(\015)5 b(;)17 b Fr([\006]\).)386 4577 y Fy(Remark)46 b(3.5.)f Fr(In)c(De\014nition)d(2.18)i(w)m(e)h(de\014ned)h(the)e(rotation)f(n)m (um)m(b)s(er)h(for)386 4693 y(Legendrian)k(curv)m(es)i(on)d(transv)m (erse)j(h)m(yp)s(ersurfaces)h(in)c(orien)m(ted)h(manifolds)386 4809 y(with)f(orien)m(ted)g(Engel)g(structures.)78 b(In)43 b(De\014nition)f(3.4)h(w)m(e)h(\014x)g(an)g(orien)m(ted)386 4925 y(trivialization)25 b(of)j(the)i(orien)m(ted)f(con)m(tact)g (structure)i(on)d(a)h(Seifert)g(surface)h(\006)f(of)386 5042 y(the)k(Legendrian)f(knot)h Fo(@)5 b Fr(\006)29 b(=)f Fo(\015)37 b Fr(and)c(compare)47 b(_)-42 b Fo(\015)38 b Fr(with)32 b(this)g(trivialization.)486 5158 y(No)m(w)f(assume)g (that)f Fp(C)37 b Fr(is)30 b(the)h(orien)m(ted)f(con)m(tact)h (structure)h(on)e(a)g(transv)m(erse)386 5274 y(h)m(yp)s(ersurface)i(of) d(an)g(orien)m(ted)h(manifold)d(with)i(orien)m(ted)h(Engel)g (structure.)43 b(In)386 5390 y(this)28 b(case)i(w)m(e)f(can)g(use)h (the)f(in)m(tersection)f(line)f(\014eld)i(as)f(the)h(\014rst)h(comp)s (onen)m(t)e(of)386 5506 y(the)36 b(trivialization)31 b(of)k Fp(C)41 b Fr(o)m(v)m(er)c(\006.)53 b(Th)m(us)37 b(in)d(this)h(situation)f(the)i(t)m(w)m(o)g(rotation)386 5623 y(n)m(um)m(b)s(ers)d(in)f(De\014nition)f(2.18)h(and)h (De\014nition)e(3.4)h(are)g(equiv)-5 b(alen)m(t.)p eop %%Page: 15 15 15 14 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(15)486 459 y Fr(Using)44 b(con)m(tact)h(framings)e(and)h(rotation)f (n)m(um)m(b)s(ers)i(one)g(can)g(distinguish)386 575 y(isotop)m(y)28 b(classes)g(of)f(Legendrian)h(curv)m(es.)44 b(This)28 b(isotop)m(y)f(t)m(yp)s(e)i(can)f(b)s(e)g(c)m(hanged)386 691 y(using)k(the)h(follo)m(wing)d(construction.)486 807 y(Let)50 b Fo(\015)55 b Fr(b)s(e)50 b(a)g(closed)h(Legendrian)f (curv)m(e)h(in)f(a)g(con)m(tact)g(manifold.)94 b(Ac-)386 924 y(cording)44 b(to)h(Corollary)f(2.3)h(there)h(are)f(co)s(ordinates) g Fo(x;)17 b(z)t(;)g(t)46 b Fr(on)f(a)g(neigh)m(b)s(or-)386 1040 y(ho)s(o)s(d)35 b(of)h Fo(\015)k Fr(suc)m(h)e(that)d(the)i(con)m (tact)f(structure)h(is)f(de\014ned)h(b)m(y)g Fo(dz)29 b Fp(\000)c Fo(x)17 b(dt)36 b Fr(and)386 1156 y Fo(\015)42 b Fr(=)37 b Fp(f)p Fo(x)h Fr(=)f(0)p Fo(;)17 b(z)41 b Fr(=)d(0)p Fp(g)p Fr(.)59 b(The)40 b Fo(t)p Fr({axis)e(in)f(b)s(oth)h (parts)g(of)g(Figure)f(1)h(represen)m(ts)386 1272 y(the)i(pro)5 b(jection)40 b(of)f(a)g(neigh)m(b)s(orho)s(o)s(d)g(of)g(a)g(segmen)m(t) i(of)e Fo(\015)44 b Fr(to)c(the)g Fo(x;)17 b(t)p Fr({plane)386 1389 y(and)42 b Fo(@)636 1404 y Fl(z)717 1389 y Fr(is)f(p)s(oin)m(ting) f(out)m(w)m(ards.)72 b(Let)42 b Fo(\033)j Fr(b)s(e)d(another)g(smo)s (oth)e(curv)m(e)j(in)e(the)386 1505 y Fo(t;)17 b(x)p Fr({plane)34 b(that)g(coincides)h(with)f(the)h Fo(t)p Fr({axis)f(near)h(its)f(endp)s(oin)m(ts.)50 b(Since)34 b(the)386 1621 y(con)m(tact)h(structure)g(near)g Fo(\015)k Fr(is)34 b(de\014ned)i(b)m(y)f Fo(dz)28 b Fp(\000)23 b Fo(x)17 b(dt)34 b Fr(the)h(segmen)m(t)g Fo(\033)j Fr(lifts)33 b(to)386 1737 y(a)e(Legendrian)g(curv)m(e)i(whose)f(endp)s(oin)m(ts)g (lie)d(on)i Fo(\015)37 b Fr(if)30 b(and)h(only)g(if)f(the)h(in)m (tegral)386 1853 y(of)d(the)g(1{form)f Fo(x)17 b(dt)28 b Fr(along)f Fo(\033)32 b Fr(is)c(zero.)42 b(Then)29 b(the)g(lifts)e(of)g(curv)m(es)k Fo(\033)2908 1817 y Fn(+)2967 1853 y Fo(\015)5 b Fr(,)29 b(resp)s(ec-)386 1970 y(tiv)m(ely)40 b Fo(\033)721 1934 y Fk(\000)780 1970 y Fo(\015)46 b Fr(as)41 b(in)e(Figure)h(1,)i(yield)e(new)h(closed) g(Legendrian)g(curv)m(es)h(whic)m(h)386 2086 y(are)32 b(close)h(to)f Fo(\015)5 b Fr(.)43 b(They)34 b(represen)m(t)g(the)f(p)s (ositiv)m(e)e(and)i(the)f(stabilization)d(of)j Fo(\015)5 b Fr(.)446 2868 y @beginspecial 0 @llx 0 @lly 370 @urx 81 @ury 3443 @rwi @setspecial %%BeginDocument: artstab.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: artstab.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Wed Oct 13 16:26:26 2004 %%For: tvogel@math2h (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 370 81 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -4.0 88.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /endash 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 2458 m -1000 -1000 l 7218 -1000 l 7218 2458 l cp clip 0.06000 0.06000 sc % Polyline 7.500 slw [60] 0 sd n 4109 332 m 4110 332 l 4114 329 l 4122 324 l 4134 318 l 4146 311 l 4159 305 l 4170 300 l 4181 296 l 4190 294 l 4201 291 l 4212 290 l 4223 288 l 4234 287 l 4246 285 l 4258 284 l 4269 282 l 4280 281 l 4291 279 l 4302 277 l 4312 275 l 4322 273 l 4332 270 l 4342 267 l 4352 265 l 4363 263 l 4373 261 l 4384 260 l 4396 260 l 4408 261 l 4421 263 l 4435 265 l 4449 268 l 4464 272 l 4478 275 l 4493 278 l 4506 282 l 4519 285 l 4531 288 l 4542 291 l 4553 294 l 4563 297 l 4572 301 l 4581 304 l 4591 307 l 4600 310 l 4609 314 l 4618 318 l 4628 322 l 4638 327 l 4648 332 l 4660 339 l 4674 346 l 4688 355 l 4701 362 l 4712 369 l 4718 373 l 4722 375 l gs col0 s gr [] 0 sd /Symbol ff 195.00 scf sf 5302 1172 m gs 1 -1 sc (-) col0 sh gr /Symbol ff 210.00 scf sf 5194 1263 m gs 1 -1 sc (s g) col0 sh gr % Polyline gs clippath 274 306 m 304 186 l 334 306 l 334 144 l 274 144 l cp clip n 304 159 m 304 1446 l gs col0 s gr gr % arrowhead n 274 306 m 304 186 l 334 306 l col0 s % Polyline gs clippath 6059 358 m 6179 388 l 6059 418 l 6221 418 l 6221 358 l cp clip n 3315 388 m 6206 388 l gs col0 s gr gr % arrowhead n 6059 358 m 6179 388 l 6059 418 l col0 s % Polyline gs clippath 3518 289 m 3548 169 l 3578 289 l 3578 127 l 3518 127 l cp clip n 3548 142 m 3548 1431 l gs col0 s gr gr % arrowhead n 3518 289 m 3548 169 l 3578 289 l col0 s % Polyline [60] 0 sd n 921 1037 m 924 1036 l 931 1033 l 944 1029 l 960 1022 l 981 1015 l 1004 1006 l 1028 997 l 1051 988 l 1072 979 l 1090 971 l 1106 964 l 1119 957 l 1130 950 l 1143 940 l 1154 930 l 1162 920 l 1170 910 l 1176 900 l 1183 889 l 1189 879 l 1195 868 l 1200 858 l 1206 847 l 1211 836 l 1215 825 l 1219 813 l 1222 802 l 1225 790 l 1228 778 l 1231 767 l 1233 755 l 1234 743 l 1234 731 l 1233 719 l 1231 706 l 1228 694 l 1225 681 l 1221 667 l 1217 654 l 1213 641 l 1209 629 l 1204 618 l 1198 607 l 1192 597 l 1184 588 l 1177 580 l 1168 571 l 1160 563 l 1151 556 l 1142 548 l 1133 541 l 1123 534 l 1113 527 l 1102 521 l 1090 515 l 1077 510 l 1064 505 l 1050 501 l 1036 496 l 1022 492 l 1009 488 l 994 484 l 980 480 l 968 477 l 955 475 l 942 472 l 928 470 l 915 468 l 901 466 l 887 464 l 874 462 l 860 460 l 847 459 l 833 458 l 820 458 l 807 458 l 793 459 l 780 460 l 766 461 l 753 463 l 739 465 l 725 467 l 712 469 l 699 471 l 686 474 l 674 477 l 663 480 l 650 485 l 639 491 l 627 497 l 616 504 l 606 511 l 595 519 l 585 526 l 576 534 l 566 541 l 558 549 l 550 557 l 544 565 l 537 573 l 532 582 l 526 590 l 521 599 l 516 607 l 511 616 l 507 626 l 503 636 l 500 647 l 498 658 l 496 670 l 494 683 l 492 696 l 491 709 l 490 721 l 490 734 l 489 745 l 490 756 l 492 769 l 495 781 l 498 792 l 502 803 l 506 814 l 510 825 l 516 836 l 522 847 l 528 856 l 535 864 l 542 873 l 550 881 l 557 889 l 565 897 l 574 906 l 583 914 l 592 922 l 603 930 l 613 937 l 623 943 l 634 950 l 646 956 l 658 962 l 670 968 l 682 974 l 695 981 l 708 987 l 720 994 l 733 1001 l 747 1008 l 759 1015 l 772 1022 l 787 1030 l 804 1039 l 823 1049 l 843 1060 l 863 1071 l 883 1082 l 901 1092 l 915 1100 l 925 1105 l 931 1108 l 934 1110 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1672 1057 m 1675 1054 l 1680 1048 l 1689 1039 l 1701 1026 l 1714 1011 l 1729 995 l 1742 980 l 1755 966 l 1765 953 l 1774 942 l 1782 932 l 1790 920 l 1798 908 l 1805 897 l 1812 885 l 1818 874 l 1824 862 l 1830 851 l 1836 839 l 1843 828 l 1850 817 l 1857 806 l 1864 795 l 1872 784 l 1879 774 l 1886 763 l 1894 752 l 1901 742 l 1908 731 l 1916 721 l 1923 712 l 1930 703 l 1938 694 l 1945 686 l 1952 678 l 1958 671 l 1965 663 l 1973 656 l 1980 648 l 1989 640 l 1998 632 l 2008 624 l 2019 616 l 2031 607 l 2043 598 l 2055 589 l 2067 580 l 2080 572 l 2092 564 l 2105 558 l 2117 553 l 2130 549 l 2143 547 l 2156 546 l 2170 545 l 2183 546 l 2197 546 l 2210 546 l 2224 546 l 2237 546 l 2250 546 l 2263 546 l 2276 546 l 2289 546 l 2302 545 l 2315 545 l 2328 545 l 2340 545 l 2353 546 l 2365 548 l 2376 551 l 2387 555 l 2397 561 l 2407 567 l 2416 574 l 2425 582 l 2434 590 l 2443 598 l 2452 605 l 2461 613 l 2470 621 l 2478 629 l 2487 638 l 2495 646 l 2502 655 l 2510 664 l 2518 673 l 2525 682 l 2533 692 l 2540 702 l 2547 712 l 2554 723 l 2560 735 l 2566 747 l 2572 759 l 2578 772 l 2583 785 l 2589 798 l 2595 810 l 2601 822 l 2607 834 l 2613 845 l 2620 857 l 2626 868 l 2632 879 l 2639 890 l 2645 901 l 2652 912 l 2658 923 l 2665 934 l 2671 944 l 2677 954 l 2683 964 l 2690 975 l 2696 985 l 2702 996 l 2708 1006 l 2714 1016 l 2720 1024 l 2726 1032 l 2733 1038 l 2742 1043 l 2752 1047 l 2764 1048 l 2778 1049 l 2790 1049 l 2799 1049 l 2803 1049 l 2804 1049 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 929 1114 m 930 1114 l 933 1117 l 942 1122 l 956 1130 l 973 1140 l 990 1150 l 1008 1159 l 1023 1167 l 1036 1174 l 1049 1179 l 1061 1183 l 1074 1187 l 1087 1191 l 1100 1194 l 1113 1197 l 1126 1199 l 1139 1202 l 1152 1205 l 1165 1207 l 1177 1209 l 1189 1211 l 1201 1212 l 1212 1213 l 1223 1214 l 1234 1215 l 1245 1216 l 1256 1216 l 1267 1216 l 1279 1216 l 1291 1216 l 1304 1215 l 1317 1214 l 1330 1212 l 1344 1211 l 1358 1209 l 1372 1207 l 1385 1205 l 1399 1202 l 1412 1199 l 1425 1196 l 1438 1192 l 1450 1187 l 1462 1182 l 1474 1177 l 1486 1171 l 1498 1165 l 1510 1159 l 1522 1153 l 1534 1147 l 1547 1140 l 1558 1134 l 1570 1127 l 1583 1118 l 1598 1108 l 1615 1098 l 1632 1086 l 1648 1076 l 1662 1066 l 1673 1059 l 1679 1055 l 1682 1053 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 3995 403 m 3999 403 l 4006 404 l 4017 405 l 4033 407 l 4051 409 l 4071 412 l 4089 414 l 4106 417 l 4122 420 l 4135 423 l 4148 426 l 4160 430 l 4173 435 l 4186 440 l 4199 445 l 4212 451 l 4225 456 l 4237 462 l 4250 468 l 4262 474 l 4274 480 l 4284 486 l 4294 492 l 4305 499 l 4314 507 l 4323 514 l 4331 522 l 4338 529 l 4346 537 l 4353 545 l 4359 553 l 4366 562 l 4371 572 l 4376 583 l 4379 595 l 4383 607 l 4386 621 l 4388 634 l 4391 648 l 4393 661 l 4394 675 l 4395 687 l 4394 700 l 4392 712 l 4389 725 l 4384 737 l 4379 750 l 4374 762 l 4368 775 l 4362 787 l 4356 799 l 4351 810 l 4345 820 l 4339 830 l 4333 838 l 4327 847 l 4321 854 l 4315 862 l 4308 869 l 4302 877 l 4294 884 l 4286 892 l 4277 899 l 4268 905 l 4259 911 l 4249 917 l 4239 924 l 4228 930 l 4217 936 l 4206 943 l 4195 949 l 4184 954 l 4172 959 l 4160 964 l 4148 968 l 4135 971 l 4122 974 l 4108 976 l 4093 979 l 4078 980 l 4063 982 l 4047 984 l 4032 985 l 4018 986 l 4003 986 l 3989 987 l 3976 986 l 3963 985 l 3950 983 l 3938 980 l 3926 977 l 3914 974 l 3902 971 l 3890 967 l 3878 964 l 3867 960 l 3855 955 l 3844 950 l 3833 945 l 3822 939 l 3812 932 l 3801 925 l 3790 918 l 3779 910 l 3768 902 l 3758 894 l 3748 886 l 3738 877 l 3729 868 l 3721 858 l 3714 847 l 3708 835 l 3703 823 l 3698 809 l 3694 795 l 3690 781 l 3686 766 l 3683 750 l 3680 735 l 3678 721 l 3678 706 l 3678 692 l 3680 678 l 3684 664 l 3690 650 l 3697 636 l 3706 621 l 3716 606 l 3727 591 l 3737 576 l 3748 562 l 3758 549 l 3768 536 l 3777 525 l 3785 515 l 3795 505 l 3806 497 l 3818 489 l 3831 482 l 3845 475 l 3858 470 l 3869 465 l 3876 462 l 3879 461 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 4764 427 m 4766 429 l 4771 435 l 4778 443 l 4789 454 l 4801 468 l 4813 482 l 4826 496 l 4837 510 l 4848 523 l 4858 536 l 4868 549 l 4877 561 l 4886 574 l 4895 589 l 4905 604 l 4915 620 l 4926 637 l 4937 654 l 4947 671 l 4957 687 l 4967 703 l 4977 718 l 4986 732 l 4994 744 l 5002 755 l 5012 768 l 5020 779 l 5027 788 l 5033 796 l 5039 803 l 5045 810 l 5051 816 l 5058 823 l 5065 831 l 5074 838 l 5083 845 l 5093 852 l 5104 859 l 5115 866 l 5126 872 l 5137 879 l 5148 885 l 5159 891 l 5171 897 l 5182 902 l 5194 906 l 5205 910 l 5218 914 l 5230 918 l 5242 921 l 5254 924 l 5267 927 l 5279 929 l 5291 931 l 5304 932 l 5317 932 l 5330 931 l 5343 930 l 5356 928 l 5370 926 l 5383 924 l 5396 922 l 5409 919 l 5422 915 l 5434 911 l 5446 906 l 5457 900 l 5468 893 l 5479 885 l 5490 877 l 5501 869 l 5512 861 l 5522 854 l 5531 846 l 5540 838 l 5548 830 l 5554 823 l 5560 816 l 5565 810 l 5569 803 l 5573 797 l 5578 789 l 5584 780 l 5591 769 l 5599 755 l 5606 743 l 5613 730 l 5621 716 l 5629 700 l 5638 683 l 5647 665 l 5657 647 l 5666 628 l 5675 610 l 5684 593 l 5693 577 l 5701 562 l 5709 548 l 5717 536 l 5727 521 l 5737 507 l 5746 494 l 5755 483 l 5764 472 l 5772 462 l 5781 452 l 5789 443 l 5798 434 l 5806 427 l 5816 420 l 5825 416 l 5834 414 l 5842 413 l 5850 412 l 5859 411 l 5869 410 l 5881 408 l 5892 407 l 5904 406 l 5919 406 l 5937 405 l 5956 405 l 5974 404 l 5989 404 l 5998 404 l 6002 404 l 6003 404 l gs col0 s gr [] 0 sd /Times-Roman-iso ff 210.00 scf sf 3370 259 m gs 1 -1 sc (x) col0 sh gr /Symbol ff 210.00 scf sf 2035 1205 m gs 1 -1 sc (g) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 2593 1234 m gs 1 -1 sc (t) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 139 242 m gs 1 -1 sc (x) col0 sh gr /Symbol ff 210.00 scf sf 1857 414 m gs 1 -1 sc (s g) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 5728 284 m gs 1 -1 sc (t) col0 sh gr /Symbol ff 210.00 scf sf 5070 288 m gs 1 -1 sc (g) col0 sh gr /Times-Roman-iso ff 195.00 scf sf 2015 334 m gs 1 -1 sc (+) col0 sh gr % Polyline n 994 1057 m 1617 1057 l gs col0 s gr % Polyline n 93 1054 m 737 1054 l gs col0 s gr % Polyline gs clippath 2844 1025 m 2964 1054 l 2844 1085 l 3006 1085 l 3006 1025 l cp clip n 1750 1054 m 2991 1055 l gs col0 s gr gr % arrowhead n 2844 1025 m 2964 1054 l 2844 1085 l col0 s $F2psEnd rs %%EndDocument @endspecial 1623 3084 a Fs(Figure)37 b(1.)486 3315 y Fr(Up)25 b(to)f(isotop)m(y)h(through)g(Legendrian)g(curv)m(es)i(the)e (stabilized)f(curv)m(es)j(do)e(not)386 3431 y(dep)s(end)i(on)f(the)h(c) m(hoices)g(made)f(ab)s(o)m(v)m(e.)42 b(In)27 b(particular,)f(p)s (ositiv)m(e)f(and)h(negativ)m(e)386 3547 y(stabilization)36 b(comm)m(ute)j(up)h(to)f(Legendrian)g(isotop)m(y)-8 b(.)64 b(The)40 b(e\013ect)h(of)e(stabi-)386 3663 y(lization)e(on)j(the)g (rotation)e(n)m(um)m(b)s(er)i(with)g(resp)s(ect)h(to)e(a)h(no)m(where)h (v)-5 b(anishing)386 3780 y(Legendrian)32 b(v)m(ector)i(\014eld)e Fo(X)41 b Fr(is)32 b(giv)m(en)g(b)m(y)1252 3941 y(rot)o(\()p Fo(\033)1473 3900 y Fn(+)1532 3941 y Fo(\015)5 b(;)17 b(X)8 b Fr(\))27 b(=)h(rot\()p Fo(\015)5 b(;)17 b(X)8 b Fr(\))21 b(+)h(1)1252 4098 y(rot)o(\()p Fo(\033)1473 4057 y Fk(\000)1532 4098 y Fo(\015)5 b(;)17 b(X)8 b Fr(\))27 b(=)h(rot\()p Fo(\015)5 b(;)17 b(X)8 b Fr(\))21 b Fp(\000)i Fr(1)32 b Fo(:)386 4023 y Fr(\(10\))386 4274 y(Hence)f Fo(\033)732 4238 y Fn(+)791 4274 y Fo(\015)5 b(;)17 b(\033)950 4238 y Fk(\000)1008 4274 y Fo(\015)35 b Fr(and)29 b Fo(\015)35 b Fr(are)29 b(pairwise)g(non{isotopic)f(as)h(Legendrian)h(curv)m(es.) 386 4391 y(F)-8 b(rom)31 b(the)i(construction)g(of)f Fo(\033)1539 4354 y Fk(\006)1598 4391 y Fo(\015)37 b Fr(it)32 b(follo)m(ws)f(that)386 4559 y(\(11\))544 b(fr)o(\()p Fo(\033)1268 4518 y Fn(+)1327 4559 y Fo(\015)5 b Fr(\))28 b(=)f(1)22 b Fp(\001)g Fr(\()p Fo( )1774 4574 y Fn(1)p Fk(\003)1866 4559 y Fr(fr)1934 4574 y Fk(C)1979 4559 y Fr(\()p Fo(\015)5 b Fr(\)\))27 b(=)h(fr)o(\()p Fo(\033)2444 4518 y Fk(\000)2503 4559 y Fo(\015)5 b Fr(\))33 b Fo(:)386 4728 y Fr(F)-8 b(rom)35 b(\(9\))g(it)g(follo)m(ws)g(that)h(this)g(is)g (coheren)m(t)h(with)f(the)h(w)m(ell)e(kno)m(wn)j(form)m(ula)386 4844 y(tb\()p Fo(\033)575 4808 y Fk(\006)634 4844 y Fo(\015)5 b Fr(\))28 b(=)f(tb\()p Fo(\015)5 b Fr(\))22 b Fp(\000)h Fr(1)32 b(for)g(n)m(ull{homologous)e(Legendrian)i(curv)m(es.)386 5042 y(3.2.)48 b Fy(F)-9 b(acts)50 b(from)e(the)h(theory)g(of)h(con)m (v)m(ex)f(surfaces.)h Fr(Let)43 b Fo(N)54 b Fr(b)s(e)43 b(a)f(3{)386 5158 y(manifold)e(with)j(an)f(orien)m(ted)h(con)m(tact)h (structure)g Fp(C)6 b Fr(.)75 b(Consider)43 b(a)g(prop)s(erly)386 5274 y(em)m(b)s(edded)38 b(orien)m(ted)e(surface)i(\006.)56 b(The)38 b(con)m(tact)f(structure)h(induces)f(the)g Fm(sin-)386 5390 y(gular)26 b(foliation)c Fp(F)37 b Fr(=)28 b Fp(C)9 b(\\)s Fo(T)14 b Fr(\006)23 b(on)g(\006.)41 b(Usually)22 b(this)h(is)f(called)g(the)i(c)m(haracteristic)386 5506 y(foliation)j(of)i(\006.)44 b(W)-8 b(e)30 b(a)m(v)m(oid)g(this)g (terminology)e(since)j(there)g(is)f(already)g(a)g(c)m(har-)386 5623 y(acteristic)i(\(non{singular\))f(foliation)e(in)j(the)h(con)m (text)g(of)f(Engel)h(structures.)p eop %%Page: 16 16 16 15 bop 386 259 a Fq(16)1096 b(THOMAS)25 b(V)n(OGEL)486 459 y Fr(The)30 b(singularities)d(of)h Fp(F)39 b Fr(are)29 b(those)h(p)s(oin)m(ts)f Fo(p)f Fp(2)g Fr(\006)i(where)g Fp(C)2743 474 y Fl(p)2811 459 y Fr(=)e Fo(T)2972 474 y Fl(p)3011 459 y Fr(\006.)43 b(F)-8 b(rom)386 575 y(the)37 b(orien)m(tations)d(of)i(\006)h(and)f Fp(C)42 b Fr(w)m(e)c(obtain)d(an) h(orien)m(tation)e(of)i Fp(F)45 b Fr(a)m(w)m(a)m(y)38 b(from)386 691 y(the)30 b(singularities)e(using)h(the)i(follo)m(wing)c (con)m(v)m(en)m(tion:)43 b(F)-8 b(or)29 b(eac)m(h)i(non{singular)386 807 y(p)s(oin)m(t)h Fo(p)g Fr(of)g Fp(F)42 b Fr(w)m(e)34 b(c)m(ho)s(ose)1051 1003 y Fo(v)e Fp(2)c(F)1296 1018 y Fl(p)1335 1003 y Fo(;)17 b(v)1426 1018 y Fn(\006)1509 1003 y Fp(2)28 b Fo(T)1660 1018 y Fl(p)1700 1003 y Fr(\006)22 b Fp(n)g(F)1936 1018 y Fl(p)2008 1003 y Fr(and)33 b Fo(v)2245 1018 y Fk(C)2318 1003 y Fp(2)28 b(C)2464 1018 y Fl(p)2526 1003 y Fp(n)22 b(F)2670 1018 y Fl(p)386 1198 y Fr(suc)m(h)44 b(that)e(\()p Fo(v)t(;)17 b(v)1017 1213 y Fk(C)1061 1198 y Fo(;)g(v)1152 1213 y Fn(\006)1207 1198 y Fr(\))43 b(is)f(the)g(con)m (tact)h(orien)m(tation,)h(\()p Fo(v)t(;)17 b(v)2643 1213 y Fn(\006)2697 1198 y Fr(\))42 b(orien)m(ts)h(\006)g(and)386 1314 y(\()p Fo(v)t(;)17 b(v)566 1329 y Fk(C)611 1314 y Fr(\))32 b(orien)m(ts)h Fp(C)6 b Fr(.)43 b(Then)34 b Fo(v)i Fr(represen)m(ts)f(the)e(orien)m(tation)e(of)h Fp(F)2771 1329 y Fl(p)2811 1314 y Fr(.)486 1431 y(Generically)-8 b(,)32 b(singular)h(p)s(oin)m(ts)h(are)h(non{degenerate.)49 b(W)-8 b(e)35 b(sa)m(y)g(that)f(a)g(sin-)386 1547 y(gular)25 b(p)s(oin)m(t)h(is)h Fm(el)5 b(liptic)26 b Fr(if)f(its)i(index)g(is)f (+1)g(and)h Fm(hyp)-5 b(erb)g(olic)26 b Fr(if)g(the)h(index)g(is)f Fp(\000)p Fr(1.)386 1663 y(When)g(the)h(orien)m(tation)c(of)j Fp(C)32 b Fr(and)25 b(the)h(orien)m(tation)e(of)h(the)i(surface)f (coincide)f(at)386 1779 y(a)k(singular)e(p)s(oin)m(t)h(of)h Fp(F)10 b Fr(,)29 b(w)m(e)h(sa)m(y)g(that)f(this)g(singularit)m(y)e(is) i Fm(p)-5 b(ositive)p Fr(,)29 b(otherwise)386 1896 y(it)42 b(is)g Fm(ne)-5 b(gative)p Fr(.)73 b(If)42 b(w)m(e)i(orien)m(t)e Fp(F)52 b Fr(according)42 b(to)h(our)f(con)m(v)m(en)m(tions,)47 b(p)s(ositiv)m(e)386 2012 y(elliptic)30 b(p)s(oin)m(ts)i(are)g(sources) j(and)d(negativ)m(e)h(elliptic)d(p)s(oin)m(ts)i(are)h(sinks.)386 2168 y Fy(De\014nition)49 b(3.6.)e Fr(\006)d(is)f(called)f Fm(c)-5 b(onvex)43 b Fr(if)f(there)i(is)f(a)g(con)m(tact)h(v)m(ector)h (\014eld)386 2284 y(transv)m(erse)35 b(to)d(\006.)486 2440 y(Giroux)24 b(studied)i(closed)g(con)m(v)m(ex)i(surfaces)f(in)e ([Gi1)n(].)42 b(If)25 b(\006)h(has)h(b)s(oundary)f(w)m(e)386 2556 y(will)21 b(usually)h(assume)i(that)f Fo(@)5 b Fr(\006)25 b(is)e(Legendrian.)40 b(In)23 b(particular)f(he)i(sho)m(w)m(ed)h(that) 386 2672 y(a)40 b(closed)h(em)m(b)s(edded)g(surface)h(is)e(generically) f(con)m(v)m(ex)j(\(with)e(resp)s(ect)i(to)e(the)386 2788 y Fo(C)463 2752 y Fk(1)538 2788 y Fr({top)s(ology\).)57 b(The)39 b(analogous)e(statemen)m(t)h(is)f(in)g(general)h(not)f(true)i (when)386 2905 y(\006)k(has)g(b)s(oundary)f(\(ev)m(en)i(when)g Fo(@)5 b Fr(\006)44 b(is)e(Legendrian\).)72 b(F)-8 b(or)42 b(eac)m(h)h(b)s(oundary)386 3021 y(comp)s(onen)m(t)33 b Fo(\015)f Fp(\032)c Fo(@)5 b Fr(\006)34 b(let)e Fo(t)p Fr(\()p Fo(\015)5 b(;)17 b Fr(fr)1615 3036 y Fn(\006)1670 3021 y Fr(\))32 b(b)s(e)h(the)g(unique)g(in)m(teger)g(suc)m(h)h(that) 1355 3216 y Fo(t)p Fr(\()p Fo(\015)5 b(;)17 b Fr(fr)1596 3231 y Fn(\006)1651 3216 y Fr(\))22 b Fp(\001)g Fr(fr)1828 3231 y Fn(\006)1883 3216 y Fr(\()p Fo(\015)5 b Fr(\))28 b(=)f(fr\()p Fo(\015)5 b Fr(\))33 b Fo(:)386 3412 y Fr(If)d Fo(\015)35 b Fr(is)30 b(a)g(Legendrian)g(knot)h(and)f(\006)h(is)f(a)g (Seifert)g(surface)h(for)e Fo(\015)5 b Fr(,)31 b(then)g Fo(t)p Fr(\()p Fo(\015)5 b(;)17 b Fr(fr)3282 3427 y Fn(\006)3337 3412 y Fr(\))386 3528 y(is)32 b(the)h(Th)m(urston{Bennequin)i(in)m(v)-5 b(arian)m(t)31 b(b)m(y)k(\(9\).)386 3684 y Fy(Prop)s(osition)24 b(3.7)i Fr(\(Honda,)f([Ho]\))p Fy(.)34 b Fm(L)-5 b(et)26 b Fr(\006)g Fm(b)-5 b(e)25 b(a)h(c)-5 b(omp)g(act,)26 b(oriente)-5 b(d,)27 b(pr)-5 b(op)g(erly)386 3800 y(emb)g(e)g(dde)g(d) 35 b(surfac)-5 b(e)36 b(with)h(L)-5 b(e)g(gendrian)36 b(b)-5 b(oundary,)36 b(and)h(assume)f Fo(t)p Fr(\()p Fo(\015)5 b(;)17 b Fr(fr)3093 3815 y Fn(\006)3148 3800 y Fr(\))31 b Fp(\024)h Fr(0)386 3916 y Fm(for)f(al)5 b(l)30 b(b)-5 b(oundary)31 b(c)-5 b(omp)g(onents)30 b Fo(\015)36 b Fm(of)31 b Fr(\006)p Fm(.)43 b(Ther)-5 b(e)31 b(exists)f(a)h Fo(C)2649 3880 y Fn(0)2689 3916 y Fm({smal)5 b(l)30 b(p)-5 b(erturb)g(a-)386 4033 y(tion)30 b(ne)-5 b(ar)29 b(the)h(b)-5 b(oundary)30 b(\(\014xing)e Fo(@)5 b Fr(\006)p Fm(\))32 b(which)c(puts)j(an)e(annular)g(neighb)-5 b(orho)g(o)g(d)386 4149 y Fo(A)28 b Fm(of)g Fo(@)5 b Fr(\006)29 b Fm(into)f(a)g(standar)-5 b(d)28 b(form,)h(and)e(a)h(subse) -5 b(quent)28 b Fo(C)2485 4113 y Fk(1)2560 4149 y Fm({smal)5 b(l)27 b(p)-5 b(erturb)g(ation)386 4265 y(ke)g(eping)34 b Fo(A)h Fm(\014xe)-5 b(d)34 b(which)g(makes)g Fr(\006)h Fm(c)-5 b(onvex.)486 4381 y(Mor)g(e)g(over,)33 b(if)h Fo(V)55 b Fm(is)33 b(a)h(c)-5 b(ontact)33 b(ve)-5 b(ctor)34 b(\014eld)f(de\014ne)-5 b(d)32 b(on)i(a)f(neighb)-5 b(orho)g(o)g(d)32 b(of)386 4497 y Fo(A)f Fm(and)g(tr)-5 b(ansverse)30 b(to)h Fo(A)d Fp(\032)g Fr(\006)p Fm(,)33 b(then)e Fo(V)52 b Fm(c)-5 b(an)31 b(b)-5 b(e)31 b(extende)-5 b(d)30 b(to)h(a)g(c)-5 b(ontact)31 b(ve)-5 b(ctor)386 4614 y(\014eld)34 b(tr)-5 b(ansverse)34 b(to)h(al)5 b(l)35 b(of)g Fr(\006)p Fm(.)486 4769 y Fr(The)i(singular)d(foliation)f(is)i(enough)i(to)f(determine)f (the)i(con)m(tact)f(structure)386 4886 y(on)c(a)h(small)d(neigh)m(b)s (orho)s(o)s(d)h(of)h(a)h(con)m(v)m(ex)h(surface.)386 5042 y Fy(Theorem)41 b(3.8)i Fr(\(Giroux,)36 b([Gi1)o(]\))p Fy(.)43 b Fm(L)-5 b(et)39 b Fr(\006)g Fm(b)-5 b(e)38 b(a)g(c)-5 b(omp)g(act)38 b(orientable)g(c)-5 b(onvex)386 5158 y(surfac)g(e)29 b(with)h(L)-5 b(e)g(gendrian)28 b(b)-5 b(oundary.)43 b(Two)29 b Fh(R)5 b Fm({invariant)34 b(c)-5 b(ontact)30 b(structur)-5 b(es)386 5274 y(on)29 b Fr(\006)10 b Fp(\002)g Fh(R)40 b Fm(which)28 b(induc)-5 b(e)29 b(the)g(same)g(orientation)f(and)h(the)g(same)g(singular)f (foli-)386 5390 y(ation)j(on)g Fr(\006)14 b Fp(\002)g(f)p Fr(0)p Fp(g)31 b Fm(ar)-5 b(e)31 b(isotopic.)43 b(They)31 b(ar)-5 b(e)31 b(c)-5 b(onjugate)31 b(by)g(a)g(di\013e)-5 b(omorphism)386 5506 y Fo(')23 b Fp(\002)g Fr(id)o Fm(,)36 b(and)f Fo(')g Fm(is)h(isotopic)f(to)g(the)h(identity)g(thr)-5 b(ough)36 b(di\013e)-5 b(omorphisms)33 b(of)i Fr(\006)386 5623 y Fm(that)g(pr)-5 b(eserve)34 b(the)h(singular)g(foliation.)p eop %%Page: 17 17 17 16 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(17)486 459 y Fr(F)-8 b(or)35 b(a)h(con)m(v)m(ex)i(surface)g(\006)e (with)g(Legendrian)g(b)s(oundary)h(w)m(e)g(\014x)g(a)f(con)m(tact)386 575 y(v)m(ector)e(\014eld)e Fo(V)54 b Fr(transv)m(erse)35 b(to)d(\006.)386 728 y Fy(De\014nition)j(3.9.)41 b Fr(The)32 b Fm(dividing)h(set)f Fr(\000)1900 743 y Fn(\006)1986 728 y Fr(of)f(\006)h(is)f(the)h(set)g(of)f(p)s(oin)m(ts)g(where)i Fo(V)386 844 y Fr(is)f(tangen)m(t)h(to)f(the)h(con)m(tact)g(structure.) 486 998 y(The)28 b(dividing)d(set)j(con)m(tains)g(m)m(uc)m(h)g (information)c(ab)s(out)j(the)g(con)m(tact)h(struc-)386 1114 y(ture)42 b(near)g(\006.)71 b(Giroux)40 b(sho)m(w)m(ed)k(in)d ([Gi1)n(])h(that)f(\000)2330 1129 y Fn(\006)2427 1114 y Fr(is)g(a)g(submanifold)f(of)h(\006)386 1230 y(whic)m(h)35 b(is)e(transv)m(erse)k(to)d(the)g(singular)f(foliation.)45 b(Its)35 b(isotop)m(y)f(class)g(dep)s(ends)386 1346 y(only)e(on)g(\006) h(and)g Fp(C)39 b Fr(but)33 b(not)f(on)h Fo(V)21 b Fr(.)386 1500 y Fy(De\014nition)40 b(3.10.)k Fr(Let)36 b Fp(F)46 b Fr(b)s(e)36 b(a)g(singular)e(foliation)f(on)j(\006)g(suc)m(h)i(that)e Fo(@)5 b Fr(\006)37 b(is)386 1616 y(tangen)m(t)g(to)f Fp(F)10 b Fr(.)54 b(A)36 b(collection)e(\000)g Fp(\032)h Fr(\006)h(of)g(closed)h(curv)m(es)h(and)e(arcs)h(with)f(end)386 1732 y(p)s(oin)m(ts)28 b(on)g Fo(@)5 b Fr(\006)30 b(is)e(said)g(to)g Fm(divide)f Fp(F)38 b Fr(if)27 b(on)i(eac)m(h)g(connected)h(comp)s (onen)m(t)e(of)g(the)386 1848 y(closure)k(of)g(\006)22 b Fp(n)f Fr(\000)32 b(there)h(is)f(a)g(smo)s(oth)f(v)m(olume)h(form)f Fo(!)k Fr(and)e(a)f(v)m(ector)h(\014eld)f Fo(X)386 1964 y Fr(tangen)m(t)h(to)f Fp(F)42 b Fr(suc)m(h)34 b(that)556 2118 y(\(i\))40 b(the)33 b(div)m(ergence)h(of)e Fo(X)41 b Fr(with)32 b(resp)s(ect)i(to)e Fo(!)k Fr(is)c(p)s(ositiv)m(e)g(ev)m (erywhere)529 2234 y(\(ii\))39 b Fo(X)53 b Fr(p)s(oin)m(ts)45 b(out)m(w)m(ards)h(along)e(those)h(parts)h(of)e(the)i(b)s(oundary)f (whic)m(h)700 2350 y(corresp)s(ond)34 b(to)e(\000.)386 2503 y Fy(Theorem)41 b(3.11)h Fr(\(Giroux,)35 b([Gi1)o(]\))p Fy(.)43 b Fm(If)38 b Fr(\006)g Fm(is)g(a)g(c)-5 b(onvex)37 b(surfac)-5 b(e)38 b(with)f(L)-5 b(e)g(gen-)386 2620 y(drian)34 b(b)-5 b(oundary,)35 b(then)f Fr(\000)1373 2635 y Fn(\006)1463 2620 y Fm(divides)g(the)h(singular)f(foliation)g (on)h Fr(\006)p Fm(.)486 2736 y(Conversely,)e(let)i Fp(F)45 b Fm(b)-5 b(e)34 b(a)h(singular)g(foliation)f(on)g(a)h(c)-5 b(omp)g(act)34 b(oriente)-5 b(d)35 b(sur-)386 2852 y(fac)-5 b(e)34 b Fr(\006)h Fm(such)g(that)g Fo(@)5 b Fr(\006)36 b Fm(is)e(tangent)h(to)g Fp(F)44 b Fm(and)34 b Fr(\000)h Fm(divides)f Fp(F)10 b Fm(.)44 b(Then)34 b(ther)-5 b(e)34 b(is)h(a)386 2968 y(p)-5 b(ositive)37 b Fh(R)5 b Fm({invariant)44 b(c)-5 b(ontact)38 b(structur)-5 b(e)39 b(on)f Fr(\006)25 b Fp(\002)g Fh(R)49 b Fm(such)38 b(that)g Fr(\006)25 b Fp(\002)g(f)p Fr(0)p Fp(g)38 b Fm(is)386 3085 y(c)-5 b(onvex,)27 b(the)f(induc)-5 b(e)g(d)26 b(singular)f(foliation)h(on)g Fr(\006)s Fp(\002)s(f)p Fr(0)p Fp(g)h Fm(is)f(pr)-5 b(e)g(cisely)25 b Fp(F)36 b Fm(and)26 b(such)386 3201 y(that)32 b Fr(\000)f Fm(c)-5 b(onsists)31 b(of)g(those)g(p)-5 b(oints)31 b(wher)-5 b(e)30 b(the)i(c)-5 b(ontact)31 b(structur)-5 b(e)32 b(is)f(tangent)h(to)386 3317 y(the)j(se)-5 b(c)g(ond)34 b(factor)g(in)h Fr(\006)22 b Fp(\002)h Fh(R)5 b Fm(.)486 3470 y Fr(This)33 b(theorem)f(implies)f(in)h(particular)f(that)i(the)h (dividing)d(set)i(of)g(a)f(con)m(v)m(ex)386 3586 y(surface)47 b(with)f(Legendrian)g(b)s(oundary)h(is)f(not)g(empt)m(y)-8 b(.)85 b(Assume)47 b(that)g Fp(C)52 b Fr(is)386 3703 y(co)s(orien)m(ted)26 b(b)m(y)h Fo(\013)g Fr(and)f(that)h(the)f(orien)m (tation)f(\006)h(follo)m(w)m(ed)g(b)m(y)h Fo(V)48 b Fr(is)25 b(the)i(con)m(tact)386 3819 y(orien)m(tation.)70 b(Then)43 b(the)g(dividing)d(set)j(\000)f(separates)h(the)g(region)e(\006)3024 3834 y Fn(+)3125 3819 y Fr(where)386 3935 y Fo(\013)q Fr(\()p Fo(V)21 b Fr(\))47 b(is)g(p)s(ositiv)m(e)g(from)f(the)i(region) e(\006)1944 3950 y Fk(\000)2051 3935 y Fr(where)j Fo(\013)q Fr(\()p Fo(V)21 b Fr(\))47 b(is)g(negativ)m(e.)88 b(If)47 b(\006)386 4051 y(is)42 b(closed,)j(then)e(the)g(Euler)g(c)m (haracteristic)f Fo(\037)p Fr(\()p Fp(C)6 b Fr(\))43 b(of)f Fp(C)48 b Fr(view)m(ed)c(as)f(orien)m(ted)386 4168 y(bundle)33 b(satis\014es)386 4358 y(\(12\))800 b Fo(\037)p Fr(\()p Fp(C)6 b Fr(\))28 b(=)f Fo(\037)p Fr(\(\006)1855 4373 y Fn(+)1915 4358 y Fr(\))22 b Fp(\000)h Fo(\037)p Fr(\(\006)2244 4373 y Fk(\000)2303 4358 y Fr(\))33 b Fo(:)386 4548 y Fr(If)25 b(\006)g(is)g(a)f(Seifert)h(surface)h(for)e (a)h(Legendrian)g(knot)g Fo(\015)5 b Fr(,)27 b(then)e(one)h(can)f (determine)386 4664 y(the)30 b(Th)m(urston{Bennequin)j(in)m(v)-5 b(arian)m(t)28 b(and)i(the)h(rotation)d(n)m(um)m(b)s(er)j(of)e Fo(\015)35 b Fr(using)386 4781 y(the)e(follo)m(wing)d(form)m(ulas)h (from)g([Ka2,)h(Ho])1362 5019 y(tb\()p Fo(\015)5 b Fr(\))27 b(=)h Fp(\000)1804 4952 y Fr(1)p 1804 4996 49 4 v 1804 5088 a(2)1880 5019 y Fp(j)o Fr(\000)23 b Fp(\\)f Fo(\015)5 b Fp(j)1329 5200 y Fr(rot\()p Fo(\015)g Fr(\))27 b(=)h Fo(\037)p Fr(\(\006)1886 5215 y Fn(+)1946 5200 y Fr(\))22 b Fp(\000)g Fo(\037)p Fr(\(\006)2274 5215 y Fk(\000)2334 5200 y Fr(\))32 b Fo(:)386 5088 y Fr(\(13\))386 5390 y(Next)42 b(w)m(e)h(consider)f(deformations)e(of)h(the)h(singular)e (foliation.)67 b(Let)41 b(\006)h(b)s(e)g(a)386 5506 y(con)m(v)m(ex)34 b(surface)e(with)f(Legendrian)h(b)s(oundary)g(and)g(\014x)g(a)f(transv) m(erse)j(con)m(tact)386 5623 y(v)m(ector)g(\014eld)e Fo(V)21 b Fr(.)p eop %%Page: 18 18 18 17 bop 386 259 a Fq(18)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fy(De\014nition)40 b(3.12.)j Fr(An)36 b(isotop)m(y)g(\010)1759 474 y Fl(s)1832 459 y Fr(of)f(a)g(surface)i(\006)f(is)f(called)g Fm(admissible)f Fr(if)386 575 y(\010)456 590 y Fl(s)493 575 y Fr(\(\006\))f(is)f(transv)m(erse)j(to)d Fo(V)54 b Fr(for)32 b(all)f Fo(s)p Fr(.)486 729 y(The)43 b(follo)m(wing)d (theorem)i(is)g(a)g(generalization)e(of)i(the)h(Giroux)e(\015exibilit)m (y)386 846 y(theorem)32 b(in)g([Gi1)o(])g(where)i(\006)f(is)f(assumed)h (to)g(b)s(e)f(closed.)386 1000 y Fy(Theorem)37 b(3.13)g Fr(\(Giroux,)31 b(Honda,[Gi1)o(,)h(Ho]\))p Fy(.)41 b Fm(L)-5 b(et)35 b Fp(F)2534 1015 y Fn(0)2607 1000 y Fm(b)-5 b(e)35 b(the)f(singular)g(fo-)386 1116 y(liation)46 b(on)g Fr(\006)h Fm(induc)-5 b(e)g(d)46 b(by)h(the)g(c)-5 b(ontact)46 b(structur)-5 b(e)48 b(and)e Fp(F)2694 1131 y Fn(1)2780 1116 y Fm(a)g(singular)g(fo-)386 1233 y(liation)g(which)f(is)h(divide) -5 b(d)46 b(by)g Fr(\000)1645 1248 y Fn(\006)1700 1233 y Fm(.)79 b(Then)46 b(ther)-5 b(e)46 b(is)g(an)g(admissible)f(isotopy) 386 1349 y Fr(\010)456 1364 y Fl(s)493 1349 y Fo(;)17 b(s)27 b Fp(2)i Fr([0)p Fo(;)17 b Fr(1])p Fm(,)34 b(of)g Fr(\006)i Fm(such)e(that)i Fp(F)1678 1364 y Fn(1)1752 1349 y Fm(is)e(the)h(singular)f(foliation)g(on)h Fr(\010)2986 1364 y Fn(1)3026 1349 y Fr(\(\006\))p Fm(.)386 1503 y Fy(Example)54 b(3.14.)c Fr(In)f(this)e(example)h(w)m(e)h(w)m(an)m(t)g (to)f(\014x)h(some)f(terminology)-8 b(.)386 1619 y(Consider)22 b(the)h Fh(R)5 b Fr({in)m(v)-5 b(arian)m(t)26 b(con)m(tact)d(form)e Fo(\013)28 b Fr(=)g(cos\()p Fo(')p Fr(\))17 b Fo(dt)q Fr(+)q(sin)o(\()p Fo(')p Fr(\))g Fo(dx)k Fr(on)h Fo(T)3257 1583 y Fn(2)3297 1619 y Fp(\002)386 1736 y Fh(R)43 b Fr(where)33 b Fo(x)g Fr(is)e(the)i(co)s(ordinate)e(on)h(the)g Fh(R)5 b Fr({factor)37 b(while)32 b Fo(')f Fr(and)i Fo(t)f Fr(corresp)s(ond)386 1852 y(to)39 b Fo(T)583 1816 y Fn(2)662 1852 y Fr(=)h Fo(S)844 1816 y Fn(1)910 1852 y Fp(\002)27 b Fo(S)1080 1816 y Fn(1)1120 1852 y Fr(.)64 b(W)-8 b(e)40 b(sa)m(y)h(that)f(the)g(singular)e(foliation)e Fp(F)49 b Fr(on)40 b Fo(T)3056 1816 y Fn(2)3122 1852 y Fp(\002)27 b(f)p Fr(0)p Fp(g)386 1968 y Fr(is)44 b(in)f Fm(standar)-5 b(d)45 b(form)p Fr(.)77 b(The)45 b(singularities)d(of)h(the)i(singular) d(foliation)f(form)386 2084 y(t)m(w)m(o)35 b(circles)e Fp(f)p Fo(')d Fr(=)g Fo(\031)t(=)p Fr(2)p Fp(g)22 b([)i(f)p Fo(')29 b Fr(=)h(3)p Fo(\031)t(=)p Fr(2)p Fp(g)p Fr(.)47 b(These)36 b(are)e(called)e(the)j Fm(L)-5 b(e)g(gendrian)386 2201 y(divides)p Fr(.)42 b(The)34 b(dividing)c(set)k(of)e Fo(T)1657 2164 y Fn(2)1718 2201 y Fp(\002)23 b(f)p Fr(0)p Fp(g)32 b Fr(has)h(t)m(w)m(o)g(connected)h(comp)s(onen)m(ts)1350 2393 y(\000)28 b(=)g Fp(f)p Fo(')f Fr(=)h(0)p Fp(g)21 b([)i(f)p Fo(')k Fr(=)h Fo(\031)t Fp(g)k Fo(:)386 2586 y Fr(The)24 b(curv)m(es)g(tangen)m(t)f(to)g Fo(@)1371 2601 y Fl(')1444 2586 y Fr(form)e(the)i Fm(L)-5 b(e)g(gendrian)25 b(ruling)p Fr(.)40 b(By)23 b(Theorem)g(3.13,)386 2702 y(the)g(Legendrian)g(ruling)e(can)j(b)s(e)f(c)m(hanged)h(in)m(to)e(an)m (y)i(foliation)19 b(b)m(y)24 b(straigh)m(t)e(lines)386 2818 y(as)33 b(long)e(as)i(the)g(straigh)m(t)f(lines)f(remain)h(transv) m(erse)i(to)f(the)g(dividing)d(set.)486 2973 y(Using)g(Theorem)i(3.13)e (w)m(e)j(can)e(isotop)s(e)g(\006)g(suc)m(h)i(that)e(a)g(submanifold)e (with)386 3089 y(certain)j(prop)s(erties)h(b)s(ecomes)g(Legendrian.)386 3244 y Fy(De\014nition)41 b(3.15.)j Fr(A)37 b(union)f Fo(C)43 b Fr(of)37 b(disjoin)m(t)e(prop)s(erly)i(em)m(b)s(edded)g(arcs) h(and)386 3360 y(closed)33 b(curv)m(es)h(on)f(\006)g(is)f(called)f Fm(non{isolating)g Fr(if)556 3514 y(\(i\))40 b Fo(C)g Fr(is)32 b(transv)m(erse)j(to)d(\000)g(and)h(ev)m(ery)i(arc)d(b)s (egins)g(and)h(ends)h(on)e(\000)529 3631 y(\(ii\))39 b(the)50 b(b)s(oundary)f(of)f(eac)m(h)i(comp)s(onen)m(t)f(of)f(\006)34 b Fp(n)f Fr(\(\000)g Fp([)g Fo(C)7 b Fr(\))49 b(con)m(tains)g(a)700 3747 y(segmen)m(t)34 b(of)e(\000.)386 3901 y Fy(Theorem)27 b(3.16)g Fr(\(Kanda,)f(Honda,)f([Ka1,)f(Ho)o(]\))p Fy(.)35 b Fm(Consider)26 b(a)h(c)-5 b(ol)5 b(le)-5 b(ction)25 b Fo(C)34 b Fm(of)386 4017 y(pr)-5 b(op)g(erly)39 b(emb)-5 b(e)g(dde)g(d)39 b(close)-5 b(d)38 b(curves)i(and)f(ar)-5 b(cs)39 b(on)g(a)h(c)-5 b(onvex)38 b(surfac)-5 b(e)40 b Fr(\006)g Fm(with)386 4134 y(L)-5 b(e)g(gendrian)45 b(b)-5 b(oundary.)79 b(If)45 b Fo(C)54 b Fm(is)46 b(non{isolating)e (ther)-5 b(e)46 b(exists)g(a)g Fo(C)3061 4098 y Fn(0)3101 4134 y Fm({smal)5 b(l)386 4250 y(admissible)33 b(isotopy)i Fr(\010)1258 4265 y Fl(s)1295 4250 y Fo(;)17 b(s)27 b Fp(2)i Fr([0)p Fo(;)17 b Fr(1])34 b Fm(such)h(that)556 4404 y Fr(\(i\))40 b(\010)770 4419 y Fn(0)838 4404 y Fr(=)28 b(id)34 b Fm(and)g Fr(\010)1317 4419 y Fn(1)1357 4404 y Fr(\(\000)1456 4419 y Fn(\006)1511 4404 y Fr(\))28 b(=)f(\000)1741 4420 y Fn(\010)1792 4429 y Fg(1)1827 4420 y Fn(\(\006\))529 4521 y Fr(\(ii\))39 b(\010)770 4536 y Fn(1)810 4521 y Fr(\()p Fo(C)7 b Fr(\))35 b Fm(is)g(L)-5 b(e)g(gendrian.)486 4675 y Fr(W)d(e)28 b(\014nish)g(this)f(section)h (with)g(an)f(imp)s(ortan)m(t)f(dic)m(hotom)m(y)i(of)f(con)m(tact)h (struc-)386 4791 y(tures)33 b(on)g(3{manifolds.)386 4946 y Fy(De\014nition)58 b(3.17.)51 b Fr(An)h(em)m(b)s(edded)h(disc)f(with) f(Legendrian)g(b)s(oundary)h(is)386 5062 y Fm(overtwiste)-5 b(d)30 b(disc)c Fr(if)h(all)e(singularities)g(on)i(the)h(b)s(oundary)g (ha)m(v)m(e)h(the)f(same)f(sign.)386 5178 y(A)41 b(con)m(tact)g (structure)i(is)d(called)g Fm(overtwiste)-5 b(d)40 b Fr(if)g(it)g(there)i(is)e(an)h(o)m(v)m(ert)m(wisted)386 5294 y(disc.)i(A)33 b(con)m(tact)g(structure)h(is)e Fm(tight)h Fr(if)e(it)h(is)g(not)g(o)m(v)m(ert)m(wisted.)486 5449 y(The)h(follo)m(wing)c(theorem)j(sho)m(ws)i(that)d(tigh)m(t)h(con)m (tact)g(structures)i(are)e(more)386 5565 y(in)m(teresting)g(than)h(o)m (v)m(ert)m(wisted)h(ones)f(in)f(man)m(y)h(asp)s(ects.)p eop %%Page: 19 19 19 18 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(19)386 459 y Fy(Theorem)45 b(3.18)h Fr(\(Eliash)m(b)s(erg,)40 b([El1)o(]\))p Fy(.)46 b Fm(If)40 b(two)i(overtwiste)-5 b(d)40 b(c)-5 b(ontact)41 b(struc-)386 575 y(tur)-5 b(es)31 b(on)f(a)g(close)-5 b(d)29 b(manifold)g(ar)-5 b(e)30 b(homotopic)f(as)h(plane)g(\014elds)f(and)h(induc)-5 b(e)30 b(the)386 691 y(same)k(orientation,)g(then)h(they)g(ar)-5 b(e)35 b(isotopic.)486 838 y Fr(F)-8 b(or)22 b(our)i(purp)s(oses)h(ho)m (w)m(ev)m(er,)j(the)c(\015exibilit)m(y)e(of)h(o)m(v)m(ert)m(wisted)i (con)m(tact)f(struc-)386 954 y(tures)40 b(will)d(b)s(e)i(v)m(ery)i (useful.)63 b(Regions)39 b(where)h(a)f(giv)m(en)g(con)m(tact)h (structure)g(is)386 1070 y(tigh)m(t)32 b(can)h(b)s(e)f(found)h(using)f (the)h(follo)m(wing)d(theorem.)386 1217 y Fy(Theorem)43 b(3.19)h Fr(\(Colin,)39 b([Col)o(]\))p Fy(.)44 b Fm(If)c Fr(\006)d Fp(6)p Fr(=)g Fo(S)2150 1181 y Fn(2)2230 1217 y Fm(is)i(a)h(c)-5 b(onvex)39 b(surfac)-5 b(e)40 b(\(close)-5 b(d)386 1333 y(or)42 b(c)-5 b(omp)g(act)40 b(with)i(L)-5 b(e)g(gendrian)41 b(b)-5 b(oundary\))41 b(in)g(a)h(c)-5 b(ontact)41 b(manifold)g Fr(\()p Fo(M)5 b(;)17 b Fp(C)6 b Fr(\))p Fm(,)386 1450 y(then)33 b Fr(\006)g Fm(has)g(a)g(tight)g (neighb)-5 b(orho)g(o)g(d)31 b(if)i(and)f(only)h(if)g(the)g(dividing)f (set)h(of)g Fr(\006)g Fm(has)386 1566 y(no)39 b(homotopic)-5 b(al)5 b(ly)38 b(trivial)h(close)-5 b(d)38 b(curves.)57 b(If)39 b Fr(\006)d(=)g Fo(S)2478 1530 y Fn(2)2517 1566 y Fm(,)k(then)f Fr(\006)h Fm(has)e(a)h(tight)386 1682 y(neighb)-5 b(orho)g(o)g(d)33 b(if)i(and)f(only)h(if)f Fr(\000)1616 1697 y Fn(\006)1706 1682 y Fm(is)h(c)-5 b(onne)g(cte)g(d.)386 1897 y Fr(3.3.)48 b Fy(Bypasses)30 b(in)f(o)m(v)m(ert)m(wisted)e(con)m(tact)h(structures.)49 b Fr(Consider)25 b(a)g(con-)386 2014 y(tact)33 b(manifold)e(\()p Fo(N)5 b(;)17 b Fp(C)6 b Fr(\))33 b(and)g(a)g(con)m(v)m(ex)i(surface)g (\006)29 b Fp(\032)g Fo(N)44 b Fr(whic)m(h)34 b(has)f(empt)m(y)h(or)386 2130 y(Legendrian)e(b)s(oundary)-8 b(.)44 b(Let)33 b(\000)1596 2145 y Fn(\006)1683 2130 y Fr(b)s(e)g(a)f(dividing)f(set)i(of)f(\006.) 386 2277 y Fy(De\014nition)48 b(3.20.)f Fr(A)c Fm(byp)-5 b(ass)42 b Fr(for)h(\006)g(is)f(an)h(em)m(b)s(edded)h(half{disc)e Fo(D)j Fr(in)d Fo(N)386 2393 y Fr(whose)34 b(singular)d(foliation)e (has)k(the)g(follo)m(wing)d(prop)s(erties:)556 2540 y(\(i\))40 b Fo(@)5 b(D)43 b Fr(is)c(the)g(union)g(of)g(t)m(w)m(o)h(Legendrian)f (arcs)g Fo(\015)2516 2555 y Fn(1)2555 2540 y Fo(;)17 b(\015)2650 2555 y Fn(2)2728 2540 y Fr(whic)m(h)40 b(in)m(tersect)700 2656 y(at)33 b(their)f(endp)s(oin)m(ts.)529 2772 y(\(ii\))39 b Fo(D)32 b Fr(in)m(tersects)e(\006)f(transv)m(ersally)g(along)f Fo(\015)2217 2787 y Fn(1)2256 2772 y Fr(.)42 b(There)30 b(are)f(no)f(other)h(in)m(ter-)700 2888 y(section)k(p)s(oin)m(ts.)502 3005 y(\(iii\))38 b Fo(D)e Fr(admits)31 b(an)h(orien)m(tation)f(suc)m (h)j(that)802 3121 y Fy({)41 b Fr(there)29 b(are)g(exactly)f(t)m(w)m(o) h(p)s(ositiv)m(e)f(singularities)e(along)h Fo(\015)3017 3136 y Fn(1)3056 3121 y Fr(.)42 b(These)899 3237 y(are)33 b(the)g(endp)s(oin)m(ts)g(of)f Fo(\015)1837 3252 y Fn(1)1876 3237 y Fr(.)43 b(They)34 b(are)f(elliptic.)802 3353 y Fy({)41 b Fr(there)26 b(is)f(exactly)h(one)f(negativ)m(e)h(singularit)m (y)d(on)i Fo(\015)2764 3368 y Fn(1)2803 3353 y Fr(.)41 b(It)26 b(is)e(elliptic.)802 3469 y Fy({)41 b Fr(there)d(are)f(only)g (p)s(ositiv)m(e)f(singularities)f(along)g Fo(\015)2767 3484 y Fn(2)2806 3469 y Fr(.)58 b(They)38 b(alter-)899 3586 y(nate)33 b(b)s(et)m(w)m(een)i(elliptic)29 b(and)k(h)m(yp)s(erb)s (olic.)505 3702 y(\(iv\))40 b Fo(\015)751 3717 y Fn(1)827 3702 y Fr(in)m(tersects)f(\000)1325 3717 y Fn(\006)1417 3702 y Fr(in)d(exactly)h(three)h(p)s(oin)m(ts.)56 b(The)38 b(in)m(tersections)g(are)700 3818 y(transv)m(erse)45 b(and)e(corresp)s(ond)g(to)g(the)g(singularities)d(of)i(the)h(singular) 700 3934 y(foliation)30 b(on)i Fo(D)j Fr(along)c Fo(\015)1645 3949 y Fn(1)1684 3934 y Fr(.)532 4051 y(\(v\))41 b(The)34 b(dividing)d(set)i(of)f Fo(D)j Fr(has)e(exactly)g(one)g(connected)h (comp)s(onen)m(t.)486 4197 y(A)f(b)m(ypass)i(for)e(\006)h(allo)m(ws)e (us)i(to)f(isotop)s(e)f(\006)i(suc)m(h)h(that)e(the)h(isotop)m(y)f(t)m (yp)s(e)h(of)386 4314 y(the)d(dividing)e(set)i(c)m(hanges)h(as)f(in)f (Lemma)g(3.21.)42 b(They)32 b(where)g(in)m(tro)s(duced)f(b)m(y)386 4430 y(K.)g(Honda)h(in)e([Ho].)44 b(Requiremen)m(t)31 b(\(v\))h(in)f(De\014nition)f(3.20)h(do)s(es)g(not)h(app)s(ear)386 4546 y(in)g([Ho])g(since)h(all)d(con)m(tact)j(structures)h(considered)f (in)f(that)g(article)f(are)h(tigh)m(t.)386 4662 y(In)26 b(this)f(situation,)h(the)g(dividing)e(set)j(\000)1851 4677 y Fl(D)1940 4662 y Fr(of)f Fo(D)i Fr(is)e(determined)f(\(up)h(to)g (isotop)m(y\))386 4779 y(b)m(y)i(\(i\){\(iv\))d(while)h(this)g(is)h (not)g(true)g(for)f(o)m(v)m(ert)m(wisted)j(con)m(tact)f(structures.)43 b(Re-)386 4895 y(quiremen)m(t)29 b(\(v\))h(is)f(necessary)i(for)e(the)h (follo)m(wing)c(b)m(ypass)31 b(attac)m(hmen)m(t)f(lemma.)386 5042 y Fy(Lemma)f(3.21)g Fr(\(Honda,)e([Ho]\))p Fy(.)36 b Fm(L)-5 b(et)29 b Fo(D)i Fm(b)-5 b(e)27 b(a)i(byp)-5 b(ass)27 b(for)h(a)g(c)-5 b(onvex)28 b(surfac)-5 b(e)27 b Fr(\006)p Fm(.)386 5158 y(Ther)-5 b(e)27 b(exists)h(a)g(neighb)-5 b(orho)g(o)g(d)26 b(of)i Fr(\006)7 b Fp([)g Fo(D)32 b Fm(which)27 b(is)h(di\013e)-5 b(omorphic)26 b(to)i Fr(\006)7 b Fp(\002)g Fr([0)p Fo(;)17 b Fr(1])386 5274 y Fm(such)39 b(that)g Fr(\006)25 b Fp(\002)h(f)p Fo(i)p Fp(g)39 b Fm(is)f(c)-5 b(onvex)38 b(for)h Fo(i)c Fr(=)g(0)p Fo(;)17 b Fr(1)p Fm(.)56 b(The)38 b(dividing)g(set)h(of)f Fr(\006)26 b Fp(\002)f(f)p Fr(1)p Fp(g)386 5390 y Fm(c)-5 b(an)36 b(b)-5 b(e)35 b(obtaine)-5 b(d)36 b(fr)-5 b(om)36 b(the)g(dividing)f (set)h(of)g Fr(\006)23 b Fp(\002)h(f)p Fr(0)p Fp(g)36 b Fm(as)f(in)h(Figur)-5 b(e)36 b(2.)48 b(\(The)386 5506 y(byp)-5 b(ass)30 b(is)g(attache)-5 b(d)30 b(to)g(the)g(fr)-5 b(ont.)44 b(Figur)-5 b(e)30 b(2)g(r)-5 b(epr)g(esents)29 b(only)h(a)g(neighb)-5 b(orho)g(o)g(d)386 5623 y(of)34 b(the)h(attaching)g(r)-5 b(e)g(gion)34 b(of)h Fo(D)s Fm(.\))p eop %%Page: 20 20 20 19 bop 386 259 a Fq(20)1096 b(THOMAS)25 b(V)n(OGEL)601 1071 y @beginspecial 0 @llx 0 @lly 307 @urx 81 @ury 3070 @rwi @setspecial %%BeginDocument: artbypass.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: bypassaom.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Wed Sep 1 11:29:05 2004 %%For: tvogel@math81 (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 307 81 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -1.0 82.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 2362 m -1000 -1000 l 6127 -1000 l 6127 2362 l cp clip 0.06000 0.06000 sc % Polyline 7.500 slw n 189 258 m 1710 258 l gs col0 s gr % Polyline n 189 695 m 1710 695 l gs col0 s gr % Polyline n 189 1131 m 1710 1131 l gs col0 s gr % Polyline 15.000 slw n 949 258 m 949 1131 l gs col0 s gr /Symbol ff 195.00 scf sf 1053 1003 m gs 1 -1 sc (g) col0 sh gr /Symbol ff 195.00 scf sf 1127 1080 m gs 1 -1 sc (1) col0 sh gr % Polyline 7.500 slw n 3429 244 m 3808 244 l gs col0 s gr % Polyline n 3429 681 m 3808 681 l gs col0 s gr % Polyline n 3429 1117 m 3808 1117 l gs col0 s gr % Polyline n 4950 244 m 4570 244 l gs col0 s gr % Polyline n 4570 681 m 4950 681 l gs col0 s gr % Polyline n 4570 1117 m 4950 1117 l gs col0 s gr % Polyline n 3808 681 m 3809 681 l 3813 682 l 3823 683 l 3839 685 l 3858 688 l 3878 691 l 3897 695 l 3913 699 l 3926 703 l 3936 708 l 3945 714 l 3953 721 l 3960 728 l 3966 737 l 3972 745 l 3978 754 l 3983 763 l 3989 772 l 3994 781 l 3999 790 l 4003 800 l 4007 810 l 4011 821 l 4015 832 l 4018 843 l 4022 855 l 4025 866 l 4027 877 l 4028 888 l 4029 899 l 4028 910 l 4027 921 l 4025 932 l 4022 944 l 4018 955 l 4015 967 l 4011 978 l 4007 989 l 4003 999 l 3999 1009 l 3994 1018 l 3989 1027 l 3983 1036 l 3978 1045 l 3972 1053 l 3966 1062 l 3960 1070 l 3953 1077 l 3945 1084 l 3936 1090 l 3926 1095 l 3913 1099 l 3897 1103 l 3878 1106 l 3858 1110 l 3839 1113 l 3823 1115 l 3813 1116 l 3809 1117 l 3808 1117 l gs col0 s gr % Polyline n 4570 244 m 4569 244 l 4565 245 l 4555 246 l 4540 249 l 4521 252 l 4501 255 l 4483 258 l 4467 262 l 4454 266 l 4443 271 l 4433 277 l 4424 283 l 4416 290 l 4408 297 l 4400 304 l 4393 312 l 4386 320 l 4379 328 l 4373 337 l 4368 346 l 4364 356 l 4360 367 l 4358 378 l 4356 390 l 4354 402 l 4352 415 l 4351 427 l 4350 439 l 4349 451 l 4349 462 l 4350 473 l 4351 484 l 4353 495 l 4356 506 l 4358 518 l 4361 529 l 4364 539 l 4368 550 l 4372 560 l 4376 570 l 4381 579 l 4387 588 l 4392 597 l 4399 606 l 4405 615 l 4411 624 l 4418 632 l 4426 640 l 4434 647 l 4443 653 l 4454 658 l 4466 662 l 4482 666 l 4501 670 l 4520 673 l 4539 677 l 4555 679 l 4565 680 l 4569 681 l 4570 681 l gs col0 s gr % Polyline n 3808 244 m 3809 244 l 3813 244 l 3823 243 l 3839 243 l 3858 242 l 3878 242 l 3896 242 l 3912 242 l 3925 243 l 3936 244 l 3948 246 l 3958 249 l 3968 252 l 3977 256 l 3985 260 l 3994 264 l 4002 269 l 4010 275 l 4017 282 l 4024 289 l 4029 294 l 4034 300 l 4038 306 l 4044 316 l 4052 330 l 4062 352 l 4067 364 l 4073 378 l 4079 393 l 4085 409 l 4092 427 l 4099 445 l 4107 464 l 4114 484 l 4122 505 l 4130 525 l 4137 546 l 4145 566 l 4153 586 l 4160 606 l 4168 625 l 4175 644 l 4182 663 l 4189 681 l 4196 699 l 4203 718 l 4210 737 l 4218 756 l 4225 776 l 4233 796 l 4241 816 l 4248 836 l 4256 857 l 4264 877 l 4271 897 l 4279 916 l 4286 935 l 4293 952 l 4299 969 l 4305 984 l 4311 997 l 4316 1009 l 4326 1030 l 4333 1045 l 4338 1054 l 4342 1060 l 4346 1065 l 4351 1071 l 4357 1077 l 4364 1084 l 4372 1090 l 4381 1095 l 4390 1100 l 4400 1104 l 4410 1108 l 4420 1111 l 4431 1114 l 4443 1117 l 4454 1118 l 4467 1119 l 4483 1119 l 4501 1119 l 4521 1119 l 4540 1118 l 4555 1118 l 4565 1117 l 4569 1117 l 4570 1117 l gs col0 s gr % Polyline n 45 39 m 1950 39 l 1950 1350 l 45 1350 l cp gs col0 s gr % Polyline n 3240 39 m 5115 39 l 5115 1350 l 3240 1350 l cp gs col0 s gr % Polyline gs clippath 2765 660 m 2885 690 l 2765 720 l 2927 720 l 2927 660 l cp clip n 2285 690 m 2912 690 l gs col0 s gr gr % arrowhead n 2765 660 m 2885 690 l 2765 720 l 2765 690 l 2765 660 l cp gs 0.00 setgray ef gr col0 s $F2psEnd rs %%EndDocument @endspecial 1623 1307 a Fs(Figure)37 b(2.)486 1516 y Fr(Gluing)g(together)j(t)m(w)m(o)h(b)m(ypasses)h(along)d(the)h(b)s (oundary)g(comp)s(onen)m(t)g(con-)386 1632 y(taining)30 b(the)j(singularities)d(with)j(alternating)d(signs)j(one)f(obtains)g(a) g(disc)h(with)386 1749 y(a)40 b(singular)g(foliation)d(lik)m(e)j(the)h (singular)e(foliation)f(on)i(an)h(o)m(v)m(ert)m(wisted)h(disc.)386 1865 y(Th)m(us)35 b(one)f(can)g(think)f(of)g(a)g(b)m(ypass)i(as)f(one)g (half)e(of)h(an)h(o)m(v)m(ert)m(wisted)h(disc)f(and)386 1981 y(it)28 b(is)g(not)h(surprising)f(that)h(it)f(is)h(m)m(uc)m(h)g (easier)g(to)g(\014nd)g(b)m(ypasses)j(if)c(the)h(con)m(tact)386 2097 y(structure)34 b(is)e(o)m(v)m(ert)m(wisted)i(rather)f(than)g(tigh) m(t.)486 2213 y(In)k(the)g(follo)m(wing)d(prop)s(osition)g(w)m(e)k (assume)f(that)g Fo(\015)2472 2228 y Fn(1)2548 2213 y Fr(is)f(a)g(Legendrian)h(arc)386 2330 y(con)m(tained)f(in)g(the)h(con)m (v)m(ex)h(surface)f(\006.)55 b(Since)37 b(ev)m(ery)h(arc)e Fo(\015)2670 2294 y Fk(0)2665 2354 y Fn(1)2740 2330 y Fr(whic)m(h)h(is)f(trans-)386 2446 y(v)m(erse)i(to)d(\000)h(and)g(in)m (tersects)h(\000)f(as)g Fo(\015)1752 2461 y Fn(1)1827 2446 y Fr(in)f(Prop)s(osition)g(3.22)g(is)g(non{isolating,)386 2562 y(w)m(e)41 b(can)g(apply)f(Theorem)g(3.16)g(to)g(\014nd)h(a)f Fo(C)2129 2526 y Fn(0)2209 2562 y Fr(small)e(admissible)g(isotop)m(y)i (of)386 2678 y(\006)45 b(suc)m(h)h(that)e Fo(\015)1012 2642 y Fk(0)1007 2703 y Fn(1)1091 2678 y Fr(b)s(ecomes)h(Legendrian,)j (i.e.)79 b(the)45 b(assumption)f(that)h Fo(\015)3226 2693 y Fn(1)3309 2678 y Fr(is)386 2795 y(Legendrian)32 b(is)g(actually)g(not)g(restrictiv)m(e.)386 2932 y Fy(Prop)s(osition)44 b(3.22.)i Fm(L)-5 b(et)42 b Fr(\006)h Fm(b)-5 b(e)41 b(a)h(c)-5 b(onvex)41 b(surfac)-5 b(e)41 b(in)h(a)f(c)-5 b(ontact)42 b(manifold)386 3048 y(and)30 b Fo(D)652 3063 y Fl(ot)746 3048 y Fm(an)g(overtwiste)-5 b(d)30 b(disc)g(disjoint)f(fr) -5 b(om)30 b Fr(\006)p Fm(.)44 b(Assume)30 b(that)h Fo(\015)2897 3063 y Fn(1)2964 3048 y Fp(\032)d Fr(\006)j Fm(is)f(an)386 3164 y(emb)-5 b(e)g(dde)g(d)38 b(L)-5 b(e)g(gendrian)38 b(ar)-5 b(c)39 b(such)g(that)g(the)h(endp)-5 b(oints)38 b(lie)h(on)g Fr(\000)g Fm(and)g Fo(\015)3158 3179 y Fn(1)3222 3164 y Fp(\\)26 b Fr(\000)386 3280 y Fm(c)-5 b(ontains)35 b(thr)-5 b(e)g(e)35 b(p)-5 b(oints.)47 b(Then)35 b(ther)-5 b(e)36 b(is)f(a)g(byp)-5 b(ass)36 b(for)f Fr(\006)h Fm(which)f(interse) -5 b(cts)35 b Fr(\006)386 3397 y Fm(along)f(the)h Fo(\015)858 3412 y Fn(1)897 3397 y Fm(.)386 3564 y(Pr)-5 b(o)g(of.)41 b Fr(Note)23 b(that)f Fo(\015)1162 3579 y Fn(1)1223 3564 y Fr(is)g(automatically)d(transv)m(erse)24 b(to)e(\000.)40 b(Let)23 b Fo(V)44 b Fr(b)s(e)22 b(a)g(con)m(tact)386 3680 y(v)m(ector)29 b(\014eld)g(transv)m(erse)h(to)f(\006.)42 b(Consider)29 b(the)g(image)e Fo(R)i Fr(of)f Fo(\015)2700 3695 y Fn(1)2768 3680 y Fr(under)h(the)g(\015o)m(w)386 3796 y Fo(')450 3811 y Fl(t)504 3796 y Fr(of)23 b Fo(V)46 b Fr(for)23 b(0)28 b Fp(\024)g Fo(t)g Fp(\024)g Fo(")c Fr(suc)m(h)h(that)f Fo(\015)1734 3811 y Fn(1)1801 3796 y Fr(=)k Fo(R)6 b Fp(\\)f Fr(\006.)41 b(The)25 b(curv)m(es)h Fo(')2740 3811 y Fl(t)2770 3796 y Fr(\()p Fo(\015)2859 3811 y Fn(1)2898 3796 y Fr(\))p Fo(;)17 b Fr(0)27 b Fp(\024)h Fo(t)g Fp(\024)g Fo(")386 3912 y Fr(are)23 b(Legendrian)h(since)g Fo(')1330 3927 y Fl(t)1383 3912 y Fr(is)f(the)h(\015o)m(w)g(of)f(a)g (con)m(tact)h(v)m(ector)h(\014eld.)40 b(If)23 b Fo(p)28 b Fp(2)g Fo(\015)3174 3927 y Fn(1)3217 3912 y Fp(\\)t Fr(\000,)386 4028 y(then)33 b(the)g(segmen)m(t)g Fo(')1217 4043 y Fl(t)1247 4028 y Fr(\()p Fo(p)p Fr(\))p Fo(;)17 b Fr(0)27 b Fp(\024)h Fo(t)g Fp(\024)g Fo(")k Fr(is)g(Legendrian.)486 4145 y(If)h Fo(q)627 4160 y Fn(1)666 4145 y Fo(;)17 b(q)753 4160 y Fn(2)826 4145 y Fr(are)33 b(p)s(oin)m(ts)g(in)g(di\013eren)m(t)g (comp)s(onen)m(ts)h(of)e Fo(\015)2483 4160 y Fn(1)2545 4145 y Fp(n)22 b Fr(\()p Fo(\015)2706 4160 y Fn(1)2768 4145 y Fp(\\)h Fr(\000\),)33 b(then)h(the)386 4261 y(union)j(\000)723 4276 y Fl(R)819 4261 y Fr(of)g(the)h(curv)m(es)i Fo(')1476 4276 y Fl(t)1506 4261 y Fr(\()p Fo(q)1587 4276 y Fl(i)1615 4261 y Fr(\))e(with)f(0)f Fp(\024)h Fo(t)g Fp(\024)g Fo(";)17 b(i)37 b Fr(=)f(1)p Fo(;)17 b Fr(2)37 b(divide)g(the)h(sin-) 386 4377 y(gular)30 b(foliation)e(on)k Fo(R)q Fr(.)43 b(Since)32 b Fo(R)h Fr(has)f(Legendrian)f(b)s(oundary)-8 b(,)32 b(Theorem)g(3.11)386 4493 y(implies)37 b(that)j Fo(R)g Fr(is)f(con)m(v)m(ex.)67 b(W)-8 b(e)40 b(orien)m(t)g Fo(R)g Fr(suc)m(h)h(that)f(the)g(singularities)d(at)386 4610 y(the)c(endp)s(oin)m(ts)g(of)f Fo(\015)1161 4625 y Fn(1)1233 4610 y Fr(are)g(p)s(ositiv)m(e.)43 b(By)34 b(\(13\))1006 4767 y(tb\()p Fo(@)5 b(R)q Fr(\))28 b(=)g Fp(\000)p Fr(2)620 b(rot\()p Fo(@)5 b(R)q Fr(\))28 b(=)g(1)k Fo(:)386 4925 y Fr(The)40 b(idea)e(is)g(to)g(p)s(erform)g(a)g (Legendrian)h(connected)h(sum)f(of)f(the)h(knots)h Fo(@)5 b(R)386 5042 y Fr(and)32 b Fo(@)5 b(D)712 5057 y Fl(ot)809 5042 y Fr(together)33 b(with)f(a)g(b)s(oundary)g(connected)i(sum)e(of)g Fo(R)h Fr(and)g Fo(D)3085 5057 y Fl(ot)3149 5042 y Fr(.)43 b(Do-)386 5158 y(ing)d(this)h(carefully)f(enough,)k(one)e(obtains)e(a)h (b)m(ypass.)72 b(Let)41 b(us)h(\014rst)g(explain)386 5274 y(the)g(Legendrian)g(connected)i(sum)e(of)f(n)m(ull{homologous)e (Legendrian)j(knots)386 5390 y Fo(K)469 5405 y Fn(1)508 5390 y Fo(;)17 b(K)635 5405 y Fn(2)707 5390 y Fr(in)32 b(a)g(con)m(tact)h(manifold,)d(cf.)j([EH2].)486 5506 y(Let)g Fo(p)710 5521 y Fn(1)779 5506 y Fp(2)c Fo(K)957 5521 y Fn(1)1030 5506 y Fr(and)34 b Fo(p)1270 5521 y Fn(2)1338 5506 y Fp(2)c Fo(K)1517 5521 y Fn(2)1557 5506 y Fr(.)46 b(Cho)s(ose)34 b(a)f(Legendrian)g(curv)m(e)i Fo(\015)k Fr(b)s(et)m(w)m(een)c Fo(p)3335 5521 y Fn(1)386 5623 y Fr(and)30 b Fo(p)622 5638 y Fn(2)691 5623 y Fr(suc)m(h)h(that)f Fo(\015)35 b Fr(do)s(es)30 b(not)g(meet)g(the)g(knots)h(an)m(ywhere)h (else)e(and)g Fo(\015)k Fr(is)c(not)p eop %%Page: 21 21 21 20 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(21)386 459 y Fr(tangen)m(t)31 b(to)g Fo(K)940 474 y Fl(i)999 459 y Fr(in)g Fo(p)1161 474 y Fl(i)1220 459 y Fr(for)g Fo(i)d Fr(=)f(1)p Fo(;)17 b Fr(2.)42 b(By)32 b(Corollary)d(2.3)i(there)h(are)f(co)s(ordinates)386 575 y Fo(x;)17 b(z)t(;)g(t)28 b Fr(on)e(a)h(tubular)f(neigh)m(b)s(orho) s(o)s(d)g(of)h Fo(\015)k Fr(suc)m(h)e(that)e Fo(\015)k Fr(corresp)s(onds)e(to)d(the)i Fo(t)p Fr({)386 691 y(axis)f(and)h(the)g (con)m(tact)g(structure)h(is)e(de\014ned)h(b)m(y)h Fo(dz)16 b Fp(\000)c Fo(x)17 b(dt)p Fr(.)42 b(After)28 b(a)f Fo(C)3064 655 y Fn(0)3103 691 y Fr({small)386 807 y(Legendrian)35 b(isotop)m(y)f(of)h Fo(K)1429 822 y Fn(1)1503 807 y Fr(and)g Fo(K)1778 822 y Fn(2)1852 807 y Fr(w)m(e)h(ma)m(y)e(assume)i(that)e Fo(p)2813 822 y Fn(1)2853 807 y Fo(;)17 b(p)2946 822 y Fn(2)3020 807 y Fr(are)34 b(cusp)386 924 y(p)s(oin)m(ts)g(of)f(the)i (fron)m(t)f(pro)5 b(jection,)34 b(i.e)f(the)i(pro)5 b(jection)34 b(to)g(the)g Fo(z)t(;)17 b(t)p Fr({plane,)35 b(and)386 1040 y(that)f(the)g(knots)h(are)f(orien)m(ted)g(as)g(in)f(Figure)g(3.) 47 b(The)35 b(Legendrian)f(connected)386 1156 y(sum)e Fo(K)675 1171 y Fn(1)715 1156 y Fr(#)p Fo(K)879 1171 y Fn(2)951 1156 y Fr(is)g(then)h(formed)f(using)g(the)h(dashed)h(curv)m (es)h(in)d(Figure)f(3.)559 2048 y @beginspecial 0 @llx 0 @lly 317 @urx 90 @ury 3170 @rwi @setspecial %%BeginDocument: artconfront.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: confrontaom.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Wed Sep 1 14:15:10 2004 %%For: tvogel@math81 (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 317 90 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -4.0 93.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 2545 m -1000 -1000 l 6337 -1000 l 6337 2545 l cp clip 0.06000 0.06000 sc % Polyline 7.500 slw n 895 223 m 989 349 l 842 336 l gs col0 s gr % Polyline n 4681 336 m 4869 336 l 4788 475 l gs col0 s gr % Polyline n 5204 1394 m 5097 1281 l 5191 1255 l gs col0 s gr % Polyline n 775 1167 m 694 1293 l 801 1331 l gs col0 s gr % Polyline gs clippath 209 217 m 239 97 l 269 217 l 269 55 l 209 55 l cp clip n 239 70 m 239 1507 l gs col0 s gr gr % arrowhead n 209 217 m 239 97 l 269 217 l col0 s % Polyline gs clippath 841 741 m 961 771 l 841 801 l 1003 801 l 1003 741 l cp clip n 90 771 m 988 771 l gs col0 s gr gr % arrowhead n 841 741 m 961 771 l 841 801 l col0 s % Polyline n 538 210 m 541 211 l 548 212 l 560 214 l 577 218 l 598 223 l 624 228 l 652 234 l 681 240 l 709 246 l 736 253 l 760 258 l 783 264 l 803 269 l 821 275 l 838 280 l 856 286 l 874 293 l 891 300 l 907 307 l 923 314 l 938 320 l 954 328 l 969 335 l 984 342 l 999 350 l 1015 358 l 1030 366 l 1046 375 l 1062 385 l 1076 394 l 1090 404 l 1104 414 l 1119 425 l 1133 437 l 1148 448 l 1163 461 l 1177 473 l 1192 485 l 1207 498 l 1221 510 l 1235 521 l 1248 532 l 1261 542 l 1274 552 l 1286 561 l 1301 572 l 1315 583 l 1329 592 l 1342 601 l 1354 610 l 1366 618 l 1378 626 l 1390 634 l 1402 642 l 1413 650 l 1425 658 l 1436 665 l 1449 673 l 1460 681 l 1471 689 l 1481 697 l 1490 704 l 1500 711 l 1510 718 l 1521 724 l 1534 730 l 1548 736 l 1561 740 l 1577 744 l 1595 748 l 1616 752 l 1639 756 l 1663 760 l 1687 764 l 1707 767 l 1722 769 l 1731 770 l 1735 771 l 1736 771 l gs col0 s gr % Polyline n 538 1332 m 541 1331 l 548 1330 l 560 1328 l 577 1324 l 598 1319 l 624 1314 l 652 1308 l 681 1302 l 709 1296 l 736 1290 l 760 1284 l 783 1278 l 803 1273 l 821 1267 l 838 1262 l 856 1256 l 874 1249 l 891 1242 l 907 1235 l 923 1228 l 938 1221 l 954 1214 l 969 1207 l 984 1199 l 999 1191 l 1015 1183 l 1030 1175 l 1046 1166 l 1062 1156 l 1076 1147 l 1090 1137 l 1104 1127 l 1119 1116 l 1133 1105 l 1148 1093 l 1163 1081 l 1177 1068 l 1192 1056 l 1207 1044 l 1221 1032 l 1235 1021 l 1248 1010 l 1261 1000 l 1274 990 l 1286 981 l 1301 970 l 1315 959 l 1329 950 l 1342 941 l 1354 932 l 1366 924 l 1378 915 l 1390 907 l 1402 899 l 1413 891 l 1425 883 l 1436 876 l 1449 868 l 1460 860 l 1471 852 l 1481 845 l 1490 838 l 1500 831 l 1510 824 l 1521 818 l 1534 812 l 1548 806 l 1561 802 l 1577 798 l 1595 794 l 1616 790 l 1639 786 l 1663 782 l 1687 778 l 1707 775 l 1722 773 l 1731 772 l 1735 771 l 1736 771 l gs col0 s gr % Polyline n 5325 210 m 5322 211 l 5315 212 l 5304 214 l 5287 218 l 5265 223 l 5239 228 l 5212 234 l 5183 240 l 5155 246 l 5128 253 l 5103 258 l 5081 264 l 5061 269 l 5043 275 l 5026 280 l 5008 286 l 4990 293 l 4973 300 l 4957 307 l 4941 314 l 4926 320 l 4911 328 l 4895 335 l 4880 342 l 4865 350 l 4850 358 l 4834 366 l 4818 375 l 4802 385 l 4788 394 l 4774 404 l 4760 414 l 4745 425 l 4730 437 l 4716 448 l 4701 461 l 4686 473 l 4671 485 l 4657 498 l 4642 510 l 4628 521 l 4615 532 l 4602 542 l 4589 552 l 4577 561 l 4562 572 l 4548 583 l 4534 592 l 4521 601 l 4509 610 l 4497 618 l 4485 626 l 4473 634 l 4461 642 l 4450 650 l 4438 658 l 4427 665 l 4414 673 l 4403 681 l 4392 689 l 4383 697 l 4373 704 l 4364 711 l 4354 718 l 4343 724 l 4330 730 l 4316 736 l 4303 740 l 4288 744 l 4270 748 l 4249 752 l 4226 756 l 4201 760 l 4178 764 l 4158 767 l 4143 769 l 4134 770 l 4130 771 l 4129 771 l gs col0 s gr % Polyline n 5325 1332 m 5322 1331 l 5315 1330 l 5304 1328 l 5287 1324 l 5265 1319 l 5239 1314 l 5212 1308 l 5183 1302 l 5155 1296 l 5128 1290 l 5103 1284 l 5081 1278 l 5061 1273 l 5043 1267 l 5026 1262 l 5008 1256 l 4990 1249 l 4973 1242 l 4957 1235 l 4941 1228 l 4926 1221 l 4911 1214 l 4895 1207 l 4880 1199 l 4865 1191 l 4850 1183 l 4834 1175 l 4818 1166 l 4802 1156 l 4788 1147 l 4774 1137 l 4760 1127 l 4745 1116 l 4730 1105 l 4716 1093 l 4701 1081 l 4686 1068 l 4671 1056 l 4657 1044 l 4642 1032 l 4628 1021 l 4615 1010 l 4602 1000 l 4589 990 l 4577 981 l 4562 970 l 4548 959 l 4534 950 l 4521 941 l 4509 932 l 4497 924 l 4485 915 l 4473 907 l 4461 899 l 4450 891 l 4438 883 l 4427 876 l 4414 868 l 4403 860 l 4392 852 l 4383 845 l 4373 838 l 4364 831 l 4354 824 l 4343 818 l 4330 812 l 4316 806 l 4303 802 l 4288 798 l 4270 794 l 4249 790 l 4226 786 l 4201 782 l 4178 778 l 4158 775 l 4143 773 l 4134 772 l 4130 771 l 4129 771 l gs col0 s gr % Polyline [60] 0 sd n 988 1192 m 990 1191 l 995 1189 l 1003 1184 l 1015 1178 l 1031 1170 l 1051 1160 l 1073 1149 l 1097 1137 l 1122 1125 l 1147 1113 l 1171 1102 l 1193 1091 l 1214 1081 l 1234 1073 l 1252 1065 l 1269 1058 l 1286 1052 l 1305 1046 l 1323 1040 l 1341 1034 l 1358 1029 l 1375 1025 l 1392 1020 l 1408 1016 l 1425 1012 l 1442 1008 l 1459 1004 l 1477 1000 l 1495 996 l 1515 992 l 1537 988 l 1560 984 l 1585 981 l 1604 979 l 1624 976 l 1645 974 l 1666 973 l 1687 971 l 1709 969 l 1731 967 l 1754 966 l 1776 965 l 1799 963 l 1822 962 l 1846 960 l 1870 959 l 1894 958 l 1918 956 l 1943 955 l 1969 954 l 1995 952 l 2022 951 l 2050 949 l 2079 948 l 2109 946 l 2133 945 l 2159 943 l 2185 942 l 2211 941 l 2239 939 l 2267 938 l 2296 937 l 2325 935 l 2355 933 l 2386 932 l 2416 930 l 2447 929 l 2479 927 l 2510 925 l 2542 924 l 2573 922 l 2604 920 l 2636 919 l 2667 918 l 2698 916 l 2728 915 l 2758 914 l 2788 913 l 2818 912 l 2847 912 l 2875 911 l 2904 911 l 2932 911 l 2962 911 l 2992 911 l 3022 912 l 3051 913 l 3081 913 l 3110 914 l 3140 915 l 3169 916 l 3198 918 l 3227 919 l 3256 921 l 3285 922 l 3314 924 l 3343 925 l 3372 927 l 3401 928 l 3429 930 l 3458 932 l 3486 933 l 3514 935 l 3542 937 l 3570 939 l 3598 940 l 3625 942 l 3652 944 l 3679 946 l 3708 948 l 3736 951 l 3765 953 l 3793 956 l 3821 959 l 3850 962 l 3879 965 l 3908 968 l 3937 971 l 3966 974 l 3995 978 l 4024 981 l 4052 984 l 4081 988 l 4108 991 l 4136 994 l 4162 997 l 4188 1000 l 4212 1003 l 4236 1006 l 4258 1009 l 4279 1011 l 4298 1014 l 4316 1016 l 4347 1020 l 4372 1024 l 4390 1026 l 4404 1028 l 4414 1030 l 4422 1031 l 4429 1032 l 4438 1034 l 4449 1036 l 4463 1040 l 4481 1045 l 4503 1052 l 4519 1057 l 4537 1063 l 4558 1071 l 4581 1079 l 4607 1088 l 4635 1099 l 4666 1111 l 4698 1123 l 4731 1135 l 4762 1148 l 4792 1159 l 4819 1169 l 4840 1178 l 4857 1184 l 4868 1188 l 4874 1191 l 4877 1192 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 988 350 m 990 351 l 995 353 l 1003 358 l 1015 364 l 1031 372 l 1051 382 l 1073 393 l 1097 405 l 1122 417 l 1147 429 l 1171 440 l 1193 451 l 1214 461 l 1234 469 l 1252 477 l 1269 484 l 1286 490 l 1305 496 l 1323 502 l 1341 508 l 1358 513 l 1375 517 l 1392 522 l 1408 526 l 1425 530 l 1442 534 l 1459 538 l 1477 542 l 1495 546 l 1515 550 l 1537 554 l 1560 558 l 1585 561 l 1604 563 l 1624 566 l 1645 568 l 1666 569 l 1687 571 l 1709 573 l 1731 575 l 1754 576 l 1776 577 l 1799 579 l 1822 580 l 1846 582 l 1870 583 l 1894 584 l 1918 586 l 1943 587 l 1969 588 l 1995 590 l 2022 591 l 2050 593 l 2079 594 l 2109 596 l 2133 597 l 2159 599 l 2185 600 l 2211 601 l 2239 603 l 2267 604 l 2296 605 l 2325 607 l 2355 609 l 2386 610 l 2416 612 l 2447 613 l 2479 615 l 2510 617 l 2542 618 l 2573 620 l 2604 622 l 2636 623 l 2667 624 l 2698 626 l 2728 627 l 2758 628 l 2788 629 l 2818 630 l 2847 630 l 2875 631 l 2904 631 l 2932 631 l 2962 631 l 2992 631 l 3022 630 l 3051 629 l 3081 629 l 3110 628 l 3140 627 l 3169 626 l 3198 624 l 3227 623 l 3256 621 l 3285 620 l 3314 618 l 3343 617 l 3372 615 l 3401 614 l 3429 612 l 3458 610 l 3486 609 l 3514 607 l 3542 605 l 3570 604 l 3598 602 l 3625 600 l 3652 598 l 3679 596 l 3708 594 l 3736 591 l 3765 589 l 3793 586 l 3821 583 l 3850 580 l 3879 577 l 3908 574 l 3937 571 l 3966 567 l 3995 564 l 4024 560 l 4052 557 l 4081 554 l 4108 550 l 4136 547 l 4162 544 l 4188 541 l 4212 538 l 4236 535 l 4258 532 l 4279 530 l 4298 527 l 4316 525 l 4347 521 l 4372 518 l 4390 515 l 4404 513 l 4414 512 l 4422 511 l 4429 510 l 4438 508 l 4449 505 l 4463 502 l 4481 497 l 4503 490 l 4519 485 l 4537 479 l 4558 471 l 4581 463 l 4607 454 l 4635 443 l 4666 431 l 4698 419 l 4731 407 l 4762 394 l 4792 383 l 4819 373 l 4840 364 l 4857 358 l 4868 354 l 4874 351 l 4877 350 l gs col0 s gr [] 0 sd /Times-Roman ff 165.00 scf sf 761 979 m gs 1 -1 sc (t) col0 sh gr /Times-Roman ff 165.00 scf sf 828 1482 m gs 1 -1 sc (K) col0 sh gr /Times-Roman ff 165.00 scf sf 963 1545 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 165.00 scf sf 4788 1419 m gs 1 -1 sc (K) col0 sh gr /Times-Roman ff 165.00 scf sf 4922 1482 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 165.00 scf sf 3916 790 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 165.00 scf sf 3983 853 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 165.00 scf sf 1902 790 m gs 1 -1 sc (p) col0 sh gr /Times-Roman ff 165.00 scf sf 1969 853 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 165.00 scf sf 291 349 m gs 1 -1 sc (z) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 1623 2314 a Fs(Figure)37 b(3.)486 2484 y Fr(No)m(w)j(consider)f(Seifert)h(surfaces)g(\006)1858 2499 y Fn(1)1938 2484 y Fr(of)f Fo(K)2139 2499 y Fn(1)2218 2484 y Fr(and)g(\006)2484 2499 y Fn(2)2564 2484 y Fr(of)g Fo(K)2765 2499 y Fn(2)2804 2484 y Fr(.)64 b(W)-8 b(e)40 b(assume)386 2600 y(that)h(\006)676 2615 y Fn(1)716 2600 y Fr(,)i(resp)s(ectiv)m(ely)g(\006)1397 2615 y Fn(2)1437 2600 y Fr(,)g(coincides)e(with)g(translates)g(of)g Fo(K)2816 2615 y Fn(1)2897 2600 y Fr(in)f(the)i(neg-)386 2717 y(ativ)m(e)f Fo(t)p Fr({direction,)h(resp)s(ectiv)m(ely)h(of)d Fo(K)1904 2732 y Fn(2)1985 2717 y Fr(in)g(the)i(p)s(ositiv)m(e)e Fo(t)p Fr({direction,)j(on)e(a)386 2833 y(neigh)m(b)s(orho)s(o)s(d)31 b(of)h Fo(p)1158 2848 y Fn(1)1198 2833 y Fr(,)h(resp)s(ectiv)m(ely)g Fo(p)1838 2848 y Fn(2)1878 2833 y Fr(.)43 b(Suc)m(h)34 b(neigh)m(b)s(orho)s(o)s(ds)e(are)g(sho)m(wn)i(in)386 2949 y(Figure)40 b(3.)67 b(If)41 b(w)m(e)g(orien)m(t)g(\006)1460 2964 y Fn(1)1540 2949 y Fr(and)g(\006)1808 2964 y Fn(2)1888 2949 y Fr(suc)m(h)i(that)d Fo(K)2419 2964 y Fn(1)2499 2949 y Fr(and)h Fo(K)2780 2964 y Fn(2)2860 2949 y Fr(are)g(orien)m(ted) 386 3065 y(as)j(b)s(oundaries,)i(then)f Fo(p)1337 3080 y Fn(1)1420 3065 y Fr(is)e(a)h(negativ)m(e)g(singularit)m(y)e(and)h Fo(p)2761 3080 y Fn(2)2845 3065 y Fr(is)g(a)g(p)s(ositiv)m(e)386 3182 y(singularit)m(y)-8 b(.)486 3298 y(W)g(e)25 b(use)h(the)f(ribb)s (on)f(whic)m(h)h(is)g(b)s(ounded)g(b)m(y)h(the)f(dashed)h(curv)m(es)h (in)d(Figure)g(3)386 3414 y(to)34 b(construct)h(a)g(Seifert)f(surface)h (\006)1729 3429 y Fn(1)1769 3414 y Fr(#\006)1920 3429 y Fn(2)1994 3414 y Fr(for)f(the)h(knot)g Fo(K)2625 3429 y Fn(1)2664 3414 y Fr(#)p Fo(K)2828 3429 y Fn(2)2868 3414 y Fr(.)49 b(According)386 3530 y(to)32 b([EH2)q(])828 3687 y(tb\()p Fo(K)1041 3702 y Fn(1)1081 3687 y Fr(#)p Fo(K)1245 3702 y Fn(2)1285 3687 y Fo(;)17 b Fr([\006)1426 3702 y Fn(1)1465 3687 y Fr(#\006)1616 3702 y Fn(2)1656 3687 y Fr(]\))28 b(=)g(tb\()p Fo(K)2066 3702 y Fn(1)2105 3687 y Fo(;)17 b Fr([\006)2246 3702 y Fn(1)2286 3687 y Fr(]\))22 b(+)g(tb\()p Fo(K)2684 3702 y Fn(2)2724 3687 y Fo(;)17 b Fr([\006)2865 3702 y Fn(2)2904 3687 y Fr(]\))22 b(+)g(1)-2752 b(\(14\))796 3838 y(rot)o(\()p Fo(K)1041 3853 y Fn(1)1081 3838 y Fr(#)p Fo(K)1245 3853 y Fn(2)1285 3838 y Fo(;)17 b Fr([\006)1426 3853 y Fn(1)1465 3838 y Fr(#\006)1616 3853 y Fn(2)1656 3838 y Fr(]\))28 b(=)g(rot)o(\()p Fo(K)2098 3853 y Fn(1)2138 3838 y Fo(;)17 b Fr([\006)2279 3853 y Fn(1)2318 3838 y Fr(]\))23 b(+)f(rot)o(\()p Fo(K)2749 3853 y Fn(2)2789 3838 y Fo(;)17 b Fr([\006)2930 3853 y Fn(2)2969 3838 y Fr(]\))33 b Fo(:)-2708 b Fr(\(15\))386 3995 y(No)m(w)43 b(w)m(e)g(apply)e(this)h(to)g(the)g(Seifert)g (surfaces)h Fo(R)i Fr(=)f(\006)2555 4010 y Fn(1)2637 3995 y Fr(of)e Fo(@)5 b(R)45 b Fr(=)f Fo(K)3136 4010 y Fn(1)3218 3995 y Fr(and)386 4112 y Fo(D)467 4127 y Fl(ot)563 4112 y Fr(=)33 b(\006)742 4127 y Fn(2)817 4112 y Fr(of)i Fo(@)5 b(D)1068 4127 y Fl(ot)1166 4112 y Fr(=)32 b Fo(K)1357 4127 y Fn(2)1432 4112 y Fr(where)37 b Fo(D)1798 4127 y Fl(ot)1897 4112 y Fr(is)e(orien)m(ted)h(suc)m(h)h(that)e(tb\()p Fo(@)5 b(D)3082 4127 y Fl(ot)3147 4112 y Fr(\))32 b(=)h(0)386 4228 y(and)27 b(rot)o(\()p Fo(@)5 b(D)869 4243 y Fl(ot)934 4228 y Fr(\))27 b(=)h Fp(\000)p Fr(1.)42 b(Then)27 b(w)m(e)h(p)s (erturb)f Fo(R)q Fr(#)p Fo(D)2270 4243 y Fl(ot)2360 4228 y Fr(to)g(a)f(con)m(v)m(ex)j(surface)e(with)386 4344 y(b)s(oundary)33 b Fo(@)5 b(R)q Fr(#)p Fo(@)g(D)1174 4359 y Fl(ot)1240 4344 y Fr(.)486 4460 y(The)40 b(di\016cult)m(y)f(in)f (sho)m(wing)i(that)f(the)g(p)s(erturb)s(ed)h(surface)g(is)f(a)g(b)m (ypass)i(is)386 4577 y(to)d(establish)g(\(v\))h(of)f(De\014nition)f (3.20.)61 b(In)38 b(order)h(to)f(do)h(this)f(w)m(e)h(reduce)h(the)386 4693 y(region)26 b(where)j(the)f(Legendrian)f(connected)i(sum)e(is)g(p) s(erformed)g(suc)m(h)i(that)e(the)386 4809 y(con)m(tact)33 b(structure)h(on)e(this)h(region)e(is)h(tigh)m(t.)486 4925 y(By)27 b(Theorem)g(3.16)f(and)h(Theorem)g(3.13)f(w)m(e)h(can)g (assume)h(that)e(the)h(singular)386 5042 y(foliation)34 b(on)k Fo(D)995 5057 y Fl(ot)1096 5042 y Fr(is)g(of)f(the)h(form)f (indicated)f(in)h(Figure)g(4)h(where)h(the)f(thic)m(k-)386 5158 y(ened)h(circle)e(is)h(the)h(dividing)d(set.)61 b(Then)39 b(w)m(e)h(can)e(decomp)s(ose)h Fo(D)2918 5173 y Fl(ot)3020 5158 y Fr(in)m(to)e(t)m(w)m(o)386 5274 y(half{discs)e(b)s (ounded)i(b)m(y)f(Legendrian)g(unknots)h(with)e(Th)m(urston{Bennequin) 386 5390 y(in)m(v)-5 b(arian)m(t)34 b Fp(\000)p Fr(1)i(and)f(rotation)f (n)m(um)m(b)s(er)i(0.)52 b(The)36 b(half{discs)f(are)g(separated)i(b)m (y)386 5506 y(straigh)m(t)31 b(Legendrian)h(arcs.)44 b(The)33 b(singular)e(foliation)e(near)j(the)g(unknots)i(is)d(in)386 5623 y(the)i(standard)g(form)e(used)j(in)e(Prop)s(osition)f(3.7.)p eop %%Page: 22 22 22 21 bop 386 259 a Fq(22)1096 b(THOMAS)25 b(V)n(OGEL)1314 1442 y @beginspecial 0 @llx 0 @lly 136 @urx 122 @ury 1360 @rwi @setspecial %%BeginDocument: artotdisc.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: otdiscaom.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Wed Sep 1 14:49:31 2004 %%For: tvogel@math81 (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 136 122 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -2.0 124.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /endash 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 3061 m -1000 -1000 l 3298 -1000 l 3298 3061 l cp clip 0.06000 0.06000 sc 30.000 slw % Ellipse n 1280 1079 536 535 0 360 DrawEllipse gs col0 s gr 7.500 slw % Ellipse n 1273 1053 1003 1002 0 360 DrawEllipse gs col0 s gr % Polyline n 1273 93 m 1273 1015 l gs col0 s gr % Polyline n 1358 104 m 1358 1015 l gs col0 s gr % Polyline n 1437 93 m 1445 93 l gs col0 s gr % Polyline n 1437 93 m 1445 93 l gs col0 s gr % Polyline n 1441 118 m 1437 1013 l gs col0 s gr % Polyline n 1189 104 m 1189 1015 l gs col0 s gr % Polyline n 1107 113 m 1107 1015 l gs col0 s gr % Polyline n 1441 2001 m 1441 1098 l gs col0 s gr % Polyline n 1107 2001 m 1107 1098 l gs col0 s gr % Polyline n 1273 2017 m 1273 1095 l gs col0 s gr % Polyline n 1191 2013 m 1189 1095 l gs col0 s gr % Polyline n 1358 2006 m 1358 1095 l gs col0 s gr % Polyline [15 45] 45 sd n 1040 1053 m 1487 1057 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 276 1049 m 383 1049 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 2171 1050 m 2271 1049 l gs col0 s gr [] 0 sd % Polyline n 1081 1015 m 1081 1013 l 1081 1008 l 1081 999 l 1081 986 l 1081 968 l 1081 946 l 1081 921 l 1081 892 l 1081 862 l 1081 831 l 1082 800 l 1082 771 l 1082 743 l 1082 717 l 1081 693 l 1081 671 l 1081 651 l 1081 632 l 1081 615 l 1081 594 l 1080 575 l 1080 556 l 1079 537 l 1079 519 l 1078 501 l 1078 483 l 1077 466 l 1076 448 l 1076 432 l 1075 415 l 1075 399 l 1074 384 l 1074 369 l 1073 355 l 1073 342 l 1073 324 l 1073 308 l 1073 294 l 1074 282 l 1075 271 l 1075 261 l 1076 250 l 1075 240 l 1075 229 l 1073 218 l 1070 205 l 1067 193 l 1064 180 l 1060 168 l 1056 156 l 1051 145 l 1043 138 l 1032 133 l 1022 132 l 1011 133 l 999 134 l 985 137 l 970 141 l 955 145 l 939 149 l 924 154 l 909 159 l 894 164 l 879 170 l 864 176 l 851 182 l 838 189 l 825 197 l 812 204 l 799 212 l 786 220 l 773 228 l 760 236 l 746 245 l 733 254 l 719 264 l 705 275 l 691 287 l 676 301 l 664 313 l 652 325 l 640 339 l 627 353 l 613 368 l 599 384 l 585 400 l 571 417 l 556 434 l 542 451 l 528 468 l 514 485 l 501 501 l 489 517 l 477 533 l 466 548 l 455 563 l 446 577 l 435 595 l 426 612 l 417 629 l 410 646 l 403 663 l 397 679 l 392 695 l 386 711 l 381 727 l 376 743 l 370 759 l 365 775 l 360 791 l 355 807 l 350 823 l 346 839 l 342 857 l 338 876 l 334 897 l 330 920 l 326 942 l 322 964 l 319 983 l 316 998 l 315 1009 l 314 1016 l 313 1019 l gs col0 s gr % Polyline n 1043 1015 m 1043 1012 l 1043 1006 l 1043 995 l 1043 979 l 1044 957 l 1044 931 l 1044 902 l 1044 870 l 1045 837 l 1045 804 l 1045 773 l 1045 743 l 1045 716 l 1045 690 l 1044 666 l 1044 644 l 1043 623 l 1042 602 l 1041 581 l 1040 560 l 1039 538 l 1038 516 l 1037 493 l 1036 470 l 1035 447 l 1033 425 l 1032 402 l 1031 380 l 1029 359 l 1027 338 l 1026 319 l 1024 301 l 1021 285 l 1019 271 l 1016 258 l 1011 243 l 1006 231 l 1000 223 l 994 218 l 988 214 l 981 212 l 974 210 l 967 209 l 960 208 l 952 207 l 944 206 l 935 206 l 925 207 l 914 210 l 903 214 l 891 219 l 878 224 l 865 230 l 853 236 l 840 243 l 827 249 l 814 256 l 802 264 l 789 272 l 778 280 l 767 288 l 756 297 l 745 307 l 734 317 l 723 327 l 712 337 l 701 348 l 689 358 l 678 369 l 667 380 l 656 391 l 645 402 l 634 414 l 623 426 l 612 439 l 601 451 l 590 465 l 578 478 l 567 492 l 555 505 l 544 519 l 532 533 l 522 547 l 511 561 l 501 575 l 492 589 l 483 603 l 475 617 l 468 632 l 461 646 l 454 661 l 448 676 l 442 691 l 437 706 l 431 721 l 426 736 l 420 751 l 415 766 l 410 781 l 405 796 l 400 811 l 396 826 l 391 842 l 387 859 l 383 878 l 379 899 l 374 920 l 370 943 l 365 964 l 362 983 l 359 999 l 357 1010 l 356 1016 l 355 1019 l gs col0 s gr % Polyline n 1078 1103 m 1078 1106 l 1078 1112 l 1078 1124 l 1078 1140 l 1079 1163 l 1079 1190 l 1079 1220 l 1080 1253 l 1080 1286 l 1080 1320 l 1081 1352 l 1081 1381 l 1081 1409 l 1081 1433 l 1081 1456 l 1081 1476 l 1081 1495 l 1081 1515 l 1081 1535 l 1080 1554 l 1080 1572 l 1079 1590 l 1078 1608 l 1078 1625 l 1077 1642 l 1076 1659 l 1076 1676 l 1075 1692 l 1075 1708 l 1074 1724 l 1074 1738 l 1073 1752 l 1073 1765 l 1073 1783 l 1073 1799 l 1073 1813 l 1074 1825 l 1075 1837 l 1075 1848 l 1076 1858 l 1075 1869 l 1075 1880 l 1073 1891 l 1070 1904 l 1067 1916 l 1064 1928 l 1060 1941 l 1056 1952 l 1051 1963 l 1043 1970 l 1032 1975 l 1022 1976 l 1011 1975 l 999 1974 l 985 1971 l 970 1968 l 955 1964 l 939 1960 l 924 1956 l 909 1951 l 894 1946 l 879 1940 l 864 1934 l 851 1928 l 838 1921 l 825 1913 l 811 1904 l 797 1895 l 783 1886 l 769 1877 l 755 1867 l 741 1857 l 727 1848 l 714 1838 l 701 1828 l 688 1819 l 676 1809 l 664 1799 l 653 1789 l 642 1779 l 631 1768 l 620 1758 l 609 1747 l 599 1736 l 588 1725 l 578 1714 l 568 1704 l 558 1693 l 548 1682 l 538 1671 l 529 1660 l 519 1647 l 509 1634 l 499 1622 l 490 1609 l 480 1596 l 471 1584 l 462 1571 l 454 1558 l 445 1545 l 436 1531 l 428 1517 l 420 1502 l 413 1489 l 407 1475 l 401 1461 l 395 1447 l 389 1433 l 383 1418 l 377 1403 l 371 1388 l 365 1373 l 360 1358 l 355 1344 l 350 1329 l 345 1314 l 341 1299 l 337 1284 l 334 1268 l 331 1250 l 328 1231 l 325 1210 l 322 1187 l 319 1164 l 316 1142 l 314 1122 l 312 1106 l 311 1095 l 310 1088 l 310 1085 l gs col0 s gr % Polyline n 1472 1098 m 1472 1101 l 1472 1107 l 1472 1118 l 1472 1135 l 1472 1157 l 1472 1183 l 1472 1213 l 1472 1245 l 1472 1278 l 1472 1311 l 1472 1342 l 1472 1372 l 1472 1399 l 1472 1423 l 1473 1445 l 1473 1465 l 1473 1484 l 1473 1504 l 1474 1524 l 1474 1542 l 1475 1561 l 1475 1579 l 1476 1597 l 1476 1614 l 1477 1632 l 1478 1649 l 1478 1666 l 1479 1682 l 1479 1698 l 1480 1713 l 1480 1728 l 1481 1742 l 1481 1755 l 1481 1773 l 1481 1789 l 1481 1803 l 1480 1815 l 1479 1826 l 1479 1837 l 1478 1848 l 1478 1858 l 1479 1869 l 1481 1880 l 1484 1893 l 1487 1905 l 1491 1918 l 1494 1930 l 1498 1942 l 1504 1952 l 1512 1960 l 1523 1965 l 1533 1966 l 1544 1966 l 1556 1964 l 1570 1961 l 1584 1958 l 1599 1954 l 1614 1950 l 1630 1946 l 1645 1941 l 1660 1936 l 1674 1930 l 1689 1924 l 1702 1918 l 1715 1910 l 1728 1903 l 1742 1894 l 1756 1885 l 1770 1876 l 1785 1866 l 1799 1856 l 1813 1847 l 1827 1837 l 1840 1827 l 1853 1818 l 1866 1808 l 1878 1798 l 1890 1788 l 1901 1778 l 1912 1767 l 1922 1756 l 1933 1746 l 1943 1735 l 1953 1724 l 1963 1713 l 1973 1701 l 1983 1690 l 1993 1679 l 2002 1668 l 2012 1657 l 2021 1646 l 2030 1635 l 2039 1624 l 2047 1612 l 2056 1601 l 2064 1590 l 2072 1578 l 2081 1567 l 2089 1556 l 2097 1544 l 2105 1533 l 2112 1521 l 2120 1509 l 2126 1497 l 2133 1485 l 2140 1470 l 2146 1456 l 2152 1441 l 2158 1426 l 2163 1411 l 2168 1396 l 2173 1380 l 2178 1365 l 2183 1349 l 2188 1334 l 2192 1317 l 2197 1301 l 2201 1286 l 2204 1271 l 2208 1253 l 2212 1234 l 2217 1212 l 2221 1189 l 2225 1166 l 2229 1143 l 2233 1123 l 2236 1107 l 2238 1095 l 2239 1088 l 2240 1085 l gs col0 s gr % Polyline n 1468 1016 m 1468 1014 l 1468 1009 l 1468 1000 l 1468 987 l 1468 969 l 1469 947 l 1469 921 l 1469 893 l 1470 863 l 1470 832 l 1470 801 l 1471 771 l 1471 743 l 1471 717 l 1472 693 l 1472 671 l 1472 651 l 1473 632 l 1473 615 l 1473 594 l 1474 575 l 1474 555 l 1475 537 l 1476 519 l 1476 501 l 1477 483 l 1478 466 l 1478 448 l 1479 432 l 1480 415 l 1480 399 l 1481 384 l 1481 369 l 1482 355 l 1482 342 l 1482 324 l 1482 308 l 1482 294 l 1481 282 l 1480 271 l 1480 261 l 1479 250 l 1480 240 l 1480 229 l 1482 218 l 1485 205 l 1488 193 l 1491 180 l 1495 168 l 1499 156 l 1504 145 l 1512 138 l 1523 133 l 1533 132 l 1544 133 l 1556 134 l 1570 137 l 1585 141 l 1600 145 l 1616 149 l 1631 154 l 1646 159 l 1661 164 l 1676 170 l 1691 176 l 1704 182 l 1717 190 l 1731 197 l 1744 206 l 1758 215 l 1773 224 l 1787 233 l 1801 243 l 1815 253 l 1829 262 l 1843 272 l 1856 282 l 1868 291 l 1880 301 l 1891 311 l 1902 321 l 1913 331 l 1923 342 l 1933 352 l 1942 363 l 1952 374 l 1961 384 l 1970 395 l 1980 406 l 1989 417 l 1998 428 l 2007 440 l 2016 451 l 2025 463 l 2033 475 l 2041 487 l 2050 500 l 2058 512 l 2066 525 l 2074 538 l 2082 551 l 2090 564 l 2098 577 l 2105 590 l 2112 602 l 2119 614 l 2126 626 l 2133 640 l 2140 653 l 2146 665 l 2152 677 l 2158 689 l 2164 701 l 2169 712 l 2175 725 l 2180 737 l 2186 751 l 2191 765 l 2196 781 l 2200 795 l 2205 812 l 2209 830 l 2214 851 l 2219 874 l 2224 899 l 2229 925 l 2234 950 l 2239 973 l 2242 991 l 2245 1004 l 2246 1011 l 2247 1015 l gs col0 s gr % Polyline n 1500 1016 m 1500 1013 l 1500 1007 l 1500 997 l 1501 981 l 1501 960 l 1501 935 l 1502 906 l 1502 875 l 1503 843 l 1503 811 l 1504 781 l 1505 752 l 1505 725 l 1506 700 l 1506 676 l 1507 654 l 1508 633 l 1509 612 l 1510 591 l 1511 569 l 1512 547 l 1512 524 l 1513 501 l 1514 477 l 1515 454 l 1515 430 l 1516 406 l 1517 384 l 1518 361 l 1519 340 l 1521 321 l 1523 302 l 1525 286 l 1527 271 l 1530 258 l 1534 246 l 1538 236 l 1542 229 l 1547 223 l 1552 219 l 1557 216 l 1562 214 l 1568 212 l 1573 211 l 1579 211 l 1585 210 l 1591 210 l 1597 209 l 1603 208 l 1610 208 l 1617 208 l 1627 209 l 1638 212 l 1649 216 l 1661 220 l 1673 225 l 1686 231 l 1698 236 l 1711 243 l 1723 249 l 1736 257 l 1749 265 l 1762 274 l 1773 283 l 1785 293 l 1797 304 l 1810 316 l 1823 329 l 1835 342 l 1848 355 l 1861 368 l 1874 382 l 1887 395 l 1899 409 l 1911 422 l 1923 435 l 1934 448 l 1945 461 l 1956 475 l 1966 489 l 1976 503 l 1987 517 l 1997 532 l 2007 546 l 2017 561 l 2027 575 l 2037 589 l 2046 604 l 2055 617 l 2063 631 l 2071 644 l 2079 659 l 2087 674 l 2094 688 l 2100 702 l 2106 715 l 2111 729 l 2116 742 l 2122 756 l 2127 770 l 2132 784 l 2138 798 l 2143 813 l 2148 828 l 2153 845 l 2158 863 l 2164 884 l 2170 907 l 2176 930 l 2182 952 l 2187 972 l 2191 986 l 2193 995 l 2194 998 l 2194 999 l gs col0 s gr % Polyline n 1500 1101 m 1500 1104 l 1500 1110 l 1500 1120 l 1501 1136 l 1501 1157 l 1501 1182 l 1502 1210 l 1503 1241 l 1503 1273 l 1504 1304 l 1504 1335 l 1505 1363 l 1506 1390 l 1506 1415 l 1507 1438 l 1507 1460 l 1508 1481 l 1509 1502 l 1509 1523 l 1510 1544 l 1510 1566 l 1511 1588 l 1511 1611 l 1511 1634 l 1511 1657 l 1511 1681 l 1511 1704 l 1512 1726 l 1512 1748 l 1513 1768 l 1514 1788 l 1516 1806 l 1518 1822 l 1520 1836 l 1523 1849 l 1528 1862 l 1533 1873 l 1539 1880 l 1546 1886 l 1553 1890 l 1561 1892 l 1568 1894 l 1576 1895 l 1584 1895 l 1593 1896 l 1601 1896 l 1609 1897 l 1617 1897 l 1626 1897 l 1637 1895 l 1648 1892 l 1659 1888 l 1671 1882 l 1683 1876 l 1695 1870 l 1708 1863 l 1720 1856 l 1733 1849 l 1745 1841 l 1757 1833 l 1770 1825 l 1781 1817 l 1792 1808 l 1803 1799 l 1815 1790 l 1826 1780 l 1838 1770 l 1849 1759 l 1861 1749 l 1873 1738 l 1884 1728 l 1895 1717 l 1907 1706 l 1917 1695 l 1928 1684 l 1938 1672 l 1949 1661 l 1959 1648 l 1969 1636 l 1979 1624 l 1990 1611 l 2000 1598 l 2010 1586 l 2020 1573 l 2029 1560 l 2039 1547 l 2048 1533 l 2056 1520 l 2064 1506 l 2071 1492 l 2078 1477 l 2085 1462 l 2091 1447 l 2097 1432 l 2103 1416 l 2108 1401 l 2114 1385 l 2119 1369 l 2124 1354 l 2129 1338 l 2135 1323 l 2139 1308 l 2144 1293 l 2148 1278 l 2152 1262 l 2157 1245 l 2161 1227 l 2165 1207 l 2170 1186 l 2174 1164 l 2178 1143 l 2182 1125 l 2185 1110 l 2187 1099 l 2188 1093 l 2189 1090 l gs col0 s gr % Polyline n 1043 1106 m 1043 1109 l 1043 1114 l 1043 1124 l 1044 1139 l 1044 1158 l 1045 1182 l 1045 1209 l 1045 1238 l 1046 1268 l 1046 1298 l 1047 1327 l 1047 1355 l 1047 1381 l 1047 1405 l 1046 1427 l 1046 1448 l 1045 1469 l 1044 1490 l 1043 1511 l 1042 1533 l 1040 1555 l 1039 1578 l 1038 1601 l 1036 1625 l 1034 1650 l 1033 1674 l 1031 1698 l 1029 1721 l 1028 1744 l 1026 1766 l 1023 1786 l 1021 1804 l 1018 1821 l 1015 1836 l 1012 1849 l 1007 1863 l 1002 1873 l 996 1881 l 990 1886 l 984 1890 l 978 1892 l 971 1893 l 965 1894 l 958 1894 l 951 1894 l 944 1895 l 936 1895 l 928 1895 l 920 1895 l 910 1893 l 899 1890 l 887 1887 l 875 1882 l 863 1877 l 850 1871 l 838 1865 l 825 1858 l 812 1851 l 799 1844 l 787 1837 l 774 1828 l 763 1820 l 752 1811 l 740 1801 l 729 1791 l 717 1781 l 705 1770 l 693 1759 l 681 1747 l 669 1736 l 658 1724 l 646 1713 l 634 1701 l 623 1689 l 612 1677 l 601 1664 l 590 1651 l 579 1638 l 568 1624 l 557 1610 l 546 1596 l 534 1582 l 523 1568 l 513 1554 l 502 1539 l 492 1525 l 482 1510 l 473 1496 l 464 1481 l 456 1466 l 448 1451 l 441 1435 l 434 1419 l 428 1403 l 422 1387 l 415 1370 l 409 1354 l 404 1338 l 398 1322 l 392 1306 l 387 1291 l 382 1276 l 378 1262 l 374 1246 l 370 1229 l 366 1212 l 363 1193 l 359 1172 l 356 1151 l 353 1132 l 351 1115 l 349 1102 l 348 1095 l 348 1092 l 348 1091 l gs col0 s gr /Times-Roman-iso ff 165.00 scf sf 631 1493 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 165.00 scf sf 1846 1493 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 165.00 scf sf 1576 1156 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 165.00 scf sf 833 1156 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 452 1100 m gs 1 -1 sc (D) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 48 1065 m gs 1 -1 sc (p) col0 sh gr /Times-Roman-iso ff 150.00 scf sf 111 1202 m gs 1 -1 sc (2) col0 sh gr /Times-Roman-iso ff 150.00 scf sf 2175 2026 m gs 1 -1 sc (ot) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 2063 1895 m gs 1 -1 sc (D) col0 sh gr /Times-Roman-iso ff 150.00 scf sf 566 1215 m gs 1 -1 sc (l) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 1623 1708 a Fs(Figure)37 b(4.)486 1933 y Fr(By)h(the)g(last)e(part)i(of)f(Prop)s(osition)f(3.7)h(w)m(e)h(can)g (pretend)h(that)e(w)m(e)i(form)d(a)386 2049 y(b)s(oundary)30 b(connected)i(sum)e(of)g(the)g(surfaces)h Fo(R)g Fr(with)f(the)g(left)g (part)f Fo(D)3065 2064 y Fl(l)3121 2049 y Fr(of)h Fo(D)3311 2064 y Fl(ot)386 2165 y Fr(and)45 b(that)f(the)i(p)s(erturbation)d(of)i Fo(R)q Fr(#)p Fo(D)1936 2180 y Fl(ot)2045 2165 y Fr(only)f(c)m(hanges)i (the)f(half{disc)f Fo(D)3322 2180 y Fl(l)3348 2165 y Fr(.)386 2282 y(The)c(presence)h(of)d(the)h(Legendrian)g(curv)m(es)i (in)d(the)h(middle)e(of)i Fo(D)2917 2297 y Fl(ot)3019 2282 y Fr(prev)m(en)m(ts)386 2398 y(an)32 b(in)m(teraction)g(b)s(et)m (w)m(een)j(the)e(left)e(and)i(the)g(righ)m(t)f(part)g(of)g Fo(D)2724 2413 y Fl(ot)2788 2398 y Fr(.)486 2514 y(The)48 b(union)e(of)h(tubular)f(neigh)m(b)s(orho)s(o)s(ds)g(of)h Fo(R)q Fr(,)k(of)46 b(the)i(Legendrian)f(arc)386 2630 y(connecting)33 b Fo(R)h Fr(with)f Fo(D)1286 2645 y Fl(ot)1382 2630 y Fr(and)g(of)g(the)g(left)f(part)h Fo(D)2316 2645 y Fl(l)2375 2630 y Fr(of)f Fo(D)2567 2645 y Fl(ot)2664 2630 y Fr(can)h(b)s(e)g(reco)m(v)m(ered)386 2747 y(in)d(tigh)m(t)f(con) m(tact)i(manifolds:)40 b Fo(D)1633 2762 y Fl(l)1689 2747 y Fr(can)31 b(b)s(e)g(obtained)e(applying)h(Theorem)g(3.13)386 2863 y(to)c(a)g(b)m(ypass)j(in)c(a)h(tigh)m(t)g(con)m(tact)h(manifold,) e(hence)j(a)e(neigh)m(b)s(orho)s(o)s(d)f(of)h Fo(R)q Fr(#)p Fo(D)3348 2878 y Fl(l)386 2979 y Fr(is)32 b(tigh)m(t.)486 3095 y(The)e(Th)m(urston{Bennequin)h(in)m(v)-5 b(arian)m(t)28 b(of)h Fo(@)5 b(R)q Fr(#)p Fo(@)g(D)2468 3110 y Fl(l)2525 3095 y Fr(is)29 b Fp(\000)p Fr(2)g(b)m(y)i(\(14\).)42 b(F)-8 b(rom)386 3211 y(\(13\))30 b(and)i(Theorem)f(3.19)f(it)g(follo)m (ws)g(that)h(the)h(dividing)d(set)j(on)f Fo(R)q Fr(#)p Fo(D)3082 3226 y Fl(l)3139 3211 y Fr(\(after)386 3328 y(this)21 b(surface)i(is)e(p)s(erturb)s(ed)i(to)e(a)h(con)m(v)m(ex)i (surface\))e(consists)h(of)e(exactly)i(t)m(w)m(o)f(arcs)386 3444 y(with)31 b(endp)s(oin)m(ts)h(on)g Fo(@)5 b(R)q Fr(#)p Fo(@)g(D)1535 3459 y Fl(l)1594 3444 y Fr(and)32 b(no)g(closed)f(comp)s(onen)m(ts.)44 b(The)33 b(notations)386 3560 y Fo(R)q Fr(#)p Fo(D)623 3575 y Fl(l)676 3560 y Fr(and)27 b Fo(R)q Fr(#)p Fo(D)1097 3575 y Fl(ot)1188 3560 y Fr(are)g(sligh)m(tly)f(misleading)e(b)s(ecause)29 b Fo(D)2596 3575 y Fl(l)2622 3560 y Fr(,)f(resp)s(ectiv)m(ely)g Fo(D)3284 3575 y Fl(ot)3348 3560 y Fr(,)386 3676 y(is)35 b Fm(not)h Fr(a)g(subset)h(of)e Fo(R)q Fr(#)p Fo(D)1398 3691 y Fl(l)1425 3676 y Fr(,)h(resp)s(ectiv)m(ely)h Fo(R)q Fr(#)p Fo(D)2260 3691 y Fl(ot)2324 3676 y Fr(,)g(after)e(these)j (surfaces)f(are)386 3793 y(smo)s(othed)32 b(and)h(made)f(con)m(v)m(ex.) 486 3909 y(W)-8 b(e)24 b(are)g(left)f(with)h(the)h(t)m(w)m(o)g(p)s (ossibilities)c(for)i(the)i(isotop)m(y)f(t)m(yp)s(e)h(of)f(the)g (divid-)386 4025 y(ing)h(set)i(of)e Fo(R)q Fr(#)p Fo(D)1029 4040 y Fl(ot)1119 4025 y Fr(sho)m(wn)i(in)f(Figure)f(5.)41 b(According)25 b(to)h(\(13\))f(the)i(b)s(oundary)f(of)526 4804 y @beginspecial 0 @llx 0 @lly 325 @urx 71 @ury 3250 @rwi @setspecial %%BeginDocument: divbypass.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: divbypass.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Thu Mar 11 14:23:12 2004 %%For: tvogel@math81 (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 325 71 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save 0.0 80.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /endash 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 2327 m -1000 -1000 l 6412 -1000 l 6412 2327 l cp clip 0.06000 0.06000 sc % Arc 7.500 slw gs n 4829.5 740.5 574.5 -90.1 90.1 arc gs col0 s gr gr % Arc gs n 1952.5 740.5 574.5 -90.0 90.0 arc gs col0 s gr gr % Arc 15.000 slw gs n 1950.0 740.0 192.0 -89.4 89.4 arc gs col0 s gr gr % Arc gs n 3868.0 740.0 192.0 -89.7 89.7 arc gs col0 s gr gr % Ellipse n 4828 740 191 191 0 360 DrawEllipse gs col0 s gr % Polyline 7.500 slw n 2910 166 m 2910 1315 l gs col0 s gr % Polyline n 2910 166 m 4828 166 l gs col0 s gr % Polyline n 35 166 m 35 1315 l gs col0 s gr % Polyline n 35 166 m 1952 166 l gs col0 s gr % Polyline n 35 1315 m 1952 1315 l gs col0 s gr % Polyline n 2897 1315 m 4815 1315 l gs col0 s gr % Polyline 15.000 slw n 1952 932 m 35 932 l gs col0 s gr % Polyline n 35 546 m 1952 546 l gs col0 s gr % Polyline n 2904 548 m 3864 548 l gs col0 s gr % Polyline n 2910 932 m 3869 932 l gs col0 s gr /Times-Roman-iso ff 180.00 scf sf 600 825 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 975 1200 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3375 750 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 4800 825 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 4275 525 m gs 1 -1 sc (+) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 1623 5070 a Fs(Figure)37 b(5.)386 5274 y Fr(the)c(left)f(part)h(of)f(Figure)g(5)g(has)i(rotation)d(n)m(um)m(b)s (er)i(1)g(while)f(in)g(the)h(righ)m(t)f(part)386 5390 y(its)27 b(rotation)g(n)m(um)m(b)s(er)h(is)f Fp(\000)p Fr(2.)43 b(By)29 b(\(15\))f(the)g(rotation)e(n)m(um)m(b)s(er)j(of)e Fo(@)5 b(R)q Fr(#)p Fo(@)g(D)3215 5405 y Fl(ot)3309 5390 y Fr(is)386 5506 y(1.)43 b(This)32 b(sho)m(ws)i(that)e(the)g(dividing)e (set)j(on)f(the)h(Seifert)e(surface)i(of)f Fo(@)5 b(R)q Fr(#)p Fo(@)g(D)3309 5521 y Fl(ot)386 5623 y Fr(con)m(tains)33 b(no)f(closed)h(comp)s(onen)m(ts.)p eop %%Page: 23 23 23 22 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(23)486 459 y Fr(The)49 b(remaining)d(conditions)h(\(i\),)k(\(ii\))c (and)h(\(iv\))g(in)f(De\014nition)g(3.20)h(are)386 575 y(satis\014ed)35 b(b)m(y)h(construction.)52 b(Condition)33 b(\(iii\))g(can)i(b)s(e)g(ac)m(hiev)m(ed)i(using)d(Theo-)386 691 y(rem)e(3.13.)43 b(Th)m(us)34 b Fo(R)q Fr(#)p Fo(D)1309 706 y Fl(ot)1406 691 y Fr(yields)e(a)g(b)m(ypass.)1228 b Fi(\003)603 992 y Fr(4.)49 b Fs(R)m(ound)38 b(handle)g (decompositions)g(and)g(model)g(Engel)1661 1109 y(str)n(uctures)486 1283 y Fr(This)29 b(section)g(pro)m(vides)h(the)g(building)d(blo)s(c)m (ks)j(for)e(our)h(construction)h(of)f(En-)386 1399 y(gel)22 b(manifolds.)39 b(W)-8 b(e)23 b(\014rst)h(recall)e(sev)m(eral)i(facts)g (from)e([As)q(])h(ab)s(out)g(round)h(handle)386 1515 y(decomp)s(ositions.)46 b(In)34 b(Section)f(4.2)h(w)m(e)g(describ)s(e)h (a)e(p)s(erturb)s(ed)i(v)m(ersion)f(of)f(the)386 1632 y(prolongation)25 b(construction)k(and)f(\014rst)g(examples)g(of)g (Engel)f(structures)j(whic)m(h)386 1748 y(are)38 b(compatible)f(with)h (a)g(round)h(handle)f(decomp)s(osition)f(of)h(the)h(underlying)386 1864 y(manifold.)47 b(P)m(erturb)s(ed)36 b(prolongation)c(will)g(b)s(e) j(used)g(for)f(the)h(construction)g(of)386 1980 y(mo)s(del)c(Engel)h (structures)i(on)f(round)g(handles)g(in)e(Sections)i(4.3)f(to)h(4.6.) 386 2227 y(4.1.)48 b Fy(Round)42 b(handle)f(decomp)s(ositions.)47 b Fr(Round)36 b(handles)f(w)m(ere)i(used)g(b)m(y)386 2343 y(D.)g(Asimo)m(v)g(\([As)q(]\))h(for)f(the)h(study)h(of)e(\015o)m (w)i(manifolds.)56 b(A)38 b(\015o)m(w)g(manifold)d(is)386 2460 y(a)44 b(manifold)d Fo(M)56 b Fr(with)43 b(a)i(non{singular)d(v)m (ector)k(\014eld)e Fo(W)58 b Fr(transv)m(erse)46 b(to)e(the)386 2576 y(b)s(oundary)33 b(whic)m(h)g(p)s(oin)m(ts)f(out)m(w)m(ards)i (along)d Fo(@)2126 2591 y Fn(+)2186 2576 y Fo(M)43 b Fr(and)33 b(in)m(w)m(ards)g(along)e Fo(@)3182 2591 y Fk(\000)3242 2576 y Fo(M)10 b Fr(.)386 2730 y Fy(De\014nition)30 b(4.1.)38 b Fo(R)1189 2745 y Fl(l)1243 2730 y Fr(=)28 b Fo(D)1431 2694 y Fl(l)1469 2730 y Fp(\002)12 b Fo(D)1642 2694 y Fl(n)p Fk(\000)p Fl(l)q Fk(\000)p Fn(1)1868 2730 y Fp(\002)g Fo(S)2023 2694 y Fn(1)2091 2730 y Fr(is)27 b(a)g Fm(r)-5 b(ound)30 b(hand)5 b(le)27 b Fr(of)g(dimension)386 2847 y Fo(n)33 b Fr(and)h(index)f Fo(l)f Fp(2)d(f)p Fr(0)p Fo(;)17 b(:)g(:)g(:)e(;)i(n)23 b Fp(\000)g Fr(1)p Fp(g)p Fr(.)46 b(The)34 b(b)s(oundary)g Fo(@)5 b(R)2528 2862 y Fl(l)2588 2847 y Fr(is)33 b(the)h(union)e(of)h(the)386 2963 y(t)m(w)m(o)g(subsets)1245 3156 y Fo(@)1296 3171 y Fk(\000)1355 3156 y Fo(R)1429 3171 y Fl(l)1483 3156 y Fr(=)28 b Fo(@)5 b(D)1727 3115 y Fl(l)1775 3156 y Fp(\002)80 b Fo(D)2016 3115 y Fl(n)p Fk(\000)p Fl(l)q Fk(\000)p Fn(1)2252 3156 y Fp(\002)22 b Fo(S)2417 3115 y Fn(1)1245 3320 y Fo(@)1296 3335 y Fn(+)1355 3320 y Fo(R)1429 3335 y Fl(l)1483 3320 y Fr(=)84 b Fo(D)1727 3279 y Fl(l)1775 3320 y Fp(\002)23 b Fo(@)5 b(D)2015 3279 y Fl(n)p Fk(\000)p Fl(l)q Fk(\000)p Fn(1)2252 3320 y Fp(\002)22 b Fo(S)2417 3279 y Fn(1)2489 3320 y Fo(:)486 3513 y Fr(A)35 b(round)h(handle)f(is)g (attac)m(hed)h(to)f(a)h(\015o)m(w)g(manifold)c(using)k(an)f(em)m(b)s (edding)386 3629 y Fo( )d Fr(:)27 b Fo(@)586 3644 y Fk(\000)646 3629 y Fo(R)720 3644 y Fl(l)774 3629 y Fp(\000)-16 b(!)27 b Fo(@)1013 3644 y Fn(+)1073 3629 y Fo(M)10 b Fr(.)40 b(One)23 b(can)g(extend)g Fo(W)36 b Fr(to)22 b(a)g(non{singular)f(v)m (ector)j(\014eld)e(on)386 3746 y Fo(M)h Fp([)13 b Fo(R)656 3761 y Fl(l)710 3746 y Fr(whic)m(h)29 b(is)e(again)g(transv)m(erse)j (to)d(the)i(b)s(oundary)-8 b(.)42 b(This)28 b(is)f(imp)s(ossible)f(if) 386 3862 y(one)i(attac)m(hes)i(an)e(ordinary)f(handle)h Fo(h)1824 3877 y Fl(l)1878 3862 y Fr(=)g Fo(D)2066 3826 y Fl(l)2105 3862 y Fp(\002)13 b Fo(D)2279 3826 y Fl(n)p Fk(\000)p Fl(l)2431 3862 y Fr(of)28 b(index)h Fo(l)g Fp(2)g(f)p Fr(0)p Fo(;)17 b(:)g(:)g(:)e(;)i(n)p Fp(g)386 3978 y Fr(to)32 b Fo(M)43 b Fr(since)33 b(this)g(c)m(hanges)h(the)f (Euler)f(c)m(haracteristic)g(b)m(y)i(\()p Fp(\000)p Fr(1\))2799 3942 y Fl(l)2825 3978 y Fr(.)486 4094 y(No)m(w)40 b(assume)h(that)g Fo(W)53 b Fr(spans)42 b(the)f(c)m(haracteristic)f(foliation)d(of)j(an)g (Engel)386 4211 y(structure)32 b(on)e Fo(M)10 b Fr(.)44 b(If)30 b(one)h(extends)h(the)f(Engel)f(structure)i(from)d Fo(M)41 b Fr(to)31 b Fo(M)d Fp([)18 b Fo(R)3348 4226 y Fl(l)386 4327 y Fr(it)45 b(is)g(clear)h(from)f(the)h(prop)s(erties)g (of)g(transv)m(erse)i(h)m(yp)s(ersurfaces)g(that)e(it)f(is)386 4443 y(useful)31 b(to)g(ensure)i(that)f(the)f(b)s(oundary)h(of)f(the)h (new)g(Engel)g(manifold)c(is)j(again)386 4559 y(transv)m(erse)36 b(to)e(the)g(c)m(haracteristic)g(foliation.)44 b(Therefore)35 b(round)f(handles)h(are)386 4675 y(suitable)d(building)e(blo)s(c)m(ks)j (for)f(the)h(construction)g(of)f(Engel)g(structures.)386 4830 y Fy(De\014nition)46 b(4.2.)g Fr(If)c Fo(M)52 b Fr(is)40 b(obtained)h(from)f(the)i(disjoin)m(t)e(union)h(of)f (\014nitely)386 4946 y(man)m(y)e(round)f(handles)h(of)f(index)h(0)g(b)m (y)g(attac)m(hing)f(round)h(handles)g(of)f(higher)386 5063 y(index)c(successiv)m(ely)-8 b(,)35 b(i.e.)1044 5282 y Fo(M)j Fr(=)1280 5171 y Fj(\020)1339 5282 y Fo(:)17 b(:)g(:)1470 5171 y Fj(\020)1530 5187 y([)1657 5282 y Fo(R)1731 5297 y Fn(0)1771 5171 y Fj(\021)1852 5282 y Fp([)1918 5297 y Fl( )1964 5306 y Fg(1)2026 5282 y Fo(R)2100 5297 y Fl(\014)2140 5306 y Fg(1)2195 5282 y Fo(:)g(:)g(:)2309 5171 y Fj(\021)2391 5282 y Fp([)2457 5297 y Fl( )2503 5305 y Ff(r)2564 5282 y Fo(R)2638 5297 y Fl(\014)2678 5305 y Ff(r)386 5506 y Fr(with)26 b Fo(\014)657 5521 y Fl(i)713 5506 y Fp(2)i(f)p Fr(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(n)11 b Fp(\000)g Fr(1)p Fp(g)26 b Fr(for)g Fo(i)i Fp(2)g(f)p Fr(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(r)s Fp(g)p Fr(,)27 b(then)g(w)m(e)h (ha)m(v)m(e)h(a)d Fm(r)-5 b(ound)30 b(hand)5 b(le)386 5623 y(de)-5 b(c)g(omp)g(osition)31 b Fr(of)h Fo(M)10 b Fr(.)p eop %%Page: 24 24 24 23 bop 386 259 a Fq(24)1096 b(THOMAS)25 b(V)n(OGEL)486 459 y Fr(W)-8 b(e)30 b(can)f(rearrange)h(a)f(giv)m(en)h(round)g(handle) f(decomp)s(osition)f(of)h(a)g(manifold)386 575 y(suc)m(h)f(that)f(the)g (round)h(handles)f(are)g(ordered)g(according)g(to)f(their)h(index.)42 b(Con-)386 691 y(trary)37 b(to)g(the)g(case)h(of)f(ordinary)f(handles,) j(t)m(w)m(o)e(round)h(handles)f(of)f(the)i(same)386 807 y(index)33 b(can)g(not)f(b)s(e)h(in)m(terc)m(hanged)g(in)f(general.)486 924 y(If)i(a)h(closed)g(manifold)d Fo(M)46 b Fr(admits)34 b(a)g(round)i(handle)e(decomp)s(osition,)g(then)386 1040 y(its)44 b(Euler)h(c)m(haracteristic)g Fo(\037)p Fr(\()p Fo(M)10 b Fr(\))45 b(has)h(to)e(v)-5 b(anish)45 b(b)s(ecause)h(w)m(e)g (can)f(use)h(the)386 1156 y(round)34 b(handle)f(decomp)s(osition)e(to)i (\014nd)h(a)f(non{singular)f(v)m(ector)i(\014eld)g(on)f Fo(M)10 b Fr(.)386 1272 y(The)33 b(con)m(v)m(erse)i(direction)d(is)g (co)m(v)m(ered)j(b)m(y)e(the)g(follo)m(wing)d(theorem.)386 1415 y Fy(Theorem)j(4.3)g Fr(\(Asimo)m(v,)c([As]\))p Fy(.)39 b Fm(A)32 b(close)-5 b(d,)31 b(c)-5 b(onne)g(cte)g(d)30 b(manifold)g(of)h(dimen-)386 1531 y(sion)40 b Fo(n)g Fp(6)p Fr(=)f(3)i Fm(admits)f(a)h(de)-5 b(c)g(omp)g(osition)40 b(into)h(r)-5 b(ound)41 b(hand)5 b(les)40 b(if)g(and)h(only)g(if)386 1648 y Fo(\037)p Fr(\()p Fo(M)10 b Fr(\))38 b(=)f(0)p Fm(.)60 b(This)39 b(de)-5 b(c)g(omp)g(osition)38 b(c)-5 b(an)40 b(b)-5 b(e)39 b(chosen)g(such)h(that)g(ther)-5 b(e)40 b(is)g(only)386 1764 y(one)34 b(r)-5 b(ound)35 b Fr(0)p Fm({hand)5 b(le)33 b(and)i(one)f(r)-5 b(ound)35 b Fr(\()p Fo(n)22 b Fp(\000)h Fr(1\))p Fm({hand)5 b(le.)486 1907 y Fr(The)50 b(analogous)f(statemen)m(t)h(is)f(wrong)h(for)g (manifolds)d(of)i(dimension)g(3,)386 2023 y(cf.)27 b([Mor].)41 b(In)27 b(order)f(to)h(pro)m(v)m(e)g(the)g(last)f(part)g(of)g(Theorem)g (4.3)g(one)h(follo)m(ws)e(the)386 2139 y(pro)s(of)i(in)h([As)q(])g (starting)f(with)h(a)g(decomp)s(osition)f(of)g Fo(M)39 b Fr(in)m(to)28 b(ordinary)f(handles)386 2255 y(with)32 b(only)g(one)h(handle)f(of)g(index)h(0,)g(resp)s(ectiv)m(ely)g Fo(n)p Fr(.)486 2372 y(One)i(can)h(decomp)s(ose)g Fo(R)1444 2387 y Fl(l)1505 2372 y Fr(in)m(to)f(one)h(ordinary)e(handle)i(of)e (index)i Fo(l)i Fr(and)d(an-)386 2488 y(other)e(one)f(of)h(index)f Fo(l)25 b Fr(+)d(1)32 b(as)h(follo)m(ws)768 2644 y Fo(R)842 2659 y Fl(l)896 2644 y Fr(=)28 b Fo(D)1084 2603 y Fl(l)1131 2644 y Fp(\002)23 b Fo(D)1315 2603 y Fl(n)p Fk(\000)p Fl(l)q Fk(\000)p Fn(1)1551 2644 y Fp(\002)f Fo(S)1716 2603 y Fn(1)1783 2644 y Fr(=)28 b Fo(D)1971 2603 y Fl(l)2019 2644 y Fp(\002)22 b Fo(D)2202 2603 y Fl(n)p Fk(\000)p Fl(l)q Fk(\000)p Fn(1)2438 2644 y Fp(\002)h Fr(\()p Fo(D)2660 2603 y Fn(1)2721 2644 y Fp([)f Fo(D)2893 2603 y Fn(1)2933 2644 y Fr(\))896 2808 y(=)1000 2727 y Fj(\000)1045 2808 y Fo(D)1129 2767 y Fl(l)1177 2808 y Fp(\002)h Fr(\()p Fo(D)1399 2767 y Fl(n)p Fk(\000)p Fl(l)q Fk(\000)p Fn(1)1634 2808 y Fp(\002)g Fo(D)1818 2767 y Fn(1)1857 2808 y Fr(\))1895 2727 y Fj(\001)1963 2808 y Fp([)2051 2727 y Fj(\000)2097 2808 y Fr(\()p Fo(D)2219 2767 y Fl(l)2267 2808 y Fp(\002)f Fo(D)2450 2767 y Fn(1)2490 2808 y Fr(\))g Fp(\002)g Fo(D)2733 2767 y Fl(n)p Fk(\000)p Fl(l)q Fk(\000)p Fn(1)2947 2727 y Fj(\001)896 2953 y Fr(=)28 b Fo(h)1056 2968 y Fl(l)1104 2953 y Fp([)22 b Fo(h)1248 2968 y Fl(l)q Fn(+1)1397 2953 y Fo(:)386 2799 y Fr(\(16\))386 3123 y(This)44 b(allo)m(ws)f(us)i(to)f (obtain)f(decomp)s(ositions)g(in)m(to)h(ordinary)f(handles)i(from)386 3239 y(round)33 b(handle)f(decomp)s(ositions.)386 3438 y(4.2.)48 b Fy(P)m(erturb)s(ed)57 b(prolongation.)48 b Fr(F)-8 b(rom)49 b(no)m(w)h(on)g(w)m(e)h(consider)f(only)f(4{)386 3555 y(dimensional)21 b(round)j(handles.)41 b(F)-8 b(or)23 b(the)h(construction)g(of)g(Engel)f(structures)j(w)m(e)386 3671 y(will)j(\014x)k(a)e(set)i(of)e(particular)f(Engel)i(structures)h (on)f Fo(R)2434 3686 y Fl(l)2492 3671 y Fr(for)f(eac)m(h)i Fo(l)d Fr(=)d(0)p Fo(;)17 b Fr(1)p Fo(;)g Fr(2)p Fo(;)g Fr(3.)386 3814 y Fy(De\014nition)37 b(4.4.)42 b Fr(A)34 b(mo)s(del)e(Engel)h(structure)i(on)e(a)g(round)h(handle)f(of)g(index) 386 3930 y Fo(l)42 b Fr(is)d(an)h(Engel)f(structure)j(on)d Fo(R)1626 3945 y Fl(l)1692 3930 y Fr(suc)m(h)i(that)f(the)g(c)m (haracteristic)g(foliation)c(is)386 4046 y(orien)m(ted)j(and)h(transv)m (erse)i(to)d Fo(@)1612 4061 y Fn(+)1671 4046 y Fo(R)1745 4061 y Fl(l)1811 4046 y Fr(and)h Fo(@)2059 4061 y Fk(\000)2118 4046 y Fo(R)2192 4061 y Fl(l)2218 4046 y Fr(.)65 b(It)39 b(p)s(oin)m(ts)g(out)m(w)m(ards)i(along)386 4162 y Fo(@)437 4177 y Fn(+)496 4162 y Fo(R)570 4177 y Fl(l)629 4162 y Fr(and)33 b(in)m(w)m(ards)g(along)e Fo(@)1488 4177 y Fk(\000)1548 4162 y Fo(R)1622 4177 y Fl(l)1648 4162 y Fr(.)486 4305 y(The)44 b(usual)e(prolongation)f(construction)i(is)g (describ)s(ed)g(in)g(Example)f(2.14)386 4421 y(and)c(deformations)f(of) h(certain)g(prolonged)g(Engel)g(structures)i(are)f(discussed)386 4538 y(in)29 b([Mo2].)43 b(W)-8 b(e)30 b(no)m(w)h(describ)s(e)f(p)s (erturbations)g(of)f(Engel)h(structures)h(whic)m(h)g(w)m(e)386 4654 y(will)f(use)k(later)d(for)h(the)h(explicit)f(construction)g(of)g (mo)s(del)f(Engel)i(structures.)486 4770 y(Let)j Fo(\013)h Fr(b)s(e)g(a)f(con)m(tact)h(form)e(on)i(a)f(compact)g(manifold)e Fo(N)47 b Fr(suc)m(h)38 b(that)e(there)386 4886 y(is)h(a)h(con)m(tact)g (v)m(ector)h(\014eld)e Fo(V)60 b Fr(for)37 b Fp(C)43 b Fr(=)36 b(k)m(er)q(\()p Fo(\013)q Fr(\))h(whic)m(h)i(is)e(transv)m (erse)j(to)d Fo(@)5 b(N)10 b Fr(.)386 5003 y(Moreo)m(v)m(er)36 b(w)m(e)f(assume)g(that)g Fp(C)40 b Fr(is)34 b(trivial)e(as)j(a)f (bundle)g(and)h(w)m(e)h(\014x)f(a)f(trivial-)386 5119 y(ization)h Fo(C)780 5134 y Fn(1)819 5119 y Fo(;)17 b(C)933 5134 y Fn(2)1009 5119 y Fr(of)36 b Fp(C)6 b Fr(.)57 b(W)-8 b(e)37 b(use)h(the)f(same)g(notation)e(for)i(the)g(horizon)m(tal)e (lifts)386 5235 y(to)d Fo(N)h Fp(\002)22 b Fo(S)781 5199 y Fn(1)821 5235 y Fr(.)386 5378 y Fy(Prop)s(osition)40 b(4.5.)j Fm(F)-7 b(or)38 b Fo(k)f Fp(2)d Fh(Z)h Fm(the)k(distribution)f Fp(D)2459 5393 y Fl(k)2539 5378 y Fm(on)g Fo(N)d Fp(\002)26 b Fo(S)2964 5342 y Fn(1)3041 5378 y Fm(sp)-5 b(anne)g(d)386 5494 y(by)900 5623 y Fo(W)41 b Fr(=)28 b Fo(@)1188 5638 y Fl(t)1240 5623 y Fr(+)22 b Fo("V)56 b Fm(and)34 b Fo(X)1767 5638 y Fl(k)1838 5623 y Fr(=)27 b(cos)q(\()p Fo(k)s(t)p Fr(\))p Fo(C)2307 5638 y Fn(1)2368 5623 y Fr(+)22 b(sin\()p Fo(k)s(t)p Fr(\))p Fo(C)2821 5638 y Fn(2)p eop %%Page: 25 25 25 24 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(25)386 459 y Fm(is)40 b(an)g(Engel)h(structur)-5 b(e)41 b(on)f Fo(N)d Fp(\002)27 b Fo(S)1764 423 y Fn(1)1844 459 y Fm(for)40 b Fo(")e(>)g Fr(0)j Fm(smal)5 b(l)40 b(enough)g(and)g Fo(k)h Fp(6)p Fr(=)d(0)p Fm(.)386 575 y(F)-7 b(or)39 b Fo(k)h Fr(=)d(0)i Fm(we)h(obtain)f(an)g(Engel)h (structur)-5 b(e)41 b(if)e(and)h(only)f(if)h Fo(C)2832 590 y Fn(1)2911 575 y Fm(and)f Fr([)p Fo(V)5 b(;)17 b(C)3308 590 y Fn(1)3348 575 y Fr(])386 691 y Fm(ar)-5 b(e)35 b(line)-5 b(arly)34 b(indep)-5 b(endent)33 b(everywher)-5 b(e.)486 807 y(Then)36 b(the)h(char)-5 b(acteristic)36 b(foliation)g(of)h Fp(D)2095 822 y Fl(k)2174 807 y Fm(is)g(sp)-5 b(anne)g(d)36 b(by)h Fo(W)14 b Fm(.)51 b(In)36 b(p)-5 b(artic-)386 924 y(ular,)40 b(the)g(b)-5 b(oundary)39 b(is)g(tr)-5 b(ansverse)39 b(and)g(b)-5 b(oth)39 b Fp(C)46 b Fm(and)39 b(the)g(c)-5 b(ontact)40 b(structur)-5 b(e)386 1040 y(on)34 b Fo(@)5 b(N)34 b Fp(\002)23 b Fo(S)859 1004 y Fn(1)933 1040 y Fm(induc)-5 b(e)34 b(the)h(same)f(singular)g (foliation)g(on)h Fo(@)5 b(N)33 b Fp(\002)23 b(f)p Fr(1)p Fp(g)p Fm(.)386 1209 y(Pr)-5 b(o)g(of.)41 b Fr(In)29 b(order)f(to)g(sho)m(w)h(that)f Fp(D)1688 1224 y Fl(k)1759 1209 y Fr(is)f(an)h(Engel)g(structure)i(w)m(e)f(\014rst)g(calculate)906 1355 y([)p Fo(W)m(;)17 b(X)1147 1370 y Fl(k)1189 1355 y Fr(])28 b(=)g Fp(\000)p Fo(k)20 b Fr(sin\()p Fo(k)s(t)p Fr(\))p Fo(C)1851 1370 y Fn(1)1912 1355 y Fr(+)i Fo(k)e Fr(cos)q(\()p Fo(k)s(t)p Fr(\))p Fo(C)2447 1370 y Fn(2)1336 1506 y Fr(+)i Fo(")p Fr(\(cos\()p Fo(k)s(t)p Fr(\)[)p Fo(V)5 b(;)17 b(C)2016 1521 y Fn(1)2056 1506 y Fr(])22 b(+)g(sin\()p Fo(k)s(t)p Fr(\)[)p Fo(V)5 b(;)17 b(C)2691 1521 y Fn(2)2730 1506 y Fr(]\))33 b Fo(:)386 1437 y Fr(\(17\))386 1671 y(If)27 b Fo(k)k Fp(6)p Fr(=)c(0)g(and)g Fo(")h(>)f Fr(0)g(is)g(small)e(enough,)k(then)e Fo(W)m(;)17 b(X)2308 1686 y Fl(k)2378 1671 y Fr(and)27 b([)p Fo(W)m(;)17 b(X)2803 1686 y Fl(k)2846 1671 y Fr(])27 b(are)h(linearly)386 1788 y(indep)s(enden)m(t.)59 b(The)38 b(case)g Fo(k)h Fr(=)c(0)i(is)g(ob)m(vious.)58 b(Since)37 b Fo(V)59 b Fr(is)37 b(a)g(con)m(tact)h(v)m(ector)386 1904 y(\014eld)43 b([)p Fp(D)712 1919 y Fl(k)755 1904 y Fo(;)17 b Fp(D)876 1919 y Fl(k)918 1904 y Fr(])44 b(is)f(generated)h(b)m(y)g Fo(W)m(;)17 b(C)1902 1919 y Fn(1)1985 1904 y Fr(and)44 b Fo(C)2256 1919 y Fn(2)2295 1904 y Fr(.)76 b(Hence)45 b Fp(E)55 b Fr(=)46 b([)p Fp(D)3033 1919 y Fl(k)3075 1904 y Fo(;)17 b Fp(D)3196 1919 y Fl(k)3239 1904 y Fr(])43 b(is)386 2020 y(de\014ned)34 b(b)m(y)386 2180 y(\(18\))920 b Fo(\014)33 b Fr(=)28 b Fo(\013)22 b Fp(\000)h Fo("\013)q Fr(\()p Fo(V)e Fr(\))c Fo(dt)32 b(:)386 2339 y Fr(F)-8 b(rom)43 b(the)i(fact)f(that)g Fo(\013)h Fr(is)f(a)g(con)m(tact)i(form) d(it)g(follo)m(ws)g(that)i Fo(\014)50 b Fr(de\014nes)c(an)386 2455 y(ev)m(en)39 b(con)m(tact)f(structure)h(for)e(all)e Fo(")g(>)h Fr(0.)58 b(This)38 b(sho)m(ws)h(that)e Fp(D)2812 2470 y Fl(k)2892 2455 y Fr(is)g(an)g(Engel)386 2572 y(structure.)486 2688 y(Notice)29 b(that)g Fo(\014)6 b Fr(\()p Fo("V)36 b Fr(+)16 b Fo(@)1378 2703 y Fl(t)1408 2688 y Fr(\))27 b(=)h(0.)42 b(Let)29 b Fo(h)h Fr(b)s(e)f(the)h(function)f(with)g(the)h (prop)s(ert)m(y)386 2804 y Fo(L)452 2819 y Fl(V)513 2804 y Fo(\013)f Fr(=)e Fo(h\013)q Fr(.)43 b(Then)869 2964 y Fo(L)935 2979 y Fn(\()p Fl("V)16 b Fn(+)p Fl(@)1144 2987 y Ff(t)1172 2979 y Fn(\))1203 2964 y Fr(\()p Fo(\013)23 b Fp(\000)g Fo("\013)q Fr(\()p Fo(V)d Fr(\))p Fo(dt)p Fr(\))28 b(=)f Fo("L)2055 2979 y Fl(V)2116 2964 y Fo(\013)c Fp(\000)g Fo(")2347 2923 y Fn(2)2386 2964 y Fr(\()p Fo(L)2490 2979 y Fl(V)2551 2964 y Fo(\013)q Fr(\()p Fo(V)e Fr(\)\))p Fo(dt)1840 3115 y Fr(=)27 b Fo("h)p Fr(\()p Fo(\013)c Fp(\000)g Fo("\013)q Fr(\()p Fo(V)d Fr(\)\))386 3275 y(implies)30 b(that)i Fo(W)47 b Fr(spans)33 b(the)g(c)m(haracteristic)g (foliation)c(of)j Fp(D)2671 3290 y Fl(k)2713 3275 y Fr(.)557 b Fi(\003)486 3444 y Fr(If)33 b Fo(N)41 b Fr(=)29 b Fo(h)864 3459 y Fl(l)891 3444 y Fr(,)34 b(then)g(w)m(e)h(obtain)e(a)g(mo)s(del)g (Engel)g(structure)i(on)f Fo(R)2902 3459 y Fl(l)2958 3444 y Fr(=)c Fo(h)3120 3459 y Fl(l)3169 3444 y Fp(\002)23 b Fo(S)3335 3408 y Fn(1)386 3560 y Fr(if)47 b Fo(V)69 b Fr(p)s(oin)m(ts)48 b(out)m(w)m(ards)h(along)e Fo(@)1686 3575 y Fn(+)1745 3560 y Fo(h)1801 3575 y Fl(l)1881 3560 y Fr(=)54 b Fo(D)2095 3524 y Fl(l)2153 3560 y Fp(\002)34 b Fo(@)5 b(D)2404 3524 y Fn(3)p Fk(\000)p Fl(l)2568 3560 y Fr(and)48 b(in)m(w)m(ards)h(along)386 3676 y Fo(@)437 3691 y Fk(\000)496 3676 y Fo(h)552 3691 y Fl(l)606 3676 y Fr(=)28 b Fo(@)5 b(D)850 3640 y Fl(l)899 3676 y Fp(\002)22 b Fo(D)1082 3640 y Fn(3)p Fk(\000)p Fl(l)1198 3676 y Fr(.)486 3793 y(Let)h(us)g(describ)s(e)h(the)g(orien)m(tation)d(of)h Fp(E)37 b Fr(=)27 b([)p Fp(D)2177 3808 y Fl(k)2220 3793 y Fo(;)17 b Fp(D)2341 3808 y Fl(k)2383 3793 y Fr(])23 b(:)39 b(If)23 b Fo(k)30 b Fr(=)e(0)23 b(w)m(e)h(obtain)e(an)386 3909 y(Engel)j(structure)h(only)f(if)f Fo(C)1420 3924 y Fn(1)1484 3909 y Fr(and)h([)p Fo(V)5 b(;)17 b(C)1869 3924 y Fn(1)1908 3909 y Fr(])25 b(are)h(linearly)d(indep)s(enden)m(t)j (and)f(then)386 4025 y Fp(E)41 b Fr(is)32 b(orien)m(ted)h(b)m(y)h Fo(W)m(;)17 b(C)1294 4040 y Fn(1)1333 4025 y Fo(;)g Fr([)p Fo(V)5 b(;)17 b(C)1580 4040 y Fn(1)1619 4025 y Fr(].)45 b(F)-8 b(or)32 b Fo(k)f(>)d Fr(0)k(the)h(ev)m(en)i(con)m(tact)e (structure)h Fp(E)386 4141 y Fr(is)d(orien)m(ted)g(b)m(y)h Fo(W)m(;)17 b(C)1195 4156 y Fn(1)1234 4141 y Fo(;)g(C)1348 4156 y Fn(2)1387 4141 y Fr(.)43 b(If)31 b Fo(k)g(<)c Fr(0,)k(then)h Fp(E)40 b Fr(has)31 b(the)h(opp)s(osite)e(orien)m (tation.)486 4257 y(If)37 b Fo(p)f Fp(2)h Fo(@)5 b(N)49 b Fr(lies)36 b(on)i(the)g(c)m(haracteristic)f(surface)h(\006)f(=)f Fp(f)p Fo(\013)q Fr(\()p Fo(V)21 b Fr(\))36 b(=)g(0)p Fp(g)h Fr(of)g Fo(V)22 b Fr(,)386 4374 y(then)28 b Fp(f)p Fo(p)p Fp(g)12 b(\002)g Fo(S)919 4338 y Fn(1)986 4374 y Fr(is)27 b(tangen)m(t)h(to)f(the)h(con)m(tact)h(structure)f(on)g Fo(@)5 b(N)23 b Fp(\002)12 b Fo(S)2901 4338 y Fn(1)2941 4374 y Fr(.)42 b(F)-8 b(or)27 b(later)386 4490 y(use)35 b(w)m(e)h(determine)e(the)g(rotation)f(n)m(um)m(b)s(er)i(along)e(these) i(Legendrian)g(curv)m(es.)386 4606 y(If)41 b Fo(k)47 b Fr(=)c(0)e(the)h(rotation)e(n)m(um)m(b)s(er)i(is)f(zero)h(since)g (the)g(in)m(tersection)g(line)e(\014eld)386 4722 y(along)28 b Fo(@)5 b(N)28 b Fp(\002)16 b Fo(S)964 4686 y Fn(1)1033 4722 y Fr(is)29 b Fo(S)1194 4686 y Fn(1)1234 4722 y Fr({in)m(v)-5 b(arian)m(t.)40 b(F)-8 b(or)29 b Fo(k)i Fp(6)p Fr(=)d(0)h(w)m(e)i(ma)m (y)e(assume)h(that)g Fo(C)3126 4737 y Fn(1)3193 4722 y Fr(=)d Fo(V)386 4839 y Fr(at)32 b Fo(p)p Fr(.)44 b(Clearly)386 4998 y(\(19\))327 b Fo(V)49 b Fr(=)28 b Fo(C)1167 5013 y Fn(1)1234 4998 y Fr(=)f(cos)q(\()p Fo(k)s(t)p Fr(\))p Fo(X)1714 5013 y Fl(k)1779 4998 y Fp(\000)22 b Fr(1)p Fo(=k)e Fr(sin)o(\()p Fo(k)s(t)p Fr(\)[)p Fo(W)m(;)d(X)2572 5013 y Fl(k)2615 4998 y Fr(])23 b(+)f Fo(R)2837 5013 y Fl(")386 5158 y Fr(where)44 b(the)g(correction)e(term)h Fo(R)1640 5173 y Fl(")1720 5158 y Fr(satis\014es)g(lim)2225 5173 y Fl(")p Fk(!)p Fn(0)2384 5158 y Fo(R)2458 5173 y Fl(")2540 5158 y Fr(=)j(0.)74 b(If)43 b(w)m(e)h(pro)5 b(ject)386 5274 y(all)36 b(v)m(ector)i(\014elds)g(in)f(\(19\))g(along)f Fo(W)52 b Fr(to)37 b Fo(@)5 b(N)37 b Fp(\002)26 b Fo(S)2282 5238 y Fn(1)2359 5274 y Fr(w)m(e)39 b(obtain)d(an)i(analogous)386 5390 y(expression)f(for)e Fo(@)1062 5405 y Fl(t)1092 5390 y Fr(.)53 b(Notice)35 b(that)g(the)h(pro)5 b(jection)36 b(of)f Fo(X)2528 5405 y Fl(k)2606 5390 y Fr(to)g Fo(@)5 b(N)36 b Fp(\002)25 b Fo(S)3066 5354 y Fn(1)3141 5390 y Fr(spans)386 5506 y(the)36 b(in)m(tersection)g(line)f(\014eld.)53 b(Hence)37 b(the)g(rotation)d(n)m(um)m(b)s(er)i(along)f Fp(f)p Fo(p)p Fp(g)24 b(\002)g Fo(S)3335 5470 y Fn(1)386 5623 y Fr(is)32 b Fp(\000j)p Fo(k)s Fp(j)g Fr(\(cf.)h(De\014nition)e (2.18\).)p eop %%Page: 26 26 26 25 bop 386 259 a Fq(26)1096 b(THOMAS)25 b(V)n(OGEL)486 459 y Fr(In)49 b(order)g(to)f(obtain)g(mo)s(del)f(Engel)i(structures)i (with)d(p)s(ositiv)m(e)g(rotation)386 575 y(n)m(um)m(b)s(er)43 b(along)e Fp(f)p Fo(p)p Fp(g)28 b(\002)i Fo(S)1371 539 y Fn(1)1453 575 y Fr(one)42 b(can)h(replace)g Fo(W)56 b Fr(b)m(y)43 b Fp(\000)p Fo(@)2592 590 y Fl(t)2652 575 y Fr(+)29 b Fo("V)21 b Fr(,)45 b(but)e(doing)386 691 y(so)32 b(one)f(c)m(hanges)i(the)f(con)m(tact)g(structure)h(on)e Fo(@)5 b(N)32 b Fp(\002)20 b Fo(S)2432 655 y Fn(1)2471 691 y Fr(.)43 b(In)32 b(the)g(cases)h Fo(N)38 b Fr(=)28 b Fo(h)3349 706 y Fl(l)386 807 y Fr(for)35 b Fo(l)h Fr(=)d(0)p Fo(;)17 b(:)g(:)g(:)f(;)h Fr(3)35 b(considered)i(later)e(w)m(e)h(will)e (construct)j(a)f(di\013eomorphism)d Fo(f)386 924 y Fr(of)38 b Fo(R)577 939 y Fl(l)642 924 y Fr(=)g Fo(h)812 939 y Fl(l)864 924 y Fp(\002)27 b Fo(S)1034 887 y Fn(1)1112 924 y Fr(whic)m(h)40 b(preserv)m(es)h(the)f(con)m(tact)f(structure)h (on)f(large)e(parts)386 1040 y(of)31 b Fo(@)547 1055 y Fk(\000)607 1040 y Fo(R)681 1055 y Fl(l)739 1040 y Fr(and)h(rev)m(erses)i(the)f(orien)m(tation)d(of)h Fp(f)p Fo(p)p Fp(g)21 b(\002)g Fo(S)2398 1004 y Fn(1)2437 1040 y Fr(.)43 b(Pushing)32 b(forw)m(ard)g(the)386 1156 y(mo)s(del)24 b(Engel)h(structures)i(with)e Fo(f)36 b Fr(one)25 b(obtains)g(mo)s(del) f(Engel)h(structures)i(with)386 1272 y(non{negativ)m(e)33 b(rotation)e(n)m(um)m(b)s(ers)i(along)e Fp(f)p Fo(p)p Fp(g)22 b(\002)h Fo(S)2341 1236 y Fn(1)2380 1272 y Fr(.)486 1389 y(Before)j(w)m(e)i(discuss)g(explicit)d(mo)s(del)g(Engel)h (structures)i(let)e(us)h(\014rst)g(describ)s(e)386 1505 y(examples)32 b(of)g(Engel)g(manifolds)e(with)h(a)h(decomp)s(osition)f (in)m(to)g(round)h(handles)386 1621 y(whic)m(h)h(carry)g(mo)s(del)e (Engel)h(structures.)486 1737 y(If)f Fo(N)43 b Fr(is)31 b(decomp)s(osed)h(in)m(to)f(ordinary)h(handles)g(w)m(e)h(obtain)d(a)i (round)g(handle)386 1853 y(decomp)s(osition)22 b(of)h Fo(N)15 b Fp(\002)t Fo(S)1362 1817 y Fn(1)1427 1853 y Fr(suc)m(h)25 b(that)f(the)g(round)g(handles)g(are)g(pro)s(ducts)h(of)e Fo(S)3335 1817 y Fn(1)386 1970 y Fr(with)30 b(ordinary)f(handles)h (from)f(the)i(decomp)s(osition)d(of)h Fo(N)10 b Fr(.)43 b(If)30 b Fo(V)52 b Fr(is)29 b(transv)m(erse)386 2086 y(to)40 b(the)g(b)s(oundaries)g(of)f(all)f(handles)j(and)f(p)s(oin)m (ts)f(in)m(to)h(the)g(righ)m(t)f(directions,)386 2202 y(then)k Fo(N)d Fp(\002)29 b Fo(S)908 2166 y Fn(1)990 2202 y Fr(carries)43 b(an)f(Engel)h(structure)g(suc)m(h)i(that)d(all)f (round)h(handles)386 2318 y(carry)d(mo)s(del)f(Engel)g(structures.)65 b(This)39 b(situation)e(arises)i(in)g(the)g(con)m(text)h(of)386 2435 y(con)m(v)m(ex)35 b(con)m(tact)e(structures)h(de\014ned)g(in)e ([ElG)o(].)386 2582 y Fy(De\014nition)55 b(4.6.)50 b Fr(A)f(con)m(tact)g(structure)h Fp(C)55 b Fr(on)49 b(a)f(manifold)e Fo(N)59 b Fr(is)49 b Fm(c)-5 b(onvex)386 2699 y Fr(if)41 b(there)j(is)e(a)h(prop)s(er)f(Morse)i(function)e Fo(g)48 b Fr(:)d Fo(N)56 b Fp(!)44 b Fr([0)p Fo(;)17 b Fp(1)p Fr(\))42 b(and)h(a)f(complete)386 2815 y(con)m(tact)35 b(v)m(ector)g(\014eld)f Fo(V)55 b Fr(whic)m(h)35 b(is)f(a)g (pseudo{gradien)m(t)g(for)g Fo(g)t Fr(,)g(i.e.)47 b(there)35 b(is)f(a)386 2931 y(Riemannian)c(metric)h(and)h(a)f(p)s(ositiv)m(e)h (function)f Fo(s)h Fr(suc)m(h)h(that)f Fo(L)2817 2946 y Fl(V)2879 2931 y Fo(g)e Fp(\025)f Fo(s)p Fp(k)p Fo(dg)t Fp(k)3310 2895 y Fn(2)3348 2931 y Fr(.)486 3079 y(According)37 b(to)h([Gi2)n(])g(ev)m(ery)i(con)m(tact)e(structure)h(on)f(a)g (3{manifold)c(is)k(con-)386 3195 y(v)m(ex.)66 b(F)-8 b(rom)37 b Fo(g)43 b Fr(one)d(obtains)f(a)g(handle)g(decomp)s(osition)f (of)g Fo(N)50 b Fr(and)40 b(applying)386 3311 y(Prop)s(osition)c(4.5)h (to)h(con)m(v)m(ex)i(con)m(tact)e(structures)i(one)e(obtains)f (examples)h(of)386 3428 y(Engel)32 b(structures)j(with)d(adapted)h (round)f(handle)h(decomp)s(osition.)386 3647 y(4.3.)48 b Fy(Mo)s(del)39 b(Engel)f(structures)g(on)h(round)h(handles)f(of)g (index)g Fr(0)p Fy(.)48 b Fr(F)-8 b(or)386 3763 y(our)29 b(pro)s(of)f(of)g(Theorem)i(6.1)e(it)g(is)h(useful)g(to)g(ha)m(v)m(e)h (o)m(v)m(ert)m(wisted)h(con)m(tact)e(struc-)386 3879 y(tures)34 b(on)e(transv)m(erse)j(b)s(oundaries)e(of)g(Engel)f (manifolds.)42 b(Since)33 b(w)m(e)h(will)d(start)386 3995 y(our)e(construction)h(of)f(Engel)h(manifolds)d(with)i(a)h(round)g (handle)f(of)g(index)h(0)g(w)m(e)386 4112 y(de\014ne)c(the)e(mo)s(del)f (Engel)h(structures)i(on)f Fo(R)2015 4127 y Fn(0)2079 4112 y Fr(suc)m(h)g(that)g(the)f(con)m(tact)h(structure)386 4228 y(on)32 b Fo(@)572 4243 y Fn(+)632 4228 y Fo(R)706 4243 y Fn(0)778 4228 y Fr(is)g(o)m(v)m(ert)m(wisted.)486 4344 y(W)-8 b(e)51 b(use)h(cylindrical)d(co)s(ordinates)h(on)h Fh(R)2108 4308 y Fn(3)2154 4344 y Fr(.)99 b(Consider)51 b(the)h(o)m(v)m(ert)m(wisted)386 4460 y(con)m(tact)c(structure)h(k)m (er)q(\(cos)q(\()p Fo(r)1562 4424 y Fn(2)1601 4460 y Fr(\))17 b Fo(dz)36 b Fr(+)d(sin)o(\()p Fo(r)2101 4424 y Fn(2)2140 4460 y Fr(\))17 b Fo(d')p Fr(\))47 b(and)h(the)g(ball)e(of) h(radius)386 4577 y Fo(r)430 4592 y Fn(0)518 4577 y Fr(=)i(3)p Fo(\031)t(=)p Fr(2.)79 b(The)46 b(singular)e(foliation)d(on)k(the)g(b)s (oundary)g Fo(S)6 b Fr(\()p Fo(r)2871 4592 y Fn(0)2911 4577 y Fr(\))44 b(admits)g(a)386 4693 y(dividing)32 b(set)j(with)f (three)h(connected)h(comp)s(onen)m(ts.)50 b(F)-8 b(rom)32 b(Theorem)j(3.11)f(it)386 4809 y(follo)m(ws)27 b(that)h Fo(S)6 b Fr(\()p Fo(r)1057 4824 y Fn(0)1096 4809 y Fr(\))29 b(is)e(con)m(v)m(ex,)32 b(i.e.)41 b(there)30 b(is)d(a)i(con)m(tact)f(v) m(ector)i(\014eld)e Fo(V)50 b Fr(trans-)386 4925 y(v)m(erse)26 b(to)f Fo(S)6 b Fr(\()p Fo(r)882 4940 y Fn(0)921 4925 y Fr(\))25 b(whic)m(h)g(p)s(oin)m(ts)f(out)h(of)f(the)h(ball.)39 b(Since)25 b(the)g(ball)e(is)h(con)m(tractible)386 5042 y(the)33 b(con)m(tact)g(structure)h(admits)d(a)h(trivialization)d Fo(C)2365 5057 y Fn(1)2404 5042 y Fo(;)17 b(C)2518 5057 y Fn(2)2557 5042 y Fr(.)486 5158 y(By)28 b(Prop)s(osition)e(4.5)h(w)m (e)i(obtain)d(mo)s(del)g(Engel)h(structures)j Fp(D)2824 5173 y Fl(k)2866 5158 y Fo(;)17 b(k)31 b Fp(2)d Fh(Z)12 b Fp(n)g(f)p Fr(0)p Fp(g)386 5274 y Fr(on)33 b Fo(R)596 5289 y Fn(0)635 5274 y Fr(.)45 b(A)m(t)33 b(this)g(p)s(oin)m(t)f(w)m(e) i(do)f(not)g(need)h(to)e(deal)h(with)f(the)i(case)g Fo(k)d Fr(=)d(0.)45 b(The)386 5390 y(singular)31 b(foliation)e(on)k Fo(S)6 b Fr(\()p Fo(r)1421 5405 y Fn(0)1460 5390 y Fr(\))27 b Fp(\032)i Fo(@)1682 5405 y Fn(+)1741 5390 y Fo(R)1815 5405 y Fn(0)1887 5390 y Fr(is)k(the)g(same)f(as)h(the)g(original)c (singular)386 5506 y(foliation)h(on)k Fo(S)6 b Fr(\()p Fo(r)1054 5521 y Fn(0)1093 5506 y Fr(\).)48 b(By)34 b(Theorem)g(3.19)f (the)i(con)m(tact)f(structure)h(on)f Fo(@)3102 5521 y Fn(+)3162 5506 y Fo(R)3236 5521 y Fn(0)3309 5506 y Fr(is)386 5623 y(o)m(v)m(ert)m(wisted.)p eop %%Page: 27 27 27 26 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(27)486 459 y Fr(Since)47 b Fo(\031)810 474 y Fn(1)849 459 y Fr(\(SO\(4\)\))52 b(=)g Fh(Z)1429 474 y Fn(2)1512 459 y Fr(there)c(are)f(exactly)g(t)m(w)m(o)h(homotop)m(y)e(classes)i (of)386 575 y(orien)m(ted)34 b(framings)e(on)i Fo(R)1375 590 y Fn(0)1415 575 y Fr(.)48 b(The)35 b(homotop)m(y)e(class)i(of)e (the)i(Engel)e(framing)f(of)386 691 y Fp(D)463 706 y Fl(k)538 691 y Fr(dep)s(ends)i(only)e(on)h(the)g(parit)m(y)f(of)g Fo(k)s Fr(.)44 b(So)32 b(w)m(e)i(ha)m(v)m(e)g(sho)m(wn)386 833 y Fy(Prop)s(osition)47 b(4.7.)g Fm(F)-7 b(or)43 b(e)-5 b(ach)43 b(of)h(the)f(two)h(homotopy)g(classes)f(of)g(oriente)-5 b(d)386 949 y(fr)g(amings)39 b(on)h Fo(R)1010 964 y Fn(0)1090 949 y Fm(ther)-5 b(e)41 b(is)f(a)g(mo)-5 b(del)40 b(Engel)g(structur)-5 b(e)41 b(on)f Fo(R)2735 964 y Fn(0)2816 949 y Fm(such)g(that)h(the)386 1065 y(Engel)33 b(fr)-5 b(aming)33 b(is)h(homotopic)e(to)i(the)g(given) f(fr)-5 b(aming)33 b(of)h(the)g(tangent)f(bund)5 b(le)386 1182 y(of)49 b Fo(R)589 1197 y Fn(0)628 1182 y Fm(.)88 b(Mor)-5 b(e)g(over,)52 b(the)d(induc)-5 b(e)g(d)48 b(c)-5 b(ontact)49 b(structur)-5 b(e)50 b(on)f(the)g(b)-5 b(oundary)49 b(is)386 1298 y(overtwiste)-5 b(d.)386 1440 y Fy(Remark)38 b(4.8.)43 b Fr(J.)34 b(Adac)m(hi)g(asks)h(in)e([Ad])h(whether)h(it)e (is)h(p)s(ossible)f(to)g(\014nd)i(an)386 1556 y(Engel)30 b(structure)i(on)e Fo(N)e Fp(\002)19 b Fr([0)p Fo(;)e Fr(1])29 b(suc)m(h)j(that)e(the)h(b)s(oundary)g(is)f(transv)m(erse)j (and)386 1672 y(the)i(induced)h(con)m(tact)g(structures)h Fp(C)1780 1687 y Fn(0)1855 1672 y Fr(on)e Fo(N)f Fp(\002)25 b(f)p Fr(0)p Fp(g)34 b Fr(and)h Fp(C)2634 1687 y Fn(1)2709 1672 y Fr(on)g Fo(N)g Fp(\002)24 b(f)p Fr(1)p Fp(g)35 b Fr(are)386 1788 y(equiv)-5 b(alen)m(t)27 b(to)f(giv)m(en)i(con)m (tact)f(structures)i(on)e Fo(N)10 b Fr(.)42 b(One)27 b(can)h(sho)m(w)g(that)f Fp(C)3151 1803 y Fn(0)3218 1788 y Fr(and)386 1904 y Fp(C)438 1919 y Fn(1)515 1904 y Fr(ha)m(v)m(e)39 b(to)e(b)s(e)h(homotopic)e(as)i(plane)f(\014elds)h(on)f Fo(N)48 b Fr(as)38 b(in)f(Lemma)f(5.7.)58 b(The)386 2021 y(follo)m(wing)30 b(example)i(sho)m(ws)i(that)e Fp(C)1723 2036 y Fn(0)1796 2021 y Fr(and)g Fp(C)2037 2036 y Fn(1)2110 2021 y Fr(do)g(not)h(ha)m(v)m(e)h(to)e(b)s(e)h(equiv)-5 b(alen)m(t.)486 2137 y(Consider)37 b(the)g(con)m(tact)g(structure)i (from)c(ab)s(o)m(v)m(e)j(on)f Fo(B)5 b Fr(\()p Fo(r)2656 2152 y Fn(0)2695 2137 y Fr(\).)56 b(W)-8 b(e)37 b(remo)m(v)m(e)h(a)386 2253 y(ball)c(con)m(tained)i(in)f(a)h(Darb)s(oux)f(c)m(hart)i(with)e (con)m(v)m(ex)j(b)s(oundary)f Fo(S)2947 2217 y Fk(0)3006 2253 y Fr(from)d(the)386 2369 y(in)m(terior)g(of)i Fo(B)5 b Fr(\()p Fo(r)1009 2384 y Fn(0)1048 2369 y Fr(\))36 b(and)g(c)m(ho)s(ose)g(the)g(con)m(tact)h(v)m(ector)g(\014eld)e Fo(V)58 b Fr(suc)m(h)37 b(that)e(it)g(is)386 2486 y(transv)m(erse)27 b(to)e(b)s(oth)f Fo(S)1240 2449 y Fk(0)1288 2486 y Fr(and)h Fo(S)6 b Fr(\()p Fo(r)1618 2501 y Fn(0)1657 2486 y Fr(\).)41 b(By)26 b(Prop)s(osition)d(4.5)h(w)m(e)i(obtain)e(an)h(Engel)386 2602 y(structure)34 b(on)e(a)h(manifold)c(di\013eomorphic)i(to)h Fo(S)2219 2566 y Fn(2)2281 2602 y Fp(\002)22 b Fr([0)p Fo(;)17 b Fr(1])22 b Fp(\002)g Fo(S)2763 2566 y Fn(1)2835 2602 y Fr(suc)m(h)34 b(that)609 2744 y Fp(\017)41 b Fr(the)33 b(c)m(haracteristic)g(foliation)c(is)j(transv)m(erse)j(to)d(the)h(b)s (oundary)609 2860 y Fp(\017)41 b Fr(the)26 b(con)m(tact)g(structure)h (on)f(one)f(b)s(oundary)h(comp)s(onen)m(t)g(is)f(tigh)m(t)g(while)700 2976 y(it)32 b(is)g(o)m(v)m(ert)m(wisted)j(on)d(the)h(other)g(b)s (oundary)g(comp)s(onen)m(t.)386 3118 y(This)g(answ)m(ers)j(the)e (question)f(completely)g(for)f Fo(N)40 b Fr(=)29 b Fo(S)2459 3082 y Fn(2)2521 3118 y Fp(\002)23 b Fo(S)2687 3082 y Fn(1)2760 3118 y Fr(b)m(y)34 b(the)g(classi\014-)386 3234 y(cation)e(of)g(con)m(tact)h(structures)h(on)f Fo(S)1792 3198 y Fn(2)1853 3234 y Fp(\002)23 b Fo(S)2019 3198 y Fn(1)2058 3234 y Fr(.)386 3429 y(4.4.)48 b Fy(Mo)s(del)41 b(Engel)e(structures)h(on)g(round)h(handles)g(of)g(index)f Fr(1)p Fy(.)49 b Fr(On)386 3545 y Fo(R)460 3560 y Fn(1)527 3545 y Fr(=)28 b Fo(D)715 3509 y Fn(1)755 3545 y Fp(\002)q Fo(D)917 3509 y Fn(2)958 3545 y Fp(\002)q Fo(S)1102 3509 y Fn(1)1164 3545 y Fr(w)m(e)c(denote)f(the)f(co)s(ordinate)g(on)g Fo(D)2438 3509 y Fn(1)2499 3545 y Fr(b)m(y)i Fo(x)p Fr(,)g(the)f(co)s (ordinates)386 3661 y(on)43 b Fo(D)616 3625 y Fn(2)699 3661 y Fr(b)m(y)h Fo(y)893 3676 y Fn(1)932 3661 y Fo(;)17 b(y)1024 3676 y Fn(2)1106 3661 y Fr(and)43 b(the)h(co)s(ordinate)f(on)g Fo(S)2188 3625 y Fn(1)2271 3661 y Fr(b)m(y)h Fo(t)p Fr(.)76 b(On)43 b Fo(h)2784 3676 y Fn(1)2870 3661 y Fr(=)j Fo(D)3076 3625 y Fn(1)3145 3661 y Fp(\002)30 b Fo(D)3336 3625 y Fn(2)386 3777 y Fr(consider)1075 3911 y Fp(C)k Fr(=)28 b(k)m(er)q(\()p Fo(\013)g Fr(=)g Fp(\000)p Fo(dy)1803 3926 y Fn(1)1864 3911 y Fp(\000)23 b Fo(y)2012 3926 y Fn(2)2067 3911 y Fo(dx)f Fp(\000)h Fr(1)p Fo(=)p Fr(2)17 b Fo(x)g(dy)2630 3926 y Fn(2)2668 3911 y Fr(\))1055 4124 y Fo(V)49 b Fr(=)28 b(1)p Fo(=)p Fr(2)17 b Fo(y)1477 4139 y Fn(1)1568 4057 y Fo(@)p 1525 4101 144 4 v 1525 4193 a(@)5 b(y)1629 4208 y Fn(1)1701 4124 y Fr(+)22 b Fo(y)1847 4139 y Fn(2)1939 4057 y Fo(@)p 1896 4101 V 1896 4193 a(@)5 b(y)2000 4208 y Fn(2)2072 4124 y Fp(\000)22 b Fr(1)p Fo(=)p Fr(2)17 b Fo(x)2427 4057 y(@)p 2400 4101 113 4 v 2400 4193 a(@)5 b(x)386 4049 y Fr(\(20\))386 4346 y(These)34 b(c)m(hoices)f(are)f(motiv)-5 b(ated)31 b(b)m(y)i([W)-8 b(ei,)32 b(El2)o(]:)44 b(The)33 b(resulting)e(ev)m(en)j (con)m(tact)386 4462 y(structure)41 b(on)f Fo(R)1028 4477 y Fn(1)1108 4462 y Fr(is)g(de\014ned)h(b)m(y)g(the)g(con)m (traction)e(of)h(the)g(symplectic)g(form)386 4579 y Fo(dy)485 4594 y Fn(1)547 4579 y Fp(^)23 b Fo(dt)g Fr(+)g Fo(dx)g Fp(^)g Fo(dy)1161 4594 y Fn(2)1233 4579 y Fr(with)34 b(the)g(Liouville)d(v)m(ector)k(\014eld)e Fo(@)2584 4594 y Fl(t)2637 4579 y Fr(+)23 b Fo(V)f Fr(.)47 b(Notice)33 b(that)386 4695 y Fo(V)57 b Fr(is)34 b(transv)m(erse)k(to)d Fo(@)1238 4710 y Fk(\006)1297 4695 y Fo(h)1353 4710 y Fn(1)1428 4695 y Fr(and)g(p)s(oin)m(ts)g(out)m(w)m(ards)i(along)d Fo(@)2651 4710 y Fn(+)2710 4695 y Fo(h)2766 4710 y Fn(1)2838 4695 y Fr(=)e Fo(D)3030 4659 y Fn(1)3093 4695 y Fp(\002)25 b Fo(@)5 b(D)3335 4659 y Fn(2)386 4811 y Fr(and)33 b(in)m(w)m(ards)g (along)e Fo(@)1245 4826 y Fk(\000)1304 4811 y Fo(h)1360 4826 y Fn(1)1428 4811 y Fr(=)c Fo(@)5 b(D)1671 4775 y Fn(1)1733 4811 y Fp(\002)23 b Fo(D)1917 4775 y Fn(2)1956 4811 y Fr(.)44 b(W)-8 b(e)33 b(apply)f(Prop)s(osition)f(4.5)h(to)825 5029 y Fo(C)895 5044 y Fn(1)962 5029 y Fr(=)1119 4961 y Fo(@)p 1075 5006 144 4 v 1075 5097 a(@)5 b(y)1179 5112 y Fn(2)1251 5029 y Fp(\000)1388 4961 y Fo(@)p 1361 5006 113 4 v 1361 5097 a(@)g(x)1505 5029 y Fr(+)22 b(\()p Fo(y)1689 5044 y Fn(2)1750 5029 y Fp(\000)h Fr(1)p Fo(=)p Fr(2)17 b Fo(x)p Fr(\))2160 4961 y Fo(@)p 2117 5006 144 4 v 2117 5097 a(@)5 b(y)2221 5112 y Fn(1)825 5290 y Fo(C)895 5305 y Fn(2)962 5290 y Fr(=)27 b([)p Fo(V)5 b(;)17 b(C)1268 5305 y Fn(1)1307 5290 y Fr(])28 b(=)g Fp(\000)1597 5223 y Fo(@)p 1553 5267 V 1553 5359 a(@)5 b(y)1657 5374 y Fn(2)1729 5290 y Fp(\000)23 b Fr(1)p Fo(=)p Fr(2)2013 5223 y Fo(@)p 1986 5267 113 4 v 1986 5359 a(@)5 b(x)2130 5290 y Fr(+)22 b(1)p Fo(=)p Fr(2\()p Fo(y)2461 5305 y Fn(2)2521 5290 y Fr(+)g Fo(x)p Fr(\))2766 5223 y Fo(@)p 2722 5267 144 4 v 2722 5359 a(@)5 b(y)2826 5374 y Fn(1)2909 5290 y Fo(:)386 5160 y Fr(\(21\))386 5506 y(with)35 b Fo(")c Fr(=)h(1.)51 b(The)36 b(prop)s(erties)f(of)g(the)h(resulting)e (distributions)f Fp(D)2925 5521 y Fl(k)3003 5506 y Fr(are)i(sum-)386 5623 y(marized)c(in)h(the)h(follo)m(wing)d(prop)s(osition.)p eop %%Page: 28 28 28 27 bop 386 259 a Fq(28)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fy(Prop)s(osition)33 b(4.9.)40 b Fm(The)32 b(distributions)h Fp(D)2037 474 y Fl(k)2112 459 y Fm(ar)-5 b(e)33 b(mo)-5 b(del)32 b(Engel)g(structur)-5 b(es)34 b(on)386 575 y Fo(R)460 590 y Fn(1)534 575 y Fm(for)h(al)5 b(l)35 b Fo(k)30 b Fp(2)e Fh(Z)p Fm(.)43 b(They)34 b(have)g(the)h(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties.)556 766 y Fr(\(i\))40 b Fm(The)c(even)g(c)-5 b(ontact)36 b(structur)-5 b(e)37 b Fp(E)i Fr(=)31 b([)p Fp(D)2182 781 y Fl(k)2224 766 y Fo(;)17 b Fp(D)2345 781 y Fl(k)2387 766 y Fr(])37 b Fm(is)f(de\014ne)-5 b(d)35 b(by)i Fo(\014)f Fr(=)30 b Fo(\013)24 b Fr(+)700 882 y(1)p Fo(=)p Fr(2)17 b Fo(y)912 897 y Fn(1)967 882 y Fo(dt)p Fm(.)43 b(The)30 b(orientation)g(of)h(the) f(c)-5 b(ontact)31 b(structur)-5 b(e)31 b(on)f Fo(@)3016 897 y Fk(\000)3076 882 y Fo(R)3150 897 y Fn(1)3220 882 y Fm(and)700 998 y Fo(@)751 1013 y Fn(+)811 998 y Fo(R)885 1013 y Fn(1)966 998 y Fm(with)40 b(r)-5 b(esp)g(e)g(ct)41 b(to)g(the)g(r)-5 b(estriction)41 b(of)g Fo(d\014)46 b Fm(is)41 b(p)-5 b(ositive)40 b(if)h Fo(k)h Fp(\025)d Fr(0)700 1114 y Fm(and)c(ne)-5 b(gative)34 b(if)h Fo(k)30 b(<)e Fr(0)34 b Fm(.)529 1230 y Fr(\(ii\))39 b Fm(The)30 b(curves)g Fo(\015)1244 1245 y Fk(\006)1330 1230 y Fr(=)e Fp(f\006)p Fr(1)p Fp(g)12 b(\002)g(f)p Fr(0)p Fp(g)g(\002)g Fo(S)2077 1194 y Fn(1)2145 1230 y Fm(ar)-5 b(e)30 b(L)-5 b(e)g(gendrian.)42 b(The)30 b(r)-5 b(otation)700 1347 y(numb)g(er)35 b(along)f(them)h(is)f Fp(\000j)p Fo(k)s Fp(j)p Fm(.)502 1463 y Fr(\(iii\))k Fm(The)j(r)-5 b(otation)42 b(numb)-5 b(er)41 b(of)g(the)g(interse)-5 b(ction)41 b(line)g(\014eld)g(with)g(r)-5 b(esp)g(e)g(ct)700 1579 y(to)36 b(the)e(L)-5 b(e)g(gendrian)34 b(ve)-5 b(ctor)35 b(\014eld)1427 1892 y Fo(Z)g Fr(=)27 b Fo(y)1680 1907 y Fn(2)1747 1825 y Fo(@)p 1729 1869 93 4 v 1729 1960 a(@)5 b(t)1853 1892 y Fr(+)22 b(1)p Fo(=)p Fr(2)17 b Fo(y)2163 1907 y Fn(1)2239 1825 y Fo(@)p 2211 1869 113 4 v 2211 1960 a(@)5 b(x)700 2196 y Fm(along)34 b Fp(f)p Fr(0)p Fp(g)22 b(\002)g(f)p Fo(y)1327 2211 y Fn(1)1394 2196 y Fr(=)27 b(0)p Fo(;)17 b(y)1638 2211 y Fn(2)1704 2196 y Fr(=)28 b(1)p Fp(g)22 b(\002)g Fo(S)2094 2160 y Fn(1)2168 2196 y Fm(is)35 b Fp(\000j)p Fo(k)s Fp(j)f Fm(and)h(it)g(e)-5 b(quals)34 b Fr(0)h Fm(along)700 2313 y Fp(f)p Fr(0)p Fp(g)22 b(\002)h Fo(S)1037 2277 y Fn(1)1098 2313 y Fp(\002)g(f)p Fr(0)p Fp(g)p Fm(.)486 2503 y Fr(This)33 b(follo)m(ws)g(from)f(the)i(discussion)g(of)f(Prop)s(osition)f(4.5)h (and)h(calculations)386 2619 y(using)g(\(20\))d(and)i(\(21\).)386 2810 y Fy(Remark)52 b(4.10.)d Fr(The)d(con)m(tact)h(structure)g(on)e Fo(@)2306 2825 y Fk(\000)2366 2810 y Fo(R)2440 2825 y Fn(1)2525 2810 y Fr(is)h(de\014ned)h(b)m(y)g Fo(\014)3190 2825 y Fk(\000)3299 2810 y Fr(=)386 2926 y Fp(\000)p Fo(dy)562 2941 y Fn(1)630 2926 y Fr(+)28 b(1)p Fo(=)p Fr(2)17 b Fo(y)946 2941 y Fn(1)1001 2926 y Fo(dt)28 b Fp(\000)h Fr(1)p Fo(=)p Fr(2)17 b Fo(x)g(dy)1556 2941 y Fn(2)1636 2926 y Fr(with)42 b Fo(x)i Fr(=)g Fp(\006)p Fr(1.)72 b(It)42 b(is)g(in)m(v)-5 b(arian)m(t)40 b(under)j(the)386 3043 y(map)35 b Fo(f)654 3058 y Fl(s)727 3043 y Fr(induced)i(b)m(y)f (\()p Fo(y)1318 3058 y Fn(1)1357 3043 y Fo(;)17 b(y)1449 3058 y Fn(2)1488 3043 y Fr(\))34 b Fp(7\000)-17 b(!)34 b Fr(\()p Fo(sy)1886 3058 y Fn(1)1925 3043 y Fo(;)17 b(sy)2063 3058 y Fn(2)2101 3043 y Fr(\))36 b(for)g Fo(s)d Fp(2)h Fr(\(0)p Fo(;)17 b Fr(1\).)53 b(In)36 b(particular,)386 3159 y(if)g Fo( )41 b Fr(:)36 b Fo(@)698 3174 y Fk(\000)758 3159 y Fo(R)832 3174 y Fn(1)908 3159 y Fp(\000)-16 b(!)36 b Fo(@)1156 3174 y Fn(+)1215 3159 y Fo(M)49 b Fr(is)37 b(an)h(em)m(b)s(edding)f(preserving)i(con)m(tact)f(structures,)386 3275 y(then)i(the)h(same)f(is)f(true)h(for)g Fo( )31 b Fp(\016)c Fo(f)1737 3290 y Fl(s)1774 3275 y Fr(.)65 b(W)-8 b(e)41 b(can)f(c)m(ho)s(ose)h Fo(s)f Fr(so)g(small)d(that)j(the) 386 3391 y(image)29 b(of)i Fo( )23 b Fp(\016)c Fo(f)979 3406 y Fl(s)1047 3391 y Fr(is)31 b(con)m(tained)g(in)f(a)h(giv)m(en)g (tubular)g(neigh)m(b)s(orho)s(o)s(d)e(of)i Fo( )t Fr(\()p Fo(\015)3251 3406 y Fk(\006)3310 3391 y Fr(\).)486 3582 y(Up)39 b(to)f(no)m(w)i(w)m(e)g(ha)m(v)m(e)g(a)f(mo)s(del)e(Engel)i (structure)h(on)f Fo(R)2671 3597 y Fn(1)2750 3582 y Fr(for)f(eac)m(h)i (orien-)386 3698 y(tation)g(of)h(the)h(con)m(tact)g(structure)g(on)f Fo(@)1953 3713 y Fk(\000)2013 3698 y Fo(R)2087 3713 y Fn(1)2168 3698 y Fr(and)g(eac)m(h)i(homotop)m(y)e(class)g(of)386 3814 y(in)m(tersection)33 b(line)e(\014elds)i(with)g(negativ)m(e)g (rotation)e(n)m(um)m(b)s(er)i(along)f Fo(\015)2993 3829 y Fk(\006)3051 3814 y Fr(.)45 b(More-)386 3931 y(o)m(v)m(er,)c(w)m(e)e (ha)m(v)m(e)g(a)f(mo)s(del)f(Engel)h(structure)h(with)f(rotation)f(n)m (um)m(b)s(er)h(0)g(along)386 4047 y Fo(\015)437 4062 y Fk(\006)529 4047 y Fr(for)33 b(one)g(of)g(the)h(t)m(w)m(o)g(p)s (ossible)f(orien)m(tations)f(of)h(the)h(con)m(tact)g(structure)h(on)386 4163 y Fo(@)437 4178 y Fk(\000)496 4163 y Fo(R)570 4178 y Fn(1)610 4163 y Fr(.)486 4279 y(In)43 b(order)h(to)f(obtain)f(the)i (missing)e(p)s(ossibilities)f(w)m(e)j(extend)h(the)f(con)m(tact)386 4395 y(di\013eomorphism)996 4660 y Fo(f)39 b Fr(:)27 b(\()p Fo(x;)17 b(y)1322 4675 y Fn(1)1361 4660 y Fo(;)g(y)1453 4675 y Fn(2)1492 4660 y Fo(;)g(t)p Fr(\))28 b Fp(7\000)-16 b(!)27 b Fr(\()p Fo(x;)17 b(y)2010 4675 y Fn(1)2049 4660 y Fo(;)g Fp(\000)p Fo(y)2218 4675 y Fn(2)2279 4660 y Fp(\000)23 b Fr(4)p Fo(xy)2531 4675 y Fn(1)2570 4660 y Fo(;)17 b Fp(\000)p Fo(t)p Fr(\))386 4925 y(of)30 b(a)g(neigh)m(b)s (orho)s(o)s(d)f(of)h Fo(\015)1344 4940 y Fk(\006)1432 4925 y Fr(in)g Fo(@)1595 4940 y Fk(\000)1654 4925 y Fo(R)1728 4940 y Fn(1)1798 4925 y Fr(to)g(a)g(di\013eomorphism)e(of)i Fo(R)2861 4940 y Fn(1)2900 4925 y Fr(.)43 b(This)30 b(map)386 5042 y(preserv)m(es)41 b Fo(\015)861 5057 y Fk(\006)957 5042 y Fr(and)d(the)h(con)m(tact)f(structure)h(near)f Fo(@)2371 5057 y Fk(\000)2431 5042 y Fo(R)2505 5057 y Fn(1)2583 5042 y Fr(but)g(it)f(rev)m(erses)j(the)386 5158 y(orien)m(tations)34 b(of)h Fo(\015)1088 5173 y Fk(\006)1182 5158 y Fr(and)h(of)f(the)h(con)m(tact)g(structure.)53 b(W)-8 b(e)36 b(push)h(forw)m(ard)e(the)386 5274 y(Engel)f(structures)i Fp(D)1195 5289 y Fl(k)1271 5274 y Fr(w)m(e)f(ha)m(v)m(e)h(obtained)d (so)i(far)e(using)h(the)g(extension)h(of)f Fo(f)11 b Fr(.)386 5390 y(Then)34 b(the)f(rotation)e(n)m(um)m(b)s(er)i(along)e Fo(\015)1849 5405 y Fk(\006)1940 5390 y Fr(c)m(hanges)j(its)e(sign.)486 5506 y(Up)c(to)h(no)m(w)g(all)e(mo)s(del)g(Engel)h(structures)j(induce) e(the)g(same)f(con)m(tact)h(fram-)386 5623 y(ing)34 b(along)f Fo(\015)864 5638 y Fk(\006)923 5623 y Fr(.)50 b(In)35 b(order)f(to)h(realize)e(other)i(con)m(tact)h(framings)d(one)i(can)g (push)p eop %%Page: 29 29 29 28 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(29)386 459 y Fr(forw)m(ard)39 b(the)g(mo)s(del)f(Engel)g(structures)j (obtained)d(so)h(far)g(with)f(the)i(self)e(dif-)386 575 y(feomorphisms)31 b(\002)1078 590 y Fl(m)1177 575 y Fr(of)h Fo(R)1362 590 y Fn(1)818 778 y Fr(\002)894 793 y Fl(m)988 778 y Fr(:)c Fo(D)1127 737 y Fn(2)1188 778 y Fp(\002)23 b Fo(I)30 b Fp(\002)23 b Fo(S)1527 737 y Fn(1)1594 778 y Fp(\000)-16 b(!)27 b Fo(D)1866 737 y Fn(2)1927 778 y Fp(\002)c Fo(I)30 b Fp(\002)22 b Fo(S)2265 737 y Fn(1)1094 996 y Fr(\()p Fo(y)1180 1011 y Fn(1)1219 996 y Fo(;)17 b(y)1311 1011 y Fn(2)1350 996 y Fo(;)g(x;)g(t)p Fr(\))28 b Fp(7\000)-16 b(!)1823 936 y Fr(\(cos)q(\()p Fo(mt)p Fr(\))p Fo(y)2236 951 y Fn(1)2297 936 y Fp(\000)23 b Fr(sin\()p Fo(mt)p Fr(\))p Fo(y)2761 951 y Fn(2)2800 936 y Fo(;)1921 1052 y Fr(sin\()p Fo(mt)p Fr(\))p Fo(y)2285 1067 y Fn(1)2346 1052 y Fr(+)f(cos)q(\()p Fo(mt)p Fr(\))p Fo(y)2819 1067 y Fn(2)2858 1052 y Fo(;)17 b(x;)g(t)p Fr(\))386 916 y(\(22\))386 1256 y(with)39 b Fo(m)h Fp(2)f Fh(Z)p Fr(.)62 b(This)39 b(map)g(preserv)m(es)j Fo(\015)1932 1271 y Fk(\006)2030 1256 y Fr(as)e(w)m(ell)e(as)i Fo(@)2539 1271 y Fk(\000)2598 1256 y Fo(R)2672 1271 y Fn(1)2712 1256 y Fr(.)64 b(No)m(w)40 b(w)m(e)g(ha)m(v)m(e)386 1373 y(sho)m(wn)34 b(the)f(follo)m(wing)d(prop)s(osition.)386 1535 y Fy(Prop)s(osition)25 b(4.11.)35 b Fm(F)-7 b(or)26 b(e)-5 b(ach)26 b(oriente)-5 b(d)26 b(fr)-5 b(aming)26 b(of)g Fo(\015)2509 1550 y Fk(\006)2595 1535 y Fp(\032)j Fo(@)2752 1550 y Fk(\000)2811 1535 y Fo(R)2885 1550 y Fn(1)2951 1535 y Fm(and)d Fo(k)31 b Fp(2)d Fh(Z)386 1651 y Fm(ther)-5 b(e)31 b(is)h(a)f(mo)-5 b(del)31 b(Engel)g(structur)-5 b(e)33 b(on)e Fo(R)1969 1666 y Fn(1)2040 1651 y Fm(such)g(that)h(the)g (c)-5 b(ontact)31 b(fr)-5 b(aming)31 b(is)386 1767 y(homotopic)g(to)h (the)g(given)f(fr)-5 b(aming)31 b(and)g(the)h(r)-5 b(otation)32 b(numb)-5 b(er)31 b(along)h Fo(\015)3099 1782 y Fk(\006)3189 1767 y Fm(is)g Fo(k)s Fm(.)486 1883 y(Given)h(one)g(such)h(mo)-5 b(del)33 b(Engel)g(structur)-5 b(e,)35 b(ther)-5 b(e)34 b(is)f(another)g(mo)-5 b(del)33 b(Engel)386 2000 y(structur)-5 b(e)42 b(which)e(induc)-5 b(es)40 b(the)h(same)f(c)-5 b(ontact)41 b(structur)-5 b(e)42 b(on)e Fo(@)2816 2015 y Fk(\000)2875 2000 y Fo(R)2949 2015 y Fn(1)3030 2000 y Fm(with)h(the)386 2116 y(same)d(r)-5 b(otation)39 b(numb)-5 b(er)39 b(along)f Fo(\015)1679 2131 y Fk(\006)1776 2116 y Fm(but)i(which)e(induc)-5 b(es)38 b(the)h(opp)-5 b(osite)38 b(orien-)386 2232 y(tation)d(of)f(the)h(c)-5 b(ontact)35 b(structur)-5 b(e)36 b(on)e Fo(@)1886 2247 y Fk(\000)1946 2232 y Fo(R)2020 2247 y Fn(1)2060 2232 y Fm(.)486 2394 y Fr(F)-8 b(or)38 b Fo(k)s(;)17 b(m)40 b Fp(2)g Fh(Z)d Fr(let)i Fp(D)1327 2409 y Fl(k)r(;m)1492 2394 y Fr(for)g Fo(k)s(;)17 b(m)39 b Fp(2)i Fh(Z)c Fr(b)s(e)i(the)i(push)f(forw)m(ard)g (with)f(\002)3308 2409 y Fl(m)386 2511 y Fr(of)45 b(a)h(mo)s(del)e (Engel)h(structure)i(with)e(rotation)f(n)m(um)m(b)s(er)i Fo(k)j Fr(along)44 b Fo(\015)3038 2526 y Fk(\006)3097 2511 y Fr(.)83 b(The)386 2627 y(orien)m(tation)37 b(of)i(the)h(con)m (tact)f(structure)i(on)e Fo(@)2149 2642 y Fk(\006)2208 2627 y Fo(R)2282 2642 y Fn(1)2361 2627 y Fr(do)s(es)h(not)f(app)s(ear)g (in)f(this)386 2743 y(notation.)386 3020 y(4.5.)48 b Fy(Mo)s(del)36 b(Engel)f(structures)g(on)g(round)i(handles)f(of)g (index)f Fr(2)p Fy(.)49 b Fr(The)386 3136 y(mo)s(del)20 b(Engel)i(structures)i(on)e Fo(R)1572 3151 y Fn(2)1640 3136 y Fr(=)27 b Fo(D)1827 3100 y Fn(2)1867 3136 y Fp(\002)q Fo(D)2029 3100 y Fn(1)2070 3136 y Fp(\002)q Fo(S)2214 3100 y Fn(1)2275 3136 y Fr(dep)s(end)d(on)e(t)m(w)m(o)h(parameters)386 3252 y Fo(n;)17 b(k)25 b Fr(whic)m(h)e(corresp)s(ond)g(to)f(the)h (homotop)m(y)f(class)h(of)f(the)g(in)m(tersection)h(line)e(\014eld)386 3368 y(near)37 b(the)h(torus)f Fo(T)1105 3332 y Fn(2)1091 3393 y(0)1179 3368 y Fr(=)e Fo(@)5 b(D)1430 3332 y Fn(2)1496 3368 y Fp(\002)25 b(f)p Fr(0)p Fp(g)g(\002)h Fo(S)1941 3332 y Fn(1)2015 3368 y Fp(\032)36 b Fo(@)2179 3383 y Fk(\000)2238 3368 y Fo(R)2312 3383 y Fn(2)2352 3368 y Fr(.)57 b(The)38 b(con)m(tact)f(structure)386 3484 y(on)32 b Fo(@)572 3499 y Fk(\000)632 3484 y Fo(R)706 3499 y Fn(2)778 3484 y Fr(is)g(essen)m(tially)g(indep)s(enden)m(t)i(of)e(the)h (mo)s(del)e(Engel)i(structure.)386 3647 y Fy(Prop)s(osition)41 b(4.12.)j Fm(Given)39 b(inte)-5 b(gers)39 b Fo(n)d Fp(2)g Fh(Z)h Fm(and)h Fo(k)h Fp(2)d Fh(Z)23 b Fp(n)i(f)p Fr(0)p Fp(g)39 b Fm(ther)-5 b(e)39 b(is)g(a)386 3763 y(mo)-5 b(del)34 b(Engel)g(structur)-5 b(e)36 b Fp(D)30 b Fr(=)e Fp(D)1634 3778 y Fl(k)r(;n)1774 3763 y Fm(on)34 b Fo(R)1987 3778 y Fn(2)2062 3763 y Fm(with)h(the)f(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties.)556 3925 y Fr(\(i\))40 b Fm(The)k(char)-5 b(acteristic)44 b(foliation)g(of)g Fp(D)j Fm(c)-5 b(an)44 b(b)-5 b(e)44 b(oriente)-5 b(d)43 b(such)i(that)f(it)700 4041 y(p)-5 b(oints)35 b(outwar)-5 b(ds)35 b(along)f Fo(@)1707 4056 y Fn(+)1766 4041 y Fo(R)1840 4056 y Fn(2)1915 4041 y Fm(and)g(inwar)-5 b(ds)34 b(along)g Fo(@)2774 4056 y Fk(\000)2834 4041 y Fo(R)2908 4056 y Fn(2)2947 4041 y Fm(.)529 4157 y Fr(\(ii\))39 b Fm(The)g(singular)h(foliation)e (on)i Fo(T)1888 4121 y Fn(2)1874 4182 y(0)1963 4157 y Fr(=)c Fo(@)5 b(D)2215 4121 y Fn(2)2281 4157 y Fp(\002)26 b(f)p Fr(0)p Fp(g)f(\002)h Fo(S)2727 4121 y Fn(1)2803 4157 y Fp(\032)37 b Fo(@)2968 4172 y Fk(\000)3027 4157 y Fo(R)3101 4172 y Fn(2)3181 4157 y Fm(is)i(in)700 4274 y(standar)-5 b(d)45 b(form.)77 b(In)44 b(p)-5 b(articular,)48 b Fo(T)2113 4237 y Fn(2)2099 4298 y(0)2198 4274 y Fm(is)d(c)-5 b(onvex.)76 b(The)45 b(L)-5 b(e)g(gendrian)700 4390 y(ruling)38 b(c)-5 b(orr)g(esp)g(onds)36 b(to)i(the)g(\014rst)g(factor)f(of)h Fo(T)2479 4354 y Fn(2)2465 4414 y(0)2551 4390 y Fr(=)32 b Fo(@)5 b(D)2799 4354 y Fn(2)2864 4390 y Fp(\002)24 b(f)p Fr(0)p Fp(g)g(\002)h Fo(S)3306 4354 y Fn(1)3345 4390 y Fm(.)700 4506 y(The)35 b(dividing)e(curves)i(ar)-5 b(e)35 b(tangent)f(to)h(the)g(last)g(factor.)502 4622 y Fr(\(iii\))j Fm(The)47 b(r)-5 b(otation)47 b(numb)-5 b(er)47 b(of)g(the)h(interse)-5 b(ction)46 b(line)h(\014eld)g(along)f Fo(\015)56 b Fr(=)700 4739 y Fo(@)5 b(D)840 4702 y Fn(2)901 4739 y Fp(\002)21 b(f)p Fr(0)p Fp(g)e(\002)i(f)p Fr(1)p Fp(g)33 b Fm(\(with)h(its)g(orientation)g(as)f(b)-5 b(oundary)34 b Fo(@)5 b(D)3020 4702 y Fn(2)3060 4739 y Fm(\))34 b(is)g Fr(2)p Fo(n)p Fm(.)505 4855 y Fr(\(iv\))40 b Fm(The)30 b(r)-5 b(otation)31 b(numb)-5 b(er)30 b(of)g(the)h(interse)-5 b(ction)30 b(line)g(\014eld)g(along)f(the)i(L)-5 b(e)g(g-)700 4971 y(endrian)28 b(divides)e(\(with)i(the)g(c)-5 b(anonic)g(al)27 b(orientation)g(of)h(the)g(last)g(factor)700 5087 y(of)35 b Fo(@)5 b(D)955 5051 y Fn(2)1017 5087 y Fp(\002)23 b(f)p Fr(0)p Fp(g)e(\002)i Fo(S)1453 5051 y Fn(1)1492 5087 y Fm(\))35 b(is)g Fo(k)30 b Fp(6)p Fr(=)e(0)p Fm(.)532 5203 y Fr(\(v\))41 b Fm(The)31 b(orientation)g(of)g(the)g(c)-5 b(ontact)31 b(structur)-5 b(e)32 b(on)f Fo(@)2595 5218 y Fn(+)2654 5203 y Fo(R)2728 5218 y Fn(2)2799 5203 y Fm(c)-5 b(an)31 b(b)-5 b(e)31 b(chosen)700 5320 y(fr)-5 b(e)g(ely.)386 5482 y(A)n(l)5 b(l)52 b(mo)-5 b(del)51 b(Engel)h(structur)-5 b(es)53 b(induc)-5 b(e)51 b(the)i(same)e(c)-5 b(ontact)52 b(structur)-5 b(e)53 b(on)f(a)386 5598 y(neighb)-5 b(orho)g(o)g(d)33 b(of)i Fo(T)1151 5562 y Fn(2)1137 5623 y(0)1217 5598 y Fp(\032)29 b Fo(@)1374 5613 y Fk(\000)1433 5598 y Fo(R)1507 5613 y Fn(2)1547 5598 y Fm(.)p eop %%Page: 30 30 30 29 bop 386 259 a Fq(30)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fm(Pr)-5 b(o)g(of.)41 b Fr(W)-8 b(e)40 b(c)m(ho)s(ose)g(the)f(ev) m(en)i(con)m(tact)e(structure)i(on)e Fo(R)2569 474 y Fn(2)2647 459 y Fr(=)g Fo(D)2846 423 y Fn(2)2912 459 y Fp(\002)27 b Fo(D)3100 423 y Fn(1)3165 459 y Fp(\002)g Fo(S)3335 423 y Fn(1)386 575 y Fr(\014rst.)58 b(The)38 b(starting)e(p)s(oin)m(t)g(is)h(a)g(singular)f(foliation)d Fp(F)47 b Fr(on)37 b(a)g(disc)g Fo(D)3049 539 y Fn(2)3125 575 y Fr(where)386 691 y(w)m(e)d(use)f(p)s(olar)e(co)s(ordinates)h(\()p Fo(r)m(;)17 b(')p Fr(\).)43 b(The)34 b(co)s(ordinate)d(on)i Fo(D)2663 655 y Fn(1)2735 691 y Fr(is)f Fo(x)p Fr(.)486 807 y(On)h Fo(A)28 b Fr(=)g Fp(f)p Fo(r)j(>)d Fr(1)p Fo(=)p Fr(2)p Fp(g)k Fr(w)m(e)i(de\014ne)g Fp(F)43 b Fr(b)m(y)34 b(cos\()p Fo(')p Fr(\))p Fo(dr)s Fr(.)44 b(In)33 b(addition)e(w)m(e)j(require)386 924 y(that)e Fp(F)42 b Fr(admits)32 b(a)g(dividing)f(set)i(\000)f(with)g(the)h (follo)m(wing)d(prop)s(erties.)556 1062 y(\(i\))40 b(The)e(straigh)m(t) e(arc)h Fo(\015)1488 1077 y Fn(0)1563 1062 y Fr(from)e(\()p Fo(r)j Fr(=)c(1)p Fo(;)17 b(')34 b Fr(=)g(0\))i(to)h(\()p Fo(r)g Fr(=)d(1)p Fo(;)17 b(')34 b Fr(=)g Fo(\031)t Fr(\))j(lies)700 1178 y(in)32 b(\000.)44 b(In)33 b(addition,)d(\000)j(con)m(tains)g Fp(j)p Fo(n)p Fp(j)f Fr(closed)g(circles.)529 1294 y(\(ii\))39 b(Except)46 b(for)c Fo(\015)1251 1309 y Fn(0)1290 1294 y Fr(,)47 b(ev)m(ery)e(comp)s(onen)m(t)e(of)g(\000)h(is)f(closed)g(and) h(b)s(ounds)g(a)700 1410 y(disc)38 b(con)m(taining)e(no)i(other)f(comp) s(onen)m(ts)h(of)f(\000.)58 b(If)38 b Fo(n)e(>)g Fr(0)h(all)e(closed) 700 1527 y(comp)s(onen)m(ts)f(lie)d(b)s(elo)m(w)i Fo(\015)1696 1542 y Fn(0)1768 1527 y Fr(in)f Fo(D)1966 1490 y Fn(2)2028 1527 y Fp(n)22 b Fo(\015)2151 1542 y Fn(0)2223 1527 y Fr(and)33 b(if)f Fo(n)c(<)g Fr(0)33 b(they)h(lie)d(in)h(the)700 1643 y(upp)s(er)i(half{disc.)386 1781 y(Figure)k(6)h(sho)m(ws)i(a)d(p)s (ossible)h Fp(F)48 b Fr(for)39 b Fo(n)g Fr(=)g(2.)62 b(The)40 b(thic)m(k)m(ened)h(curv)m(es)g(divide)386 1897 y Fp(F)10 b Fr(.)44 b(Similar)29 b(singular)i(foliations)f(can)j(b)s(e) g(found)g(other)g Fo(n)p Fr(.)45 b(By)33 b(Theorem)g(3.11)1289 3253 y @beginspecial 0 @llx 0 @lly 142 @urx 142 @ury 1420 @rwi @setspecial %%BeginDocument: artr2.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: r2.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Wed Aug 18 15:13:23 2004 %%For: tvogel@math81 (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 142 142 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -3.0 144.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /endash 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 3395 m -1000 -1000 l 3409 -1000 l 3409 3395 l cp clip 0.06000 0.06000 sc 7.500 slw % Ellipse n 1220 1613 12 12 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr 15.000 slw % Ellipse n 1489 1603 216 216 0 360 DrawEllipse gs col0 s gr % Ellipse n 960 1603 206 206 0 360 DrawEllipse gs col0 s gr % Polyline 7.500 slw n 1196 1613 m 840 1613 l gs col0 s gr % Polyline n 1253 1613 m 1609 1613 l gs col0 s gr % Polyline n 1218 2193 m 1218 1649 l gs col0 s gr % Polyline n 1228 1585 m 1228 343 l gs col0 s gr % Polyline [15 45] 45 sd n 1230 59 m 1230 299 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 1218 2214 m 1218 2375 l gs col0 s gr [] 0 sd % Polyline 15.000 slw n 75 1194 m 2387 1189 l gs col0 s gr % Polyline 7.500 slw n 1172 1132 m 1172 1315 l gs col0 s gr % Polyline n 1023 1132 m 1023 1315 l gs col0 s gr % Polyline n 833 1132 m 833 1315 l gs col0 s gr % Polyline n 643 1132 m 641 1323 l gs col0 s gr % Polyline n 840 1611 m 843 1611 l 849 1610 l 861 1610 l 877 1609 l 898 1607 l 922 1606 l 948 1604 l 975 1602 l 1001 1600 l 1025 1598 l 1047 1596 l 1066 1593 l 1082 1591 l 1095 1588 l 1106 1585 l 1120 1579 l 1130 1573 l 1136 1565 l 1141 1557 l 1144 1549 l 1146 1541 l 1148 1532 l 1151 1524 l 1154 1515 l 1157 1507 l 1160 1496 l 1162 1485 l 1163 1474 l 1163 1463 l 1164 1451 l 1164 1440 l 1165 1428 l 1166 1416 l 1167 1406 l 1168 1395 l 1169 1382 l 1171 1368 l 1172 1353 l 1174 1338 l 1175 1326 l 1176 1319 l 1176 1316 l 1176 1315 l gs col0 s gr % Polyline n 855 1603 m 855 1604 l 858 1608 l 864 1616 l 872 1628 l 882 1642 l 890 1654 l 898 1666 l 906 1678 l 913 1688 l 921 1699 l 930 1712 l 940 1725 l 950 1739 l 960 1753 l 969 1766 l 976 1778 l 981 1788 l 982 1796 l 980 1802 l 976 1808 l 969 1812 l 961 1815 l 951 1818 l 941 1820 l 931 1822 l 922 1824 l 912 1824 l 902 1824 l 891 1822 l 880 1819 l 867 1814 l 854 1809 l 840 1803 l 826 1797 l 813 1791 l 801 1785 l 789 1778 l 779 1772 l 770 1765 l 762 1758 l 754 1751 l 747 1743 l 740 1736 l 734 1728 l 728 1720 l 721 1711 l 715 1702 l 709 1692 l 703 1681 l 698 1669 l 693 1656 l 688 1642 l 684 1628 l 679 1614 l 675 1600 l 670 1587 l 667 1575 l 663 1563 l 659 1550 l 656 1538 l 653 1527 l 651 1516 l 649 1506 l 647 1495 l 645 1482 l 643 1468 l 642 1457 l 641 1445 l 641 1431 l 640 1415 l 640 1397 l 639 1379 l 639 1361 l 638 1345 l 638 1334 l 638 1326 l 638 1323 l gs col0 s gr % Polyline n 855 1593 m 855 1592 l 853 1587 l 850 1575 l 845 1559 l 840 1542 l 836 1526 l 833 1513 l 831 1502 l 830 1492 l 829 1483 l 829 1474 l 830 1465 l 830 1457 l 831 1448 l 831 1438 l 831 1427 l 831 1417 l 831 1405 l 832 1391 l 832 1375 l 832 1358 l 832 1341 l 833 1328 l 833 1319 l 833 1316 l 833 1315 l gs col0 s gr % Polyline n 1024 1312 m 1024 1313 l 1024 1317 l 1024 1328 l 1024 1344 l 1024 1363 l 1024 1384 l 1024 1403 l 1023 1419 l 1022 1433 l 1020 1444 l 1017 1456 l 1013 1466 l 1010 1475 l 1006 1484 l 1002 1492 l 997 1500 l 990 1509 l 981 1519 l 973 1526 l 964 1533 l 952 1542 l 938 1551 l 922 1561 l 906 1572 l 889 1582 l 875 1591 l 865 1597 l 858 1601 l 855 1603 l gs col0 s gr % Polyline n 1222 305 m 1174 1129 l gs col0 s gr % Polyline n 447 1130 m 447 1126 l 448 1119 l 449 1107 l 450 1092 l 451 1073 l 453 1054 l 455 1035 l 457 1017 l 459 1002 l 461 987 l 464 974 l 467 961 l 471 947 l 476 933 l 480 919 l 485 905 l 491 891 l 496 877 l 501 863 l 506 849 l 512 836 l 517 824 l 522 813 l 528 801 l 535 789 l 541 779 l 547 770 l 553 761 l 560 752 l 566 743 l 573 733 l 581 723 l 589 713 l 596 704 l 604 695 l 612 686 l 621 676 l 629 667 l 638 657 l 647 648 l 656 638 l 665 629 l 674 620 l 683 612 l 691 604 l 701 595 l 711 587 l 721 580 l 731 573 l 740 567 l 750 560 l 760 553 l 770 547 l 781 539 l 792 532 l 801 526 l 811 519 l 820 513 l 829 508 l 837 502 l 845 496 l 854 490 l 863 484 l 874 477 l 886 470 l 901 462 l 917 453 l 929 446 l 943 439 l 959 431 l 977 422 l 998 412 l 1021 400 l 1045 388 l 1071 376 l 1097 363 l 1123 350 l 1147 338 l 1168 328 l 1185 319 l 1199 313 l 1208 309 l 1213 306 l 1215 305 l gs col0 s gr % Polyline n 643 1132 m 643 1129 l 644 1122 l 645 1111 l 647 1095 l 649 1076 l 651 1054 l 654 1031 l 657 1010 l 660 989 l 662 971 l 665 956 l 668 942 l 671 930 l 676 915 l 681 901 l 687 888 l 694 877 l 700 865 l 707 855 l 714 844 l 721 833 l 728 821 l 735 810 l 743 799 l 751 788 l 759 778 l 767 768 l 776 758 l 784 748 l 793 738 l 802 727 l 811 717 l 821 706 l 829 697 l 838 688 l 846 680 l 853 672 l 860 665 l 868 658 l 876 651 l 884 642 l 894 633 l 905 621 l 918 608 l 933 592 l 943 581 l 955 569 l 968 555 l 983 540 l 999 523 l 1018 503 l 1037 482 l 1059 460 l 1081 437 l 1103 413 l 1125 390 l 1146 368 l 1165 348 l 1181 331 l 1194 317 l 1203 307 l 1210 301 l 1213 297 l 1215 295 l gs col0 s gr % Polyline n 1023 1127 m 1023 1125 l 1023 1120 l 1023 1111 l 1023 1099 l 1023 1084 l 1023 1066 l 1024 1047 l 1025 1027 l 1027 1006 l 1030 984 l 1033 961 l 1038 937 l 1044 909 l 1048 891 l 1053 871 l 1059 850 l 1065 825 l 1073 798 l 1081 769 l 1090 736 l 1100 701 l 1111 664 l 1122 626 l 1133 586 l 1145 546 l 1156 506 l 1167 468 l 1178 433 l 1187 401 l 1195 374 l 1202 351 l 1207 333 l 1211 320 l 1213 312 l 1214 307 l 1215 305 l gs col0 s gr % Polyline n 1277 1132 m 1277 1315 l gs col0 s gr % Polyline n 1426 1132 m 1426 1315 l gs col0 s gr % Polyline n 1615 1132 m 1615 1315 l gs col0 s gr % Polyline n 1808 1132 m 1811 1323 l gs col0 s gr % Polyline n 2001 1132 m 2001 1315 l gs col0 s gr % Polyline n 1609 1611 m 1606 1611 l 1600 1610 l 1588 1610 l 1572 1609 l 1552 1607 l 1528 1606 l 1502 1604 l 1475 1602 l 1449 1600 l 1425 1598 l 1404 1596 l 1385 1593 l 1369 1591 l 1356 1588 l 1345 1585 l 1331 1579 l 1321 1573 l 1314 1565 l 1310 1557 l 1307 1549 l 1304 1541 l 1302 1532 l 1299 1524 l 1295 1515 l 1292 1507 l 1289 1496 l 1287 1485 l 1286 1474 l 1285 1462 l 1285 1451 l 1284 1439 l 1284 1428 l 1283 1416 l 1283 1404 l 1282 1390 l 1282 1373 l 1283 1355 l 1283 1338 l 1283 1326 l 1283 1320 l 1283 1319 l gs col0 s gr % Polyline n 1593 1603 m 1593 1604 l 1593 1607 l 1594 1615 l 1594 1627 l 1594 1642 l 1594 1658 l 1593 1673 l 1591 1687 l 1587 1699 l 1582 1710 l 1576 1719 l 1568 1728 l 1558 1738 l 1547 1748 l 1534 1758 l 1521 1768 l 1508 1778 l 1496 1788 l 1485 1797 l 1476 1806 l 1470 1813 l 1466 1820 l 1465 1825 l 1465 1829 l 1467 1833 l 1469 1836 l 1473 1840 l 1477 1843 l 1482 1846 l 1487 1849 l 1492 1851 l 1497 1853 l 1503 1855 l 1509 1855 l 1515 1856 l 1522 1855 l 1531 1853 l 1542 1850 l 1554 1845 l 1567 1840 l 1581 1834 l 1595 1828 l 1610 1821 l 1624 1814 l 1637 1808 l 1649 1801 l 1661 1794 l 1671 1787 l 1682 1778 l 1692 1768 l 1701 1757 l 1710 1746 l 1718 1734 l 1726 1723 l 1734 1711 l 1741 1700 l 1748 1689 l 1755 1678 l 1761 1667 l 1766 1657 l 1770 1646 l 1774 1636 l 1778 1626 l 1781 1616 l 1785 1605 l 1788 1595 l 1791 1584 l 1794 1574 l 1796 1564 l 1798 1553 l 1800 1543 l 1801 1533 l 1803 1523 l 1804 1513 l 1805 1502 l 1806 1492 l 1807 1480 l 1808 1468 l 1809 1457 l 1809 1445 l 1810 1431 l 1810 1415 l 1811 1397 l 1811 1378 l 1811 1360 l 1812 1345 l 1812 1333 l 1812 1326 l 1812 1323 l gs col0 s gr % Polyline n 1593 1593 m 1593 1592 l 1595 1587 l 1599 1575 l 1603 1559 l 1608 1542 l 1613 1526 l 1616 1513 l 1618 1502 l 1619 1492 l 1620 1483 l 1620 1474 l 1619 1465 l 1619 1457 l 1618 1448 l 1618 1438 l 1618 1427 l 1618 1417 l 1618 1405 l 1617 1391 l 1617 1375 l 1616 1358 l 1616 1341 l 1615 1328 l 1615 1319 l 1615 1316 l 1615 1315 l gs col0 s gr % Polyline n 1427 1315 m 1427 1316 l 1427 1320 l 1427 1330 l 1427 1346 l 1427 1365 l 1427 1385 l 1428 1404 l 1429 1420 l 1430 1433 l 1432 1444 l 1435 1456 l 1438 1466 l 1442 1475 l 1445 1484 l 1450 1492 l 1455 1500 l 1461 1509 l 1470 1519 l 1477 1526 l 1487 1533 l 1498 1542 l 1512 1551 l 1527 1561 l 1544 1572 l 1559 1582 l 1573 1591 l 1584 1597 l 1590 1601 l 1593 1603 l gs col0 s gr % Polyline n 1228 305 m 1283 1130 l gs col0 s gr % Polyline n 1998 1127 m 1998 1126 l 1998 1123 l 1998 1116 l 1997 1103 l 1996 1087 l 1996 1068 l 1995 1048 l 1993 1028 l 1992 1011 l 1990 995 l 1988 981 l 1985 968 l 1982 956 l 1978 944 l 1973 931 l 1968 920 l 1963 908 l 1957 896 l 1952 884 l 1946 873 l 1941 861 l 1935 850 l 1930 839 l 1925 828 l 1919 815 l 1912 803 l 1906 792 l 1900 780 l 1894 769 l 1888 758 l 1882 747 l 1875 735 l 1868 724 l 1860 713 l 1853 704 l 1846 694 l 1838 685 l 1830 676 l 1822 666 l 1813 656 l 1805 647 l 1796 638 l 1788 629 l 1780 620 l 1771 612 l 1763 604 l 1753 595 l 1743 587 l 1733 580 l 1723 573 l 1713 567 l 1703 560 l 1692 554 l 1682 547 l 1671 540 l 1659 532 l 1649 526 l 1639 519 l 1629 513 l 1620 507 l 1610 501 l 1601 494 l 1591 488 l 1581 482 l 1569 475 l 1556 467 l 1542 459 l 1525 450 l 1513 443 l 1499 436 l 1483 428 l 1465 419 l 1445 409 l 1423 398 l 1399 386 l 1374 374 l 1349 361 l 1324 349 l 1300 338 l 1280 327 l 1263 319 l 1250 313 l 1241 309 l 1236 306 l 1234 305 l gs col0 s gr % Polyline n 1808 1132 m 1808 1129 l 1807 1122 l 1805 1110 l 1803 1094 l 1801 1074 l 1798 1051 l 1795 1028 l 1792 1006 l 1789 985 l 1786 967 l 1783 951 l 1780 937 l 1777 925 l 1772 910 l 1767 896 l 1761 884 l 1755 873 l 1749 863 l 1742 853 l 1736 842 l 1729 832 l 1722 821 l 1715 810 l 1707 799 l 1699 789 l 1691 778 l 1683 768 l 1674 758 l 1666 748 l 1657 738 l 1648 727 l 1639 717 l 1629 706 l 1621 697 l 1612 688 l 1604 680 l 1597 672 l 1590 665 l 1582 658 l 1575 651 l 1566 642 l 1557 633 l 1545 621 l 1532 608 l 1517 592 l 1507 581 l 1495 569 l 1482 555 l 1467 540 l 1451 523 l 1432 503 l 1412 482 l 1391 460 l 1369 437 l 1346 413 l 1324 390 l 1303 368 l 1285 348 l 1268 331 l 1255 317 l 1246 307 l 1239 301 l 1236 297 l 1234 295 l gs col0 s gr % Polyline n 1615 1127 m 1615 1126 l 1614 1122 l 1612 1112 l 1608 1096 l 1604 1076 l 1600 1056 l 1595 1036 l 1591 1018 l 1586 1002 l 1582 987 l 1578 975 l 1575 963 l 1571 953 l 1569 944 l 1567 935 l 1565 927 l 1562 918 l 1558 907 l 1553 894 l 1546 878 l 1537 857 l 1524 832 l 1517 818 l 1508 801 l 1498 783 l 1488 763 l 1475 741 l 1462 716 l 1447 689 l 1431 659 l 1413 628 l 1395 595 l 1377 562 l 1358 528 l 1339 495 l 1321 462 l 1305 433 l 1290 406 l 1277 382 l 1266 363 l 1257 348 l 1251 337 l 1247 330 l 1245 326 l 1244 324 l gs col0 s gr % Polyline n 1426 1127 m 1426 1125 l 1426 1119 l 1426 1111 l 1426 1098 l 1425 1082 l 1425 1064 l 1424 1043 l 1422 1022 l 1420 1001 l 1418 979 l 1414 955 l 1409 931 l 1403 903 l 1399 885 l 1394 866 l 1388 844 l 1382 820 l 1374 793 l 1366 763 l 1357 731 l 1347 697 l 1337 660 l 1326 622 l 1315 583 l 1303 543 l 1292 504 l 1281 467 l 1271 432 l 1262 400 l 1254 373 l 1247 350 l 1242 333 l 1238 320 l 1236 312 l 1235 307 l 1234 305 l gs col0 s gr % Polyline n 1186 59 m 1185 59 l 1178 60 l 1165 62 l 1151 64 l 1138 66 l 1125 68 l 1115 70 l 1103 72 l 1091 74 l 1078 76 l 1065 78 l 1052 80 l 1040 82 l 1028 84 l 1016 86 l 1005 88 l 994 90 l 983 91 l 971 93 l 960 95 l 949 97 l 937 100 l 928 103 l 918 105 l 908 109 l 899 112 l 889 115 l 879 119 l 869 122 l 860 126 l 850 129 l 840 132 l 830 135 l 820 138 l 810 141 l 800 144 l 790 147 l 780 149 l 770 152 l 760 156 l 750 159 l 740 163 l 730 167 l 720 172 l 710 177 l 700 183 l 690 188 l 680 194 l 670 199 l 661 205 l 652 210 l 643 215 l 633 221 l 624 226 l 616 231 l 608 236 l 601 241 l 592 246 l 582 253 l 571 261 l 563 267 l 553 274 l 543 281 l 533 289 l 522 298 l 511 306 l 499 315 l 488 323 l 477 332 l 467 340 l 457 348 l 448 356 l 437 366 l 427 375 l 418 385 l 408 394 l 399 403 l 390 413 l 381 422 l 372 432 l 362 442 l 353 453 l 345 462 l 337 472 l 329 482 l 321 492 l 313 502 l 304 512 l 296 522 l 288 533 l 280 543 l 273 553 l 265 564 l 258 575 l 251 586 l 245 598 l 238 609 l 232 622 l 226 634 l 220 646 l 214 659 l 208 671 l 202 683 l 196 695 l 191 706 l 185 717 l 179 730 l 172 742 l 166 754 l 160 765 l 154 777 l 148 788 l 142 799 l 137 810 l 132 821 l 127 832 l 123 842 l 119 853 l 116 863 l 113 872 l 110 882 l 108 891 l 105 901 l 102 910 l 100 920 l 97 930 l 94 940 l 92 950 l 89 960 l 87 970 l 84 980 l 82 989 l 80 999 l 78 1009 l 76 1019 l 74 1029 l 72 1041 l 71 1054 l 71 1066 l 70 1078 l 69 1090 l 69 1103 l 68 1115 l 68 1127 l 68 1139 l 68 1152 l 68 1164 l 68 1176 l 68 1189 l 68 1201 l 68 1213 l 68 1225 l 68 1237 l 68 1248 l 67 1258 l 67 1269 l 67 1279 l 67 1291 l 67 1304 l 68 1319 l 69 1330 l 70 1343 l 72 1357 l 73 1371 l 75 1386 l 77 1401 l 79 1417 l 81 1432 l 83 1447 l 85 1461 l 87 1475 l 90 1488 l 93 1501 l 96 1513 l 99 1525 l 103 1537 l 106 1548 l 109 1559 l 113 1570 l 117 1581 l 121 1593 l 125 1605 l 130 1618 l 136 1632 l 141 1644 l 147 1658 l 154 1671 l 161 1686 l 168 1701 l 176 1717 l 183 1732 l 191 1748 l 199 1763 l 206 1778 l 214 1792 l 221 1805 l 227 1817 l 233 1828 l 241 1842 l 248 1854 l 255 1864 l 262 1873 l 268 1881 l 274 1890 l 281 1898 l 287 1906 l 294 1916 l 301 1925 l 308 1934 l 316 1944 l 323 1953 l 330 1962 l 337 1970 l 345 1979 l 352 1988 l 360 1998 l 368 2007 l 377 2017 l 385 2026 l 393 2034 l 401 2044 l 410 2053 l 419 2063 l 428 2073 l 437 2083 l 447 2093 l 456 2102 l 465 2111 l 473 2119 l 482 2127 l 492 2135 l 503 2143 l 513 2150 l 524 2156 l 534 2162 l 545 2168 l 555 2174 l 565 2180 l 575 2185 l 584 2191 l 595 2197 l 606 2203 l 616 2209 l 625 2214 l 635 2219 l 645 2224 l 655 2229 l 666 2235 l 675 2240 l 684 2244 l 693 2249 l 703 2253 l 712 2258 l 722 2262 l 731 2267 l 741 2272 l 751 2276 l 762 2281 l 773 2286 l 785 2291 l 797 2296 l 809 2301 l 822 2306 l 835 2311 l 847 2316 l 859 2321 l 871 2325 l 882 2329 l 893 2332 l 903 2335 l 913 2337 l 922 2339 l 931 2341 l 941 2343 l 950 2345 l 960 2347 l 970 2349 l 981 2351 l 992 2353 l 1004 2356 l 1017 2359 l 1030 2362 l 1044 2365 l 1057 2368 l 1070 2370 l 1082 2373 l 1093 2375 l 1103 2377 l 1114 2379 l 1125 2380 l 1136 2381 l 1147 2382 l 1157 2382 l 1164 2383 l 1167 2383 l 1168 2383 l gs col0 s gr % Polyline n 1174 243 m 1173 243 l 1169 244 l 1162 246 l 1153 248 l 1145 250 l 1136 252 l 1128 253 l 1119 255 l 1110 256 l 1099 258 l 1089 259 l 1079 261 l 1069 262 l 1059 264 l 1049 266 l 1040 268 l 1030 270 l 1020 271 l 1011 273 l 1001 275 l 991 278 l 981 280 l 971 283 l 961 286 l 951 289 l 940 292 l 930 295 l 920 299 l 909 302 l 899 305 l 889 308 l 878 311 l 868 315 l 858 318 l 847 321 l 837 324 l 826 328 l 816 332 l 806 336 l 795 341 l 784 346 l 773 351 l 763 357 l 752 362 l 742 367 l 733 372 l 725 377 l 717 382 l 711 386 l 705 390 l 699 395 l 692 399 l 684 405 l 675 412 l 666 418 l 657 425 l 647 433 l 636 441 l 625 449 l 613 458 l 602 466 l 592 475 l 581 483 l 572 491 l 563 499 l 554 508 l 546 516 l 538 524 l 531 533 l 523 541 l 515 550 l 507 558 l 499 567 l 491 576 l 483 585 l 475 595 l 466 604 l 458 614 l 450 624 l 441 633 l 433 643 l 425 653 l 417 663 l 409 674 l 401 685 l 394 696 l 386 708 l 378 720 l 370 732 l 362 744 l 355 756 l 348 768 l 341 780 l 335 791 l 330 802 l 325 814 l 320 825 l 316 837 l 312 848 l 309 859 l 305 871 l 301 881 l 298 891 l 295 901 l 292 912 l 289 922 l 287 931 l 285 939 l 283 948 l 281 956 l 279 966 l 277 975 l 274 985 l 272 995 l 269 1005 l 266 1015 l 263 1025 l 260 1035 l 258 1045 l 256 1055 l 254 1065 l 253 1076 l 252 1086 l 251 1097 l 250 1107 l 249 1118 l 248 1129 l 247 1139 l 246 1149 l 246 1160 l 246 1170 l 245 1181 l 245 1191 l 245 1202 l 245 1212 l 245 1222 l 245 1232 l 245 1241 l 245 1250 l 245 1259 l 245 1268 l 245 1277 l 246 1288 l 247 1301 l 248 1313 l 250 1326 l 252 1339 l 254 1354 l 256 1370 l 258 1385 l 261 1400 l 263 1415 l 266 1429 l 269 1442 l 272 1455 l 276 1467 l 279 1479 l 283 1490 l 286 1501 l 290 1513 l 294 1524 l 298 1536 l 303 1549 l 309 1563 l 314 1575 l 320 1588 l 327 1602 l 333 1616 l 341 1631 l 348 1646 l 355 1661 l 363 1676 l 370 1690 l 377 1703 l 383 1715 l 389 1726 l 396 1738 l 403 1748 l 409 1757 l 415 1765 l 421 1773 l 427 1780 l 433 1787 l 439 1795 l 445 1803 l 451 1811 l 459 1821 l 466 1830 l 473 1839 l 480 1848 l 488 1857 l 495 1866 l 504 1876 l 513 1886 l 521 1895 l 530 1904 l 540 1914 l 550 1924 l 560 1934 l 570 1945 l 581 1955 l 590 1964 l 600 1973 l 609 1980 l 620 1988 l 631 1995 l 641 2000 l 652 2005 l 662 2010 l 672 2015 l 682 2021 l 691 2026 l 700 2031 l 708 2037 l 715 2042 l 722 2047 l 729 2052 l 737 2058 l 744 2063 l 753 2068 l 762 2073 l 772 2079 l 781 2084 l 791 2089 l 801 2094 l 811 2100 l 822 2105 l 833 2110 l 842 2114 l 852 2118 l 862 2122 l 873 2127 l 884 2131 l 895 2135 l 906 2139 l 916 2143 l 926 2147 l 935 2150 l 946 2153 l 956 2156 l 965 2158 l 974 2160 l 983 2162 l 993 2164 l 1003 2167 l 1014 2169 l 1023 2171 l 1033 2173 l 1044 2175 l 1055 2177 l 1067 2179 l 1079 2181 l 1090 2183 l 1100 2185 l 1109 2186 l 1117 2188 l 1128 2190 l 1139 2192 l 1149 2195 l 1158 2197 l 1162 2198 l 1163 2198 l gs col0 s gr % Polyline n 1268 243 m 1269 243 l 1274 244 l 1284 246 l 1295 248 l 1305 250 l 1315 252 l 1323 253 l 1332 255 l 1342 256 l 1352 258 l 1362 259 l 1372 261 l 1382 262 l 1392 264 l 1402 266 l 1411 268 l 1421 270 l 1431 271 l 1441 273 l 1450 275 l 1460 278 l 1470 280 l 1480 283 l 1490 286 l 1500 289 l 1510 292 l 1520 295 l 1531 299 l 1541 302 l 1551 305 l 1561 308 l 1572 311 l 1582 315 l 1593 318 l 1604 321 l 1614 324 l 1625 328 l 1635 332 l 1645 336 l 1656 341 l 1666 346 l 1677 351 l 1687 357 l 1697 362 l 1707 367 l 1716 372 l 1724 377 l 1732 382 l 1739 386 l 1745 390 l 1752 395 l 1759 399 l 1767 405 l 1777 412 l 1786 418 l 1795 425 l 1805 433 l 1816 441 l 1828 450 l 1839 458 l 1850 467 l 1861 475 l 1871 483 l 1880 491 l 1889 499 l 1898 507 l 1906 514 l 1914 522 l 1922 530 l 1929 537 l 1937 545 l 1945 553 l 1953 561 l 1961 569 l 1969 577 l 1977 585 l 1985 594 l 1992 601 l 2000 609 l 2008 618 l 2015 626 l 2023 635 l 2031 645 l 2039 655 l 2046 664 l 2053 675 l 2060 686 l 2067 697 l 2074 709 l 2082 721 l 2089 733 l 2096 745 l 2103 757 l 2109 768 l 2115 778 l 2120 788 l 2126 802 l 2131 814 l 2136 826 l 2139 837 l 2142 848 l 2146 859 l 2149 870 l 2153 882 l 2157 894 l 2160 906 l 2164 917 l 2168 929 l 2171 941 l 2175 953 l 2178 965 l 2181 976 l 2184 987 l 2186 999 l 2188 1010 l 2190 1021 l 2191 1032 l 2193 1043 l 2195 1054 l 2196 1064 l 2197 1074 l 2198 1084 l 2199 1093 l 2199 1102 l 2200 1111 l 2200 1120 l 2201 1129 l 2201 1139 l 2201 1149 l 2201 1159 l 2201 1170 l 2201 1180 l 2201 1191 l 2201 1201 l 2201 1212 l 2201 1222 l 2201 1232 l 2201 1241 l 2202 1250 l 2202 1259 l 2202 1268 l 2202 1277 l 2202 1288 l 2201 1301 l 2200 1313 l 2198 1326 l 2197 1339 l 2195 1354 l 2193 1370 l 2191 1385 l 2189 1400 l 2186 1415 l 2184 1429 l 2181 1442 l 2178 1455 l 2174 1467 l 2171 1479 l 2167 1490 l 2164 1501 l 2160 1513 l 2156 1524 l 2152 1536 l 2147 1549 l 2141 1563 l 2136 1575 l 2130 1588 l 2123 1602 l 2116 1616 l 2108 1631 l 2101 1646 l 2093 1661 l 2086 1676 l 2078 1690 l 2071 1703 l 2065 1715 l 2059 1726 l 2051 1740 l 2043 1752 l 2036 1762 l 2029 1770 l 2022 1779 l 2016 1788 l 2008 1797 l 2001 1807 l 1995 1815 l 1989 1823 l 1983 1831 l 1978 1838 l 1972 1846 l 1966 1854 l 1960 1861 l 1954 1869 l 1947 1878 l 1939 1886 l 1931 1895 l 1921 1904 l 1912 1914 l 1901 1924 l 1891 1934 l 1880 1944 l 1869 1954 l 1859 1964 l 1850 1972 l 1841 1980 l 1831 1989 l 1821 1997 l 1811 2004 l 1802 2010 l 1793 2017 l 1784 2023 l 1775 2029 l 1766 2035 l 1757 2040 l 1748 2045 l 1740 2049 l 1731 2052 l 1723 2056 l 1714 2059 l 1705 2063 l 1696 2068 l 1686 2073 l 1677 2078 l 1667 2083 l 1657 2089 l 1647 2094 l 1637 2100 l 1626 2105 l 1615 2110 l 1606 2114 l 1596 2118 l 1585 2122 l 1575 2127 l 1564 2131 l 1553 2135 l 1542 2139 l 1532 2143 l 1522 2147 l 1513 2150 l 1502 2153 l 1492 2156 l 1482 2158 l 1473 2160 l 1464 2162 l 1454 2164 l 1444 2167 l 1433 2169 l 1424 2171 l 1414 2173 l 1403 2175 l 1392 2177 l 1381 2179 l 1370 2181 l 1360 2183 l 1349 2185 l 1340 2186 l 1331 2188 l 1321 2190 l 1310 2191 l 1299 2193 l 1287 2195 l 1276 2196 l 1268 2197 l 1265 2198 l 1264 2198 l gs col0 s gr % Polyline n 1261 68 m 1262 68 l 1265 68 l 1272 68 l 1283 67 l 1294 67 l 1305 67 l 1315 67 l 1325 68 l 1336 69 l 1347 71 l 1360 73 l 1373 75 l 1386 78 l 1399 80 l 1411 82 l 1423 84 l 1435 86 l 1446 88 l 1457 90 l 1468 91 l 1479 93 l 1490 95 l 1501 97 l 1513 100 l 1522 103 l 1532 105 l 1541 109 l 1551 112 l 1560 115 l 1570 119 l 1580 122 l 1589 126 l 1599 129 l 1609 132 l 1619 135 l 1629 138 l 1640 141 l 1650 144 l 1661 147 l 1671 149 l 1681 152 l 1692 156 l 1702 159 l 1712 163 l 1722 167 l 1732 172 l 1742 177 l 1752 183 l 1761 188 l 1771 194 l 1781 199 l 1790 205 l 1799 210 l 1808 215 l 1818 221 l 1827 226 l 1835 231 l 1843 236 l 1851 241 l 1859 246 l 1869 253 l 1880 261 l 1888 267 l 1897 274 l 1907 281 l 1918 289 l 1928 298 l 1939 306 l 1950 315 l 1961 323 l 1972 332 l 1982 340 l 1992 348 l 2001 356 l 2012 366 l 2022 375 l 2032 385 l 2041 394 l 2051 403 l 2060 413 l 2069 422 l 2079 432 l 2088 442 l 2098 453 l 2106 462 l 2114 472 l 2122 482 l 2130 492 l 2138 502 l 2146 512 l 2154 522 l 2162 533 l 2170 543 l 2177 553 l 2185 564 l 2192 575 l 2199 586 l 2206 598 l 2213 609 l 2220 621 l 2227 633 l 2234 646 l 2240 658 l 2247 670 l 2253 682 l 2259 694 l 2265 706 l 2271 717 l 2277 730 l 2283 744 l 2288 757 l 2294 770 l 2299 783 l 2304 796 l 2308 809 l 2313 821 l 2318 833 l 2322 844 l 2327 857 l 2331 868 l 2335 878 l 2339 888 l 2343 898 l 2347 908 l 2350 918 l 2354 930 l 2357 940 l 2359 950 l 2362 961 l 2365 972 l 2367 983 l 2370 994 l 2372 1005 l 2374 1016 l 2376 1026 l 2378 1036 l 2380 1048 l 2381 1059 l 2383 1069 l 2384 1080 l 2385 1090 l 2385 1100 l 2386 1111 l 2387 1123 l 2387 1133 l 2388 1143 l 2388 1153 l 2388 1163 l 2388 1174 l 2387 1184 l 2387 1195 l 2387 1205 l 2387 1215 l 2387 1225 l 2387 1237 l 2387 1248 l 2387 1259 l 2387 1269 l 2387 1279 l 2386 1291 l 2385 1304 l 2384 1319 l 2383 1330 l 2381 1343 l 2380 1357 l 2378 1371 l 2376 1386 l 2374 1401 l 2372 1417 l 2370 1432 l 2368 1447 l 2365 1461 l 2363 1475 l 2360 1488 l 2357 1501 l 2354 1513 l 2350 1525 l 2347 1537 l 2344 1548 l 2340 1559 l 2337 1570 l 2333 1581 l 2329 1593 l 2325 1605 l 2320 1618 l 2314 1632 l 2309 1644 l 2303 1658 l 2296 1671 l 2289 1686 l 2282 1701 l 2274 1716 l 2267 1732 l 2259 1747 l 2251 1763 l 2244 1777 l 2236 1792 l 2230 1805 l 2223 1817 l 2217 1828 l 2209 1842 l 2201 1854 l 2194 1865 l 2187 1875 l 2180 1885 l 2174 1894 l 2167 1903 l 2160 1912 l 2153 1921 l 2146 1931 l 2139 1940 l 2132 1949 l 2125 1958 l 2118 1966 l 2112 1974 l 2105 1982 l 2098 1990 l 2091 1998 l 2083 2007 l 2074 2017 l 2066 2025 l 2058 2034 l 2049 2043 l 2040 2053 l 2031 2063 l 2021 2073 l 2012 2083 l 2002 2093 l 1993 2102 l 1984 2111 l 1975 2120 l 1966 2127 l 1956 2135 l 1946 2142 l 1935 2149 l 1925 2155 l 1915 2160 l 1905 2166 l 1896 2171 l 1886 2177 l 1876 2183 l 1867 2188 l 1856 2195 l 1845 2201 l 1835 2207 l 1825 2212 l 1815 2218 l 1805 2223 l 1794 2229 l 1783 2235 l 1774 2240 l 1765 2244 l 1755 2249 l 1746 2253 l 1736 2258 l 1727 2263 l 1717 2267 l 1707 2272 l 1697 2276 l 1686 2281 l 1675 2286 l 1663 2291 l 1651 2296 l 1639 2301 l 1626 2306 l 1614 2311 l 1602 2316 l 1590 2321 l 1578 2325 l 1567 2329 l 1554 2333 l 1542 2336 l 1530 2339 l 1519 2341 l 1507 2343 l 1496 2346 l 1483 2348 l 1470 2351 l 1459 2353 l 1447 2356 l 1434 2359 l 1421 2362 l 1408 2365 l 1395 2368 l 1382 2370 l 1370 2373 l 1358 2375 l 1348 2377 l 1336 2379 l 1324 2380 l 1311 2381 l 1298 2382 l 1286 2382 l 1278 2383 l 1274 2383 l 1273 2383 l gs col0 s gr % Polyline n 1039 1665 m 1040 1665 l 1045 1668 l 1055 1674 l 1068 1681 l 1083 1690 l 1096 1699 l 1105 1707 l 1112 1716 l 1116 1724 l 1119 1732 l 1122 1741 l 1124 1751 l 1125 1760 l 1127 1770 l 1128 1780 l 1129 1791 l 1130 1802 l 1131 1815 l 1131 1827 l 1131 1840 l 1131 1856 l 1130 1875 l 1129 1896 l 1128 1917 l 1127 1938 l 1126 1956 l 1125 1970 l 1124 1978 l 1124 1981 l 1124 1982 l gs col0 s gr % Polyline n 2002 1312 m 2002 1315 l 2001 1322 l 2001 1334 l 1999 1350 l 1998 1370 l 1996 1393 l 1994 1416 l 1992 1439 l 1990 1460 l 1987 1480 l 1984 1497 l 1981 1513 l 1977 1528 l 1972 1542 l 1967 1556 l 1961 1570 l 1955 1584 l 1948 1598 l 1941 1611 l 1934 1625 l 1927 1638 l 1920 1652 l 1912 1665 l 1905 1678 l 1897 1691 l 1890 1704 l 1882 1716 l 1874 1728 l 1866 1740 l 1857 1752 l 1849 1763 l 1841 1775 l 1832 1786 l 1823 1798 l 1815 1809 l 1806 1820 l 1797 1831 l 1787 1841 l 1778 1851 l 1768 1860 l 1757 1869 l 1744 1878 l 1730 1886 l 1716 1894 l 1701 1900 l 1686 1907 l 1670 1913 l 1655 1919 l 1639 1924 l 1624 1930 l 1609 1935 l 1595 1940 l 1581 1945 l 1567 1949 l 1554 1953 l 1540 1957 l 1526 1960 l 1512 1963 l 1499 1967 l 1485 1970 l 1472 1972 l 1459 1974 l 1446 1976 l 1435 1976 l 1424 1975 l 1412 1972 l 1401 1968 l 1390 1962 l 1380 1955 l 1370 1948 l 1360 1941 l 1351 1933 l 1343 1925 l 1336 1916 l 1331 1907 l 1328 1897 l 1326 1885 l 1325 1873 l 1326 1859 l 1327 1845 l 1328 1832 l 1330 1818 l 1332 1806 l 1334 1794 l 1336 1784 l 1340 1773 l 1347 1762 l 1355 1750 l 1365 1738 l 1375 1728 l 1382 1720 l 1385 1717 l 1386 1716 l gs col0 s gr % Polyline n 1215 320 m 1213 323 l 1209 329 l 1202 340 l 1193 354 l 1181 373 l 1168 393 l 1155 414 l 1142 434 l 1130 453 l 1119 470 l 1110 486 l 1102 499 l 1095 511 l 1087 524 l 1081 537 l 1075 548 l 1069 559 l 1064 570 l 1059 580 l 1054 591 l 1049 601 l 1044 612 l 1038 623 l 1032 634 l 1026 646 l 1019 658 l 1012 671 l 1005 684 l 998 697 l 990 711 l 982 724 l 974 738 l 966 751 l 959 764 l 952 777 l 945 788 l 939 799 l 932 811 l 925 822 l 919 833 l 913 843 l 907 853 l 901 863 l 895 872 l 890 881 l 885 889 l 881 896 l 876 906 l 872 914 l 868 922 l 866 928 l 864 931 l 864 932 l gs col0 s gr % Polyline n 448 1132 m 448 1315 l gs col0 s gr % Polyline n 447 1322 m 447 1325 l 448 1330 l 449 1340 l 450 1355 l 452 1373 l 454 1395 l 456 1419 l 459 1444 l 461 1468 l 465 1491 l 468 1512 l 471 1532 l 475 1549 l 479 1565 l 484 1580 l 490 1596 l 498 1612 l 505 1628 l 514 1643 l 523 1659 l 532 1674 l 542 1689 l 552 1704 l 562 1718 l 572 1733 l 582 1747 l 592 1761 l 602 1774 l 612 1787 l 623 1799 l 634 1811 l 645 1823 l 656 1834 l 668 1845 l 679 1856 l 691 1866 l 703 1877 l 715 1887 l 727 1897 l 739 1907 l 751 1917 l 763 1926 l 776 1934 l 789 1942 l 802 1949 l 817 1956 l 831 1963 l 846 1969 l 862 1975 l 877 1981 l 893 1987 l 908 1992 l 923 1998 l 936 2003 l 949 2007 l 960 2011 l 970 2015 l 985 2020 l 996 2022 l 1005 2023 l 1012 2024 l 1019 2024 l 1025 2024 l 1032 2024 l 1039 2024 l 1047 2024 l 1053 2023 l 1058 2021 l 1063 2020 l 1069 2018 l 1075 2015 l 1082 2011 l 1090 2004 l 1100 1996 l 1109 1989 l 1114 1985 l 1115 1984 l gs col0 s gr /Times-Roman-iso ff 195.00 scf sf 932 1920 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 195.00 scf sf 1414 1713 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 195.00 scf sf 964 1734 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 195.00 scf sf 803 1080 m gs 1 -1 sc (+) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 1623 3469 a Fs(Figure)k(6.)386 3719 y Fr(there)k(is)f(an)h Fh(R)t Fr({in)m(v)-5 b(arian)m(t)45 b(p)s(ositiv)m(e)40 b(con)m(tact)h(structure)h Fp(C)47 b Fr(on)40 b Fo(D)2903 3682 y Fn(2)2970 3719 y Fp(\002)28 b Fh(R)51 b Fr(suc)m(h)386 3835 y(that)39 b(the)i(induced)f(singular)e(foliation)f(on)i Fo(D)2142 3799 y Fn(2)2208 3835 y Fp(\002)28 b(f)p Fr(0)p Fp(g)39 b Fr(is)g Fp(F)10 b Fr(.)65 b(W)-8 b(e)40 b(c)m(ho)s(ose)h(an) 386 3951 y Fh(R)5 b Fr({in)m(v)-5 b(arian)m(t)37 b(con)m(tact)c(form)e Fo(\013)i Fr(for)g Fp(C)6 b Fr(.)43 b(suc)m(h)34 b(that)1340 4111 y Fo(\013)28 b Fr(=)g(cos\()p Fo(')p Fr(\))17 b Fo(dr)24 b Fr(+)e(sin\()p Fo(')p Fr(\))17 b Fo(dx)386 4271 y Fr(on)39 b Fo(A)27 b Fp(\002)g Fo(R)q Fr(.)64 b(The)41 b(co)s(ordinate)d(corresp)s(onding)i(to)f(the)h Fh(R)t Fr({factor)45 b(is)39 b Fo(x)p Fr(.)64 b(This)386 4387 y(c)m(hoice)35 b(\014xes)h(an)f(orien)m(tation)d(of)j(the)g(con)m (tact)g(structure.)51 b(In)35 b(order)g(to)f(\014nd)h(a)386 4503 y(con)m(tact)26 b(v)m(ector)g(\014eld)e Fo(V)47 b Fr(and)25 b(a)g(2{handle)f Fo(h)2031 4518 y Fn(2)2098 4503 y Fp(\032)29 b Fo(D)2288 4467 y Fn(2)2333 4503 y Fp(\002)7 b Fh(R)37 b Fr(suc)m(h)26 b(that)f Fo(V)46 b Fr(is)25 b(trans-)386 4619 y(v)m(erse)38 b(to)e Fo(@)5 b(h)869 4634 y Fn(2)945 4619 y Fr(w)m(e)37 b(need)h(to)d(tak)m(e)i (some)f(care)h(since)f(w)m(e)i(kno)m(w)f(nothing)e(ab)s(out)386 4736 y(the)e(region)e Fo(r)f(<)e Fr(1)p Fo(=)p Fr(2)k(except)h(that)g Fo(@)1771 4751 y Fl(x)1847 4736 y Fr(is)f(a)g(con)m(tact)h(v)m(ector)g (\014eld)f(ev)m(erywhere.)386 4852 y(W)-8 b(e)33 b(fo)s(cus)g(\014rst)g (on)f Fo(A)23 b Fp(\002)f Fh(R)5 b Fr(.)486 4968 y(Let)37 b Fo(g)712 4983 y Fn(1)751 4968 y Fo(;)17 b(g)842 4983 y Fn(2)918 4968 y Fr(b)s(e)38 b(smo)s(oth)e(functions)i(on)f Fo(A)25 b Fp(\002)h Fh(R)48 b Fr(dep)s(ending)38 b(only)f(on)g Fo(x)p Fr(.)58 b(By)386 5084 y(the)33 b(pro)s(of)f(of)g(Prop)s(osition) f(2.4)462 5292 y Fo(V)50 b Fr(=)27 b Fo(g)719 5307 y Fn(1)758 5292 y Fr(\()p Fo(x)p Fr(\))923 5225 y Fo(@)p 899 5269 104 4 v 899 5360 a(@)5 b(r)1035 5292 y Fp(\000)1135 5211 y Fj(\000)1181 5292 y Fo(g)1232 5251 y Fk(0)1228 5317 y Fn(1)1267 5292 y Fr(\()p Fo(x)p Fr(\))17 b(cos)1545 5251 y Fn(2)1585 5292 y Fr(\()p Fo(')p Fr(\))22 b(+)g Fo(g)1896 5251 y Fk(0)1892 5317 y Fn(2)1931 5292 y Fr(\()p Fo(x)p Fr(\))17 b(sin)o(\()p Fo(')p Fr(\))g(cos\()p Fo(')p Fr(\))2625 5211 y Fj(\001)2729 5225 y Fo(@)p 2697 5269 121 4 v 2697 5360 a(@)5 b(')2850 5292 y Fr(+)22 b Fo(g)2995 5307 y Fn(2)3035 5292 y Fr(\()p Fo(x)p Fr(\))3204 5225 y Fo(@)p 3176 5269 113 4 v 3176 5360 a(@)5 b(x)386 5506 y Fr(is)28 b(the)g(con)m(tact)h(v)m(ector)g(\014eld)f(asso)s(ciated)g (to)g(the)g(function)g Fo(h)g Fr(=)f Fo(g)2828 5521 y Fn(1)2867 5506 y Fr(\()p Fo(x)p Fr(\))17 b(cos)q(\()p Fo(')p Fr(\))c(+)386 5623 y Fo(g)433 5638 y Fn(2)472 5623 y Fr(\()p Fo(x)p Fr(\))k(sin\()p Fo(')p Fr(\))p Fo(:)p eop %%Page: 31 31 31 30 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(31)486 459 y Fr(W)-8 b(e)33 b(c)m(ho)s(ose)g(the)g(functions)g Fo(g)1599 474 y Fn(1)1638 459 y Fo(;)17 b(g)1729 474 y Fn(2)1800 459 y Fr(suc)m(h)34 b(that)432 760 y Fo(g)479 775 y Fn(1)518 760 y Fr(\()p Fo(x)p Fr(\))28 b(=)781 619 y Fj(\032)897 700 y Fr(0)193 b(for)32 b Fp(j)p Fo(x)p Fp(j)27 b(\025)i Fr(1)897 816 y Fp(\000)p Fr(1)116 b(for)32 b Fp(j)p Fo(x)p Fp(j)27 b(\024)1541 777 y Fn(3)p 1541 794 36 4 v 1541 851 a(4)1792 760 y Fo(g)1839 775 y Fn(2)1878 760 y Fr(\()p Fo(x)p Fr(\))h(=)2141 555 y Fj(8)2141 645 y(<)2141 824 y(:)2271 641 y Fo(a)193 b Fr(for)32 b Fo(x)c Fp(\025)2862 602 y Fn(3)p 2862 619 V 2862 676 a(4)2271 758 y Fp(\000)p Fo(a)116 b Fr(for)32 b Fo(x)c Fp(\024)g(\000)2939 719 y Fn(3)p 2939 735 V 2939 793 a(4)2271 875 y Fr(0)195 b(for)54 b Fp(\000)2796 836 y Fn(1)p 2796 852 V 2796 910 a(2)2869 875 y Fp(\024)28 b Fo(x)g Fp(\024)3172 836 y Fn(1)p 3172 852 V 3172 910 a(2)3250 875 y Fo(:)386 1061 y Fr(for)33 b Fo(a)d(>)f Fr(0.)46 b(F)-8 b(or)33 b(these)i(c)m(hoices)f(the)g(con)m(tact)h(v)m(ector)f(\014eld)g Fo(V)55 b Fr(asso)s(ciated)33 b(to)h Fo(h)386 1177 y Fr(can)e(b)s(e)h(extended)h(b)m(y)f Fo(a)21 b Fp(\001)g Fr(sgn)q(\()p Fo(x)p Fr(\))p Fo(@)1691 1192 y Fl(x)1767 1177 y Fr(on)33 b Fp(j)p Fo(x)p Fp(j)27 b(\025)h Fr(1)k(to)g(a)g(smo)s (oth)f(con)m(tact)i(v)m(ector)386 1293 y(\014eld)41 b(whic)m(h)h(w)m(e) g(still)d(denote)i(b)m(y)i Fo(V)21 b Fr(.)69 b(Finally)-8 b(,)41 b(w)m(e)h(extend)h Fo(V)62 b Fr(to)41 b(a)g(con)m(tact)386 1409 y(v)m(ector)30 b(\014eld)g(on)f(the)h(whole)f(of)g Fo(D)1646 1373 y Fn(2)1701 1409 y Fp(\002)16 b Fh(R)5 b Fr(.)49 b(\(F)-8 b(or)28 b(this)i(it)e(is)h(enough)h Fo(\013)q Fr(\()p Fo(V)21 b Fr(\))29 b(and)h(to)386 1526 y(apply)g(Prop)s(osition)f(2.4.\))43 b(The)32 b(resulting)d(con)m(tact) j(v)m(ector)g(\014eld)e(is)g(transv)m(erse)386 1642 y(to)i Fo(@)5 b(D)645 1606 y Fn(2)707 1642 y Fp(\002)23 b Fr([)p Fp(\000)p Fr(3)p Fo(=)p Fr(4)p Fo(;)17 b Fr(3)p Fo(=)p Fr(4])32 b(and)g(p)s(oin)m(ts)h(in)m(w)m(ards)g(there.)44 b(No)m(w)34 b(consider)f(the)g(pair)386 1758 y(of)f(h)m(yp)s (ersurfaces)j(de\014ned)f(b)m(y)g(the)f(equation)1504 1943 y Fp(j)p Fo(x)p Fp(j)27 b Fr(=)h(5)p Fo(=)p Fr(4)21 b Fp(\000)i Fo(r)2061 1902 y Fn(2)2100 1943 y Fo(=)p Fr(2)32 b Fo(:)386 2134 y Fr(Since)f Fo(r)g Fp(\024)d Fr(1,)j(b)s(oth)g(are)g(con)m(tained)h(in)e(the)i(region)e Fp(j)p Fo(x)p Fp(j)e(\025)g Fr(3)p Fo(=)p Fr(4)i(where)j Fo(g)3076 2149 y Fn(2)3142 2134 y Fr(=)28 b Fp(\006)p Fo(a)386 2251 y Fr(dep)s(ending)33 b(on)f(the)h(sign)f(of)g Fo(x)p Fr(.)486 2367 y(If)37 b(w)m(e)i(\014x)f Fo(a)g Fr(big)e(enough,)k Fo(V)59 b Fr(is)37 b(transv)m(erse)j(to)d(the)h(h)m (yp)s(ersurfaces)i Fp(fj)p Fo(x)p Fp(j)c Fr(=)386 2483 y(5)p Fo(=)p Fr(4)21 b Fp(\000)i Fo(r)701 2447 y Fn(2)740 2483 y Fo(=)p Fr(2)p Fp(g)32 b Fr(and)h(it)e(p)s(oin)m(ts)h(out)m(w)m (ards.)45 b(Then)1166 2670 y Fo(h)1222 2685 y Fn(2)1290 2670 y Fr(=)1393 2589 y Fj(\010)1451 2670 y Fr(\()p Fo(r)m(;)17 b(';)g(x)p Fr(\))1775 2585 y Fj(\014)1775 2645 y(\014)1841 2670 y Fp(j)p Fo(x)p Fp(j)27 b(\024)h Fr(5)p Fo(=)p Fr(4)22 b Fp(\000)h Fo(r)2400 2629 y Fn(2)2439 2670 y Fo(=)p Fr(2)2537 2589 y Fj(\011)386 2855 y Fr(is)j(di\013eomorphic)e(to)i(an)g (ordinary)f(handle)h(of)g(index)g(2)g(and)g Fo(V)48 b Fr(is)26 b(transv)m(erse)i(to)386 2971 y(b)s(oth)34 b(b)s(oundary)g (comp)s(onen)m(ts,)h(cf.)f(Figure)f(7.)47 b(Moreo)m(v)m(er)35 b Fo(V)56 b Fr(has)34 b(the)g(desired)386 3088 y(orien)m(tations)d (along)g Fo(@)1231 3103 y Fk(\006)1291 3088 y Fo(h)1347 3103 y Fn(2)1387 3088 y Fr(.)1334 4060 y @beginspecial 0 @llx 0 @lly 131 @urx 92 @ury 1310 @rwi @setspecial %%BeginDocument: henkel.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: henkel.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Mon Mar 15 19:11:43 2004 %%For: tvogel@math81 (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 131 92 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save 1.0 93.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 2536 m -1000 -1000 l 3158 -1000 l 3158 2536 l cp clip 0.06000 0.06000 sc % Arc 7.500 slw gs n 1040.0 780.0 743.0 -126.8 126.8 arcn gs col0 s gr gr % Arc gs n 1043.0 780.0 743.0 -53.2 53.2 arc gs col0 s gr gr % Polyline n 595 185 m 1042 185 l gs col0 s gr % Polyline n 595 1375 m 1042 1375 l gs col0 s gr % Polyline gs clippath 960 336 m 930 456 l 900 336 l 900 498 l 960 498 l cp clip n 930 185 m 930 483 l gs col0 s gr gr % arrowhead n 960 336 m 930 456 l 900 336 l col0 s % Polyline gs clippath 763 365 m 682 458 l 709 338 l 636 483 l 690 510 l cp clip n 819 185 m 670 483 l gs col0 s gr gr % arrowhead n 763 365 m 682 458 l 709 338 l col0 s % Polyline gs clippath 296 588 m 176 558 l 296 528 l 134 528 l 134 588 l cp clip n 149 558 m 447 558 l gs col0 s gr gr % arrowhead n 296 588 m 176 558 l 296 528 l col0 s % Polyline gs clippath 296 1034 m 176 1004 l 296 974 l 134 974 l 134 1034 l cp clip n 149 1004 m 447 1004 l gs col0 s gr gr % arrowhead n 296 1034 m 176 1004 l 296 974 l col0 s % Polyline gs clippath 900 1224 m 930 1104 l 960 1224 l 960 1062 l 900 1062 l cp clip n 930 1077 m 930 1375 l gs col0 s gr gr % arrowhead n 900 1224 m 930 1104 l 960 1224 l col0 s % Polyline gs clippath 671 1222 m 644 1101 l 725 1195 l 652 1050 l 598 1077 l cp clip n 782 1375 m 632 1077 l gs col0 s gr gr % arrowhead n 671 1222 m 644 1101 l 725 1195 l col0 s % Polyline gs clippath 559 383 m 463 461 l 511 347 l 414 477 l 462 513 l cp clip n 670 185 m 447 483 l gs col0 s gr gr % arrowhead n 559 383 m 463 461 l 511 347 l col0 s % Polyline gs clippath 511 1213 m 463 1098 l 559 1177 l 462 1047 l 414 1083 l cp clip n 670 1375 m 447 1077 l gs col0 s gr gr % arrowhead n 511 1213 m 463 1098 l 559 1177 l col0 s % Polyline gs clippath 415 1319 m 321 1239 l 442 1266 l 297 1193 l 270 1247 l cp clip n 595 1375 m 297 1227 l gs col0 s gr gr % arrowhead n 415 1319 m 321 1239 l 442 1266 l col0 s % Polyline gs clippath 442 295 m 321 321 l 415 241 l 270 314 l 297 368 l cp clip n 595 185 m 297 334 l gs col0 s gr gr % arrowhead n 442 295 m 321 321 l 415 241 l col0 s % Polyline n 1488 185 m 1042 185 l gs col0 s gr % Polyline n 1488 1375 m 1042 1375 l gs col0 s gr % Polyline gs clippath 1183 336 m 1153 456 l 1123 336 l 1123 498 l 1183 498 l cp clip n 1153 185 m 1153 483 l gs col0 s gr gr % arrowhead n 1183 336 m 1153 456 l 1123 336 l col0 s % Polyline gs clippath 1375 338 m 1401 458 l 1321 365 l 1394 510 l 1448 483 l cp clip n 1265 185 m 1414 483 l gs col0 s gr gr % arrowhead n 1375 338 m 1401 458 l 1321 365 l col0 s % Polyline gs clippath 1788 528 m 1908 558 l 1788 588 l 1950 588 l 1950 528 l cp clip n 1935 558 m 1637 558 l gs col0 s gr gr % arrowhead n 1788 528 m 1908 558 l 1788 588 l col0 s % Polyline gs clippath 1788 974 m 1908 1004 l 1788 1034 l 1950 1034 l 1950 974 l cp clip n 1935 1004 m 1637 1004 l gs col0 s gr gr % arrowhead n 1788 974 m 1908 1004 l 1788 1034 l col0 s % Polyline gs clippath 1123 1224 m 1153 1104 l 1183 1224 l 1183 1062 l 1123 1062 l cp clip n 1153 1077 m 1153 1375 l gs col0 s gr gr % arrowhead n 1123 1224 m 1153 1104 l 1183 1224 l col0 s % Polyline gs clippath 1358 1195 m 1438 1101 l 1412 1222 l 1485 1077 l 1431 1050 l cp clip n 1302 1375 m 1451 1077 l gs col0 s gr gr % arrowhead n 1358 1195 m 1438 1101 l 1412 1222 l col0 s % Polyline gs clippath 1573 347 m 1620 461 l 1525 383 l 1622 513 l 1670 477 l cp clip n 1414 185 m 1637 483 l gs col0 s gr gr % arrowhead n 1573 347 m 1620 461 l 1525 383 l col0 s % Polyline gs clippath 1525 1177 m 1620 1098 l 1573 1213 l 1670 1083 l 1622 1047 l cp clip n 1414 1375 m 1637 1077 l gs col0 s gr gr % arrowhead n 1525 1177 m 1620 1098 l 1573 1213 l col0 s % Polyline gs clippath 1641 1266 m 1761 1239 l 1668 1319 l 1813 1247 l 1786 1193 l cp clip n 1488 1375 m 1786 1227 l gs col0 s gr gr % arrowhead n 1641 1266 m 1761 1239 l 1668 1319 l col0 s % Polyline gs clippath 1668 241 m 1761 321 l 1641 295 l 1786 368 l 1813 314 l cp clip n 1488 185 m 1786 334 l gs col0 s gr gr % arrowhead n 1668 241 m 1761 321 l 1641 295 l col0 s % Polyline gs clippath 296 736 m 176 706 l 296 676 l 134 676 l 134 736 l cp clip n 149 706 m 447 706 l gs col0 s gr gr % arrowhead n 296 736 m 176 706 l 296 676 l col0 s % Polyline gs clippath 1788 676 m 1908 706 l 1788 736 l 1950 736 l 1950 676 l cp clip n 1935 706 m 1637 706 l gs col0 s gr gr % arrowhead n 1788 676 m 1908 706 l 1788 736 l col0 s % Polyline gs clippath 296 885 m 176 855 l 296 825 l 134 825 l 134 885 l cp clip n 149 855 m 447 855 l gs col0 s gr gr % arrowhead n 296 885 m 176 855 l 296 825 l col0 s % Polyline gs clippath 1788 825 m 1908 855 l 1788 885 l 1950 885 l 1950 825 l cp clip n 1935 855 m 1637 855 l gs col0 s gr gr % arrowhead n 1788 825 m 1908 855 l 1788 885 l col0 s % Polyline n 1038 37 m 1046 37 l gs col0 s gr % Polyline gs clippath 1012 184 m 1042 64 l 1072 184 l 1072 22 l 1012 22 l cp clip n 1042 37 m 1042 1524 l gs col0 s gr gr % arrowhead n 1012 184 m 1042 64 l 1072 184 l col0 s % Polyline gs clippath 1937 750 m 2057 780 l 1937 810 l 2099 810 l 2099 750 l cp clip n 0 780 m 2084 780 l gs col0 s gr gr % arrowhead n 1937 750 m 2057 780 l 1937 810 l col0 s /Times-Roman ff 180.00 scf sf 1903 1359 m gs 1 -1 sc (h) col0 sh gr /Times-Roman ff 180.00 scf sf 2012 1467 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 180.00 scf sf 2066 924 m gs 1 -1 sc (x) col0 sh gr /Times-Roman ff 180.00 scf sf 870 109 m gs 1 -1 sc (r) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 1623 4276 a Fs(Figure)37 b(7.)486 4555 y Fr(W)-8 b(e)29 b(c)m(hose)i Fp(F)39 b Fr(and)29 b Fp(C)35 b Fr(suc)m(h)c(that)e Fo(@)5 b(D)1854 4519 y Fn(2)1922 4555 y Fp(\032)28 b Fo(h)2083 4570 y Fn(2)2152 4555 y Fr(is)h(Legendrian.)42 b(Using)30 b(\(13\))f(w)m(e)386 4671 y(can)k(determine)f(the)h(rotation)e(n)m(um)m(b)s(er)i(of)f Fo(@)5 b(D)2168 4635 y Fn(2)2208 4671 y Fr(.)44 b(W)-8 b(e)32 b(obtain)386 4856 y(\(23\))959 b(rot)1643 4871 y Fk(C)1688 4856 y Fr(\()p Fo(@)5 b(D)1866 4815 y Fn(2)1906 4856 y Fr(\))28 b(=)f(2)p Fo(n)33 b(:)386 5042 y Fr(By)j(Remark)f(3.5)g (this)g(is)g(\(up)g(to)g(sign\))g(the)h(rotation)e(n)m(um)m(b)s(er)h (of)g(an)m(y)h(mo)s(del)386 5158 y(Engel)22 b(structure)h(on)e Fo(R)1253 5173 y Fn(2)1315 5158 y Fr(whose)i(ev)m(en)h(con)m(tact)e (structure)h(induces)g(the)f(singular)386 5274 y(foliation)29 b Fp(F)42 b Fr(on)32 b Fo(D)1101 5238 y Fn(2)1168 5274 y Fp(\032)c Fo(h)1329 5238 y Fn(2)1391 5274 y Fp(\002)23 b Fo(S)1557 5238 y Fn(1)1596 5274 y Fr(.)486 5390 y(No)m(w)36 b(\014x)g(an)f(orien)m(ted)g(trivialization)c Fo(C)2019 5405 y Fn(1)2059 5390 y Fo(;)17 b(C)2173 5405 y Fn(2)2212 5390 y Fr(.)51 b(W)-8 b(e)36 b(denote)g(the)g(horizon)m(tal)386 5506 y(lifts)d(of)h Fo(C)764 5521 y Fn(1)804 5506 y Fo(;)17 b(C)918 5521 y Fn(2)957 5506 y Fo(;)g(V)56 b Fr(on)34 b Fo(R)1325 5521 y Fn(2)1396 5506 y Fr(=)e Fo(h)1560 5521 y Fn(2)1623 5506 y Fp(\002)24 b Fo(S)1790 5470 y Fn(1)1864 5506 y Fr(b)m(y)36 b(the)f(same)f(sym)m(b)s(ols.)50 b(As)36 b(usual)e Fo(t)h Fr(is)386 5623 y(the)d(co)s(ordinate)g(on)g Fo(S)1234 5587 y Fn(1)1273 5623 y Fr(.)43 b(By)33 b(Prop)s(osition)d (4.5)i(the)h(distribution)d Fp(D)2953 5638 y Fl(k)3027 5623 y Fr(spanned)p eop %%Page: 32 32 32 31 bop 386 259 a Fq(32)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fr(b)m(y)791 678 y Fo(W)41 b Fr(=)1055 610 y Fo(@)p 1038 655 93 4 v 1038 746 a(@)5 b(t)1162 678 y Fr(+)22 b Fo("V)148 b Fr(and)127 b Fo(X)1876 693 y Fl(k)1947 678 y Fr(=)27 b(cos)q(\()p Fo(k)s(t)p Fr(\))p Fo(C)2416 693 y Fn(1)2477 678 y Fr(+)22 b(sin\()p Fo(k)s(t)p Fr(\))p Fo(C)2930 693 y Fn(2)386 884 y Fr(is)37 b(a)g(mo)s(del)e(Engel)j (structure)g(if)e Fo(")g(>)f Fr(0)i(is)g(small)e(enough)j(and)f Fo(k)i Fp(2)d Fh(Z)23 b Fp(n)i(f)p Fr(0)p Fp(g)p Fr(.)386 1000 y(The)38 b(c)m(haracteristic)f(foliation)d(of)i Fp(D)1774 1015 y Fl(k)1854 1000 y Fr(is)h(spanned)h(b)m(y)g Fo(W)14 b Fr(.)57 b(This)38 b(v)m(ector)g(\014eld)386 1116 y(is)31 b(transv)m(erse)i(to)e Fo(@)1112 1131 y Fk(\006)1172 1116 y Fo(R)1246 1131 y Fn(2)1317 1116 y Fr(and)g(it)g(p)s(oin)m(ts)g(in)f(the)i(desired)g(directions.)43 b(The)32 b(ev)m(en)386 1232 y(con)m(tact)j(structure)i Fp(E)j Fr(=)32 b([)p Fp(D)1458 1247 y Fl(k)1500 1232 y Fo(;)17 b Fp(D)1621 1247 y Fl(k)1663 1232 y Fr(])35 b(is)g(de\014ned)h(b)m(y)g Fo(\014)i Fr(=)31 b Fo(\013)25 b Fp(\000)f Fo("\013)q Fr(\()p Fo(V)d Fr(\))c Fo(dt)p Fr(.)50 b(Using)386 1348 y(the)33 b(expressions)h(for)e Fo(V)5 b(;)17 b(\013)q(;)g(h)32 b Fr(and)h(our)f(c)m(hoices)i(of)e Fo(g)2362 1363 y Fn(1)2401 1348 y Fo(;)17 b(g)2492 1363 y Fn(2)2563 1348 y Fr(w)m(e)34 b(obtain)1034 1519 y Fo(\014)f Fr(=)27 b(cos)q(\()p Fo(')p Fr(\))17 b Fo(dr)24 b Fr(+)e(sin\()p Fo(')p Fr(\))17 b Fo(dx)k Fr(+)h Fo(")17 b Fr(cos\()p Fo(')p Fr(\))g Fo(dt)32 b(:)386 1718 y Fr(on)546 1693 y Fj(e)523 1718 y Fo(A)g Fr(=)f Fo(A)24 b Fp(\002)g(f\000)p Fr(1)p Fo(=)p Fr(2)30 b Fp(\024)i Fo(x)g Fp(\024)g Fr(1)p Fo(=)p Fr(2)p Fp(g)22 b(\002)i Fo(S)1928 1682 y Fn(1)1967 1718 y Fr(.)50 b(The)36 b(con)m(tact)f(structure)h(on)e Fo(@)3201 1733 y Fk(\000)3261 1718 y Fo(R)3335 1733 y Fn(2)386 1835 y Fr(is)e(de\014ned)i(b)m(y)386 2005 y(\(24\))237 b Fo(\014)858 1921 y Fj(\014)858 1980 y(\014)891 2044 y Fl(@)928 2053 y Fe(\000)980 2044 y Fl(R)1033 2053 y Fg(2)1100 2005 y Fr(=)27 b(sin\()p Fo(')p Fr(\))17 b Fo(dx)k Fr(+)h Fo(")17 b Fr(cos\()p Fo(')p Fr(\))g Fo(dt)22 b Fp(\000)g Fo("g)2355 2020 y Fn(2)2394 2005 y Fr(\()p Fo(x)p Fr(\))17 b(sin\()p Fo(')p Fr(\))g Fo(dt)32 b(:)386 2202 y Fr(Restricting)27 b Fo(\014)34 b Fr(to)28 b Fo(T)1157 2166 y Fn(2)1143 2227 y(0)1224 2202 y Fr(=)f Fo(@)5 b(D)1467 2166 y Fn(2)1520 2202 y Fp(\002)13 b(f)p Fr(0)p Fp(g)g(\002)g Fo(S)1928 2166 y Fn(1)1997 2202 y Fr(w)m(e)29 b(see)h(that)e(the)g (singular)f(foliation)386 2318 y(on)33 b Fo(T)593 2282 y Fn(2)579 2343 y(0)666 2318 y Fr(is)g(in)g(standard)h(form.)45 b(The)34 b(curv)m(es)i Fo(')29 b Fr(=)g Fo(\031)t(=)p Fr(2)k(and)h Fo(')29 b Fr(=)g(3)p Fo(\031)t(=)p Fr(2)k(are)g(the)386 2435 y(Legendrian)24 b(divides)h(and)f(the)i(Legendrian)e(ruling)f(is)h (tangen)m(t)h(to)f(the)h(foliation)386 2551 y(giv)m(en)33 b(b)m(y)g(the)g(\014rst)g(factor)f(in)g Fo(T)1609 2515 y Fn(2)1595 2575 y(0)1676 2551 y Fr(=)27 b Fo(@)5 b(D)1919 2515 y Fn(2)1982 2551 y Fp(\002)22 b(f)p Fr(0)p Fp(g)g(\002)g Fo(S)2417 2515 y Fn(1)2457 2551 y Fr(.)486 2667 y(F)-8 b(or)29 b Fo(k)h(>)e Fr(0,)i(the)g(orien)m(tation)e(of)i(the)g(ev)m(en) i(con)m(tact)e(structure)h(is)e Fo(W)m(;)17 b(C)3155 2682 y Fn(1)3195 2667 y Fo(;)g(C)3309 2682 y Fn(2)3348 2667 y Fr(.)386 2783 y(This)34 b(is)g(the)h(orien)m(tation)d(used)k(in) d(\(23\))o(,)i(hence)h(the)e(rotation)f(n)m(um)m(b)s(er)h(of)g(the)386 2899 y(in)m(tersection)d(line)f(\014eld)i(along)d Fo(@)5 b(D)1700 2863 y Fn(2)1772 2899 y Fr(is)31 b(2)p Fo(n)p Fr(.)43 b(If)31 b Fo(k)g(<)d Fr(0)j(w)m(e)h(obtain)e(the)i(opp)s(osite) 386 3016 y(orien)m(tation)d Fo(W)m(;)17 b(C)1083 3031 y Fn(1)1122 3016 y Fo(;)g Fp(\000)p Fo(C)1313 3031 y Fn(2)1383 3016 y Fr(and)31 b(therefore)g(the)g(rotation)e(n)m(um)m(b)s (er)i(of)f(the)h(in)m(ter-)386 3132 y(section)42 b(line)f(\014eld)h(on) g Fo(@)1332 3147 y Fk(\000)1391 3132 y Fo(R)1465 3147 y Fn(2)1547 3132 y Fr(along)f Fo(@)5 b(D)1957 3096 y Fn(2)2039 3132 y Fr(has)42 b(no)m(w)h(the)f(opp)s(osite)g(sign,)i(i.e.) 386 3248 y(rot\()p Fo(@)5 b(D)689 3212 y Fn(2)729 3248 y Fr(\))28 b(=)g Fp(\000)p Fr(2)p Fo(n)p Fr(.)44 b(The)34 b(rotation)d(n)m(um)m(b)s(er)j(along)d(the)i(Legendrian)g(divide)f(is) 386 3364 y Fp(\000j)p Fo(k)s Fp(j)g Fr(b)m(y)i(Prop)s(osition)d(4.5.) 486 3481 y(Let)39 b(us)h(summarize)d(the)j(prop)s(erties)f(of)g(the)g (mo)s(del)f(Engel)h(structures)i(w)m(e)386 3597 y(ha)m(v)m(e)e (obtained)d(up)i(to)f(no)m(w.)58 b(Recall)36 b(that)h Fp(D)2138 3612 y Fl(k)2218 3597 y Fr(dep)s(ends)i(not)e(only)g(on)g Fo(k)j Fr(but)386 3713 y(also)27 b(on)h(the)g(c)m(hoice)g(of)g Fp(F)37 b Fr(at)28 b(the)g(b)s(eginning)f(of)g(the)h(pro)s(of)g(and)g (that)f Fp(j)p Fo(n)p Fp(j)h Fr(is)f(the)386 3829 y(n)m(um)m(b)s(er)i (of)f(closed)i(comp)s(onen)m(ts)f(of)f(the)h(dividing)e(set)j(of)e Fp(F)10 b Fr(.)42 b(In)29 b(the)g(follo)m(wing)386 3946 y(table)j Fp(j)p Fo(n)p Fp(j)g Fr(is)g(the)h(n)m(um)m(b)s(er)g(of)f (closed)h(comp)s(onen)m(ts)g(of)f(comp)s(onen)m(ts.)p 462 4049 2838 4 v 460 4281 4 233 v 777 4281 V 862 4130 a(Orien)m(tation)943 4246 y(of)g Fp(E)9 b Fo(=)p Fp(W)p 1434 4281 V 1518 4130 a Fr(Rotation)31 b(n)m(um)m(b)s(er)1523 4246 y Fo(@)5 b(D)1663 4210 y Fn(2)1725 4246 y Fp(\002)23 b(f)p Fr(0)p Fp(g)f(\002)g(f)p Fr(1)p Fp(g)p 2330 4281 V 2450 4130 a Fr(Rotation)31 b(n)m(um)m(b)s(er)2415 4246 y(Legendrian)h(divides)p 3297 4281 V 462 4285 2838 4 v 460 4401 4 117 v 503 4366 a Fo(k)f(>)d Fr(0)p 777 4401 V 238 w Fo(C)1046 4381 y Fn(1)1085 4366 y Fo(;)17 b(C)1199 4381 y Fn(2)p 1434 4401 V 1830 4366 a Fr(2)p Fo(n)p 2330 4401 V 785 w Fp(\000j)p Fo(k)s Fp(j)p 3297 4401 V 460 4517 V 503 4482 a Fo(k)31 b(<)d Fr(0)p 777 4517 V 199 w Fo(C)1007 4497 y Fn(1)1046 4482 y Fo(;)17 b Fp(\000)p Fo(C)1237 4497 y Fn(2)p 1434 4517 V 1791 4482 a Fp(\000)p Fr(2)p Fo(n)p 2330 4517 V 747 w Fp(\000j)p Fo(k)s Fp(j)p 3297 4517 V 462 4520 2838 4 v 386 4665 a Fr(Notice)30 b(that)g(since)h(w)m(e)g(\014xed)h Fo(V)51 b Fr(and)31 b Fo(\013)g Fr(on)f Fo(A)18 b Fp(\002)g Fh(R)5 b Fr(,)37 b(the)30 b(con)m(tact)h(structure)h(on)386 4781 y Fo(@)437 4796 y Fk(\000)496 4781 y Fo(R)570 4796 y Fn(2)643 4781 y Fr(is)g(indep)s(enden)m(t)i(from)d Fo(k)s(;)17 b(n)p Fr(.)486 4897 y(The)45 b(mo)s(del)d(Engel)i(structures)i(with)d(p)s (ositiv)m(e)h(rotation)e(n)m(um)m(b)s(ers)j(along)386 5013 y(the)33 b(Legendrian)f(divides)h(can)g(b)s(e)f(obtained)g(b)m(y)i (applying)d(the)i(in)m(v)m(olution)1410 5184 y Fo(f)39 b Fr(:)27 b Fo(R)1625 5199 y Fn(2)1693 5184 y Fp(\000)-16 b(!)27 b Fo(R)1955 5199 y Fn(2)1262 5335 y Fr(\()p Fo(r)m(;)17 b(';)g(x;)g(t)p Fr(\))28 b Fp(7\000)-16 b(!)27 b Fr(\()p Fo(r)m(;)17 b(';)g Fp(\000)p Fo(x;)g Fp(\000)p Fo(t)p Fr(\))33 b Fo(:)386 5506 y Fr(to)39 b(the)i(mo)s(del)d(Engel)h (structures)j(w)m(e)f(ha)m(v)m(e)g(obtained)e(so)i(far.)64 b(The)41 b(con)m(tact)386 5623 y(structure)48 b(on)f Fp(f\000)p Fr(1)p Fo(=)p Fr(2)k Fp(\024)h Fo(x)h Fp(\024)f Fr(1)p Fo(=)p Fr(2)p Fp(g)f(\032)h Fo(@)2087 5638 y Fk(\000)2147 5623 y Fo(R)2221 5638 y Fn(2)2307 5623 y Fr(is)47 b(preserv)m(ed)i(b)m (y)f Fo(f)57 b Fr(but)47 b Fo(f)p eop %%Page: 33 33 33 32 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(33)386 459 y Fr(rev)m(erses)34 b(the)d(orien)m(tation)e(of)i(the)h (Legendrian)e(divides,)i(cf.)f(\(24\))o(.)43 b(In)32 b(particu-)386 575 y(lar,)j(w)m(e)i(can)f(compare)f(the)h(orien)m (tations)e(of)h(the)h(con)m(tact)h(structure)g(and)e(the)386 691 y(homotop)m(y)41 b(class)i(of)e(the)h(in)m(tersection)g(line)f (\014elds)h(with)f(the)i(corresp)s(onding)386 807 y(prop)s(erties)h(of) g Fp(D)1056 822 y Fl(k)1098 807 y Fr(.)78 b(The)45 b(mo)s(del)e(Engel)g (structures)j Fo(f)2516 822 y Fk(\003)2556 807 y Fp(D)2633 822 y Fl(k)2719 807 y Fr(co)m(v)m(er)g(the)e(cases)386 924 y(missing)31 b(in)h(the)h(table)f(ab)s(o)m(v)m(e)h(\(with)f(the)h (exception)g(of)f Fo(k)f Fr(=)d(0\).)486 b Fi(\003)486 1123 y Fr(Using)41 b(a)h(singular)f(foliation)e(whose)k(dividing)d(set) j(has)g(more)f(non{closed)386 1240 y(comp)s(onen)m(ts)36 b(than)g Fp(F)45 b Fr(at)35 b(the)h(b)s(eginning)e(of)h(the)h(pro)s(of) e(of)h(Prop)s(osition)f(4.12)386 1356 y(one)29 b(can)g(construct)h(mo)s (del)d(Engel)i(structures)h(on)f Fo(R)2374 1371 y Fn(2)2442 1356 y Fr(suc)m(h)i(that)d(the)i(dividing)386 1472 y(set)j(of)f Fo(T)720 1436 y Fn(2)706 1497 y(0)792 1472 y Fr(has)h(4)p Fo(;)17 b Fr(6)p Fo(;)g(:)g(:)g(:)31 b Fr(comp)s(onen)m(ts.)386 1622 y Fy(Remark)k(4.13.)40 b Fr(This)31 b(remark)g(is)f(the)h (analogue)f(of)h(Remark)f(4.10)g(for)g(round)386 1739 y(handles)24 b(of)g(index)h(2.)40 b(Equip)25 b Fo(R)1556 1754 y Fn(2)1620 1739 y Fr(with)e(a)h(mo)s(del)f(Engel)h(structure)h (from)e(Prop)s(o-)386 1855 y(sition)28 b(4.12.)41 b(If)30 b Fo( )h Fr(:)d Fo(@)1186 1870 y Fk(\000)1246 1855 y Fo(R)1320 1870 y Fn(2)1387 1855 y Fp(\000)-16 b(!)27 b Fo(@)1626 1870 y Fn(+)1686 1855 y Fo(M)40 b Fr(is)28 b(an)h(em)m(b)s(edding)g(suc)m(h)i(that)e(the)g(image)386 1971 y(of)36 b Fo(T)572 1935 y Fn(2)558 1996 y(0)647 1971 y Fr(is)g(con)m(v)m(ex)i(and)f(suc)m(h)h(that)e Fo( )k Fr(preserv)m(es)f(the)e(isotop)m(y)f(class)g(of)g(the)h(di-)386 2087 y(viding)g(set,)k(then)e(one)g(can)g(isotop)s(e)f Fo( )43 b Fr(suc)m(h)d(that)e(it)g(preserv)m(es)j(the)e(con)m(tact)386 2204 y(structure.)79 b(This)44 b(follo)m(ws)f(almost)f(immediately)f (from)i(Theorem)h(3.13)g(and)386 2320 y(Theorem)33 b(3.8,)g(but)h (notice)f(that)g(the)g(con)m(tact)h(structure)h(on)e Fo(@)2782 2335 y Fk(\000)2841 2320 y Fo(R)2915 2335 y Fn(2)2988 2320 y Fr(is)g(not)g(in-)386 2436 y(v)-5 b(arian)m(t)34 b(under)i Fo(@)1049 2451 y Fl(x)1094 2436 y Fr(.)51 b(Since)36 b(it)e(is)h(homotopic)f(through)h(con)m(tact)h(structures)h(to)386 2552 y(a)43 b Fo(@)529 2567 y Fl(x)573 2552 y Fr({in)m(v)-5 b(arian)m(t)42 b(con)m(tact)h(structure,)k(w)m(e)d(can)g(use)g(Gra)m (y's)g(theorem)e(\(Theo-)386 2669 y(rem)32 b(2.2\))g(to)g(circum)m(v)m (en)m(t)i(this)e(problem.)386 2898 y(4.6.)48 b Fy(Mo)s(del)40 b(Engel)f(structures)g(on)h(round)g(handles)g(of)g(index)g Fr(3)p Fy(.)48 b Fr(W)-8 b(e)386 3014 y(no)m(w)29 b(come)f(to)g(the)h (construction)f(of)g(mo)s(del)f(Engel)h(structures)i(on)e(round)h(han-) 386 3130 y(dles)46 b(of)f(index)i(3.)83 b(In)46 b(the)g(pro)s(of)g(of)f (Theorem)h(6.1)g(w)m(e)h(will)c(attac)m(h)j Fo(R)3202 3145 y Fn(3)3288 3130 y Fr(to)386 3247 y(an)33 b(Engel)g(manifold)d Fo(M)44 b Fr(with)33 b(transv)m(erse)i(b)s(oundary)f(suc)m(h)h(that)d (the)i(con)m(tact)386 3363 y(structure)29 b(on)f Fo(@)981 3378 y Fn(+)1040 3363 y Fo(M)39 b Fr(is)28 b(o)m(v)m(ert)m(wisted.)43 b(Hence)30 b(the)e(con)m(tact)g(structure)i(on)d Fo(@)3201 3378 y Fk(\000)3261 3363 y Fo(R)3335 3378 y Fn(3)386 3479 y Fr(should)37 b(also)f(b)s(e)i(o)m(v)m(ert)m(wisted.)59 b(Con)m(trary)38 b(to)f(round)g(handles)h(of)e(index)i(0)f(w)m(e)386 3595 y(will)44 b(need)j Fm(al)5 b(l)46 b Fr(p)s(ossible)f(homotop)m(y)h (classes)h(of)e(in)m(tersection)i(line)d(\014elds)j(on)386 3712 y Fo(@)437 3727 y Fk(\000)496 3712 y Fo(R)570 3727 y Fn(3)610 3712 y Fr(.)d(In)32 b(particular)f(w)m(e)j(also)e(need)h (the)g(case)h Fo(k)c Fr(=)e(0)k(in)g(Prop)s(osition)f(4.5.)486 3828 y(If)f(it)g(w)m(ere)h(enough)g(to)g(ha)m(v)m(e)h(the)f(unique)g(p) s(ositiv)m(e)f(tigh)m(t)f(con)m(tact)i(structure)386 3944 y(on)46 b Fo(@)586 3959 y Fk(\000)646 3944 y Fo(R)720 3959 y Fn(3)811 3944 y Fp(')52 b Fo(S)1006 3908 y Fn(2)1077 3944 y Fp(\002)33 b Fo(S)1253 3908 y Fn(1)1338 3944 y Fr(it)46 b(w)m(ould)h(b)s(e)g(easy)g(to)g(describ)s(e)g(the)g(mo)s(del) e(Engel)386 4060 y(structures)37 b(explicitly)-8 b(.)51 b(In)35 b(the)h(o)m(v)m(ert)m(wisted)i(case)e(w)m(e)h(w)m(ere)g(not)e (able)g(to)g(\014nd)386 4176 y(explicit)26 b(form)m(ulas)g(but)h(w)m(e) i(still)24 b(can)k(construct)g(the)g(mo)s(del)e(Engel)h(structures.)386 4293 y(W)-8 b(e)23 b(use)h(ordinary)f(handles)g(of)f(dimension)g(3)g (to)h(construct)h(a)f(con)m(tact)g(structure)386 4409 y(together)30 b(with)g(a)g(con)m(tact)h(v)m(ector)g(\014eld)f Fo(V)52 b Fr(and)31 b(a)f(Legendrian)g(v)m(ector)h(\014eld)f Fo(X)386 4525 y Fr(suc)m(h)k(that)e Fo(X)r(;)17 b Fr([)p Fo(V)5 b(;)17 b(X)8 b Fr(])33 b(are)f(linearly)f(indep)s(enden)m(t.)486 4641 y(The)36 b(follo)m(wing)d(construction)j(is)f({)g(up)h(to)g(a)f (small)e(mo)s(di\014cation)g({)j(an)f(ex-)386 4758 y(ample)c(of)h (Giroux's)g(construction)h(of)f(con)m(v)m(ex)j(con)m(tact)e(structures) h(in)e([Gi1)o(].)386 4987 y(4.6.1.)48 b Fm(Contact)27 b(structur)-5 b(es)28 b(on)e(or)-5 b(dinary)26 b(hand)5 b(les)26 b(of)g(dimension)f Fr(3)p Fm(.)49 b Fr(On)24 b Fo(h)3232 5002 y Fn(1)3299 4987 y Fr(=)386 5103 y Fo(D)470 5067 y Fn(1)528 5103 y Fp(\002)19 b Fo(D)708 5067 y Fn(2)779 5103 y Fr(let)30 b Fo(x)h Fr(b)s(e)h(the)f(co)s(ordinate)f(on)h(the)g (\014rst)h(factor)e(and)i Fo(y)2794 5118 y Fn(1)2833 5103 y Fo(;)17 b(y)2925 5118 y Fn(2)2994 5103 y Fr(the)32 b(co)s(or-)386 5219 y(dinates)27 b(on)h(the)g(second)g(factor.)42 b(Consider)27 b(the)h(con)m(tact)g(structure)h Fp(C)3004 5234 y Fn(1)3071 5219 y Fr(de\014ned)386 5336 y(b)m(y)k Fo(\013)c Fr(=)e Fo(dy)814 5351 y Fn(1)875 5336 y Fr(+)22 b Fo(y)1021 5351 y Fn(2)1077 5336 y Fo(dx)32 b Fr(and)h(the)g(con)m (tact)g(v)m(ector)h(\014eld)386 5568 y(\(25\))687 b Fo(V)1304 5583 y Fn(1)1371 5568 y Fr(=)27 b(2)p Fo(y)1571 5583 y Fn(2)1663 5501 y Fo(@)p 1620 5545 144 4 v 1620 5637 a(@)5 b(y)1724 5652 y Fn(2)1796 5568 y Fp(\000)23 b Fo(x)1989 5501 y(@)p 1961 5545 113 4 v 1961 5637 a(@)5 b(x)2105 5568 y Fr(+)22 b Fo(y)2251 5583 y Fn(1)2344 5501 y Fo(@)p 2300 5545 144 4 v 2300 5637 a(@)5 b(y)2404 5652 y Fn(1)2487 5568 y Fo(:)p eop %%Page: 34 34 34 33 bop 386 259 a Fq(34)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fr(A)i(simple)f(calculation)f(sho)m(ws)k(that)e(the)h(Legendrian) f(v)m(ector)i(\014eld)e Fo(X)2992 474 y Fn(1)3059 459 y Fr(=)g Fo(@)3213 474 y Fl(y)3248 483 y Fg(2)3299 459 y Fr(+)386 575 y Fo(y)434 590 y Fn(2)473 575 y Fo(@)524 590 y Fl(y)559 599 y Fg(1)622 575 y Fp(\000)e Fo(@)775 590 y Fl(x)856 575 y Fr(has)36 b(the)g(prop)s(ert)m(y)h(that)f Fo(X)1902 590 y Fn(1)1941 575 y Fo(;)17 b Fr([)p Fo(V)2069 590 y Fn(1)2108 575 y Fo(;)g(X)2233 590 y Fn(1)2273 575 y Fr(])36 b(are)f(linearly)f(indep)s(enden)m(t)386 691 y(ev)m(erywhere.)90 b(The)49 b(c)m(haracteristic)e(surface)h(\(i.e.)87 b(the)47 b(p)s(oin)m(ts)g(where)h Fo(V)69 b Fr(is)386 807 y(tangen)m(t)33 b(to)f(the)h(con)m(tact)g(structure\))h(of)e Fo(V)54 b Fr(is)32 b(the)h(strip)1154 984 y Fo(S)1214 999 y Fn(1)1281 984 y Fr(=)27 b Fp(f)p Fr(\()p Fo(x;)17 b(y)1619 999 y Fn(1)1658 984 y Fo(;)g(y)1750 999 y Fn(2)1789 984 y Fr(\))28 b Fp(2)g Fo(h)2005 999 y Fn(1)2077 984 y Fp(j)k Fo(y)2185 999 y Fn(1)2252 984 y Fr(=)27 b Fo(xy)2458 999 y Fn(2)2498 984 y Fp(g)32 b Fo(:)386 1161 y Fr(The)d(con)m(tact)g (v)m(ector)g(\014eld)e Fo(V)50 b Fr(is)28 b(transv)m(erse)i(to)d Fo(@)2239 1176 y Fk(\006)2299 1161 y Fo(h)2355 1176 y Fn(1)2423 1161 y Fr(and)h(it)f(p)s(oin)m(ts)g(out)m(w)m(ards)386 1277 y(\(resp)s(ectiv)m(ely)41 b(in)m(w)m(ards\))g(along)e Fo(@)1686 1292 y Fn(+)1746 1277 y Fo(h)1802 1292 y Fn(1)1882 1277 y Fr(=)i Fo(D)2083 1241 y Fn(1)2149 1277 y Fp(\002)28 b Fo(@)5 b(D)2394 1241 y Fn(2)2475 1277 y Fr(\(resp)s(ectiv)m(ely)41 b Fo(@)3103 1292 y Fk(\000)3162 1277 y Fo(h)3218 1292 y Fn(1)3299 1277 y Fr(=)386 1393 y Fo(@)5 b(D)526 1357 y Fn(1)573 1393 y Fp(\002)i Fo(D)741 1357 y Fn(2)780 1393 y Fr(\).)41 b(The)26 b(in)m(tersection)f(of)g Fo(S)31 b Fr(with)24 b Fo(@)2055 1408 y Fk(\000)2115 1393 y Fo(h)2171 1408 y Fn(1)2210 1393 y Fr(,)j(resp)s(ectiv)m(ely)f Fo(@)2839 1408 y Fn(+)2899 1393 y Fo(h)2955 1408 y Fn(1)2994 1393 y Fr(,)h(consists)386 1509 y(of)32 b(t)m(w)m(o)h(arcs.)486 1626 y(On)h Fo(h)706 1641 y Fn(2)776 1626 y Fr(=)d Fo(D)967 1589 y Fn(2)1030 1626 y Fp(\002)23 b Fo(D)1214 1589 y Fn(1)1284 1626 y Fp(')32 b Fo(h)1449 1641 y Fn(1)1523 1626 y Fr(w)m(e)j(use)g(the)g(same)g(con)m(tact)g(structure)g(as)g(on)f Fo(h)3335 1641 y Fn(1)386 1742 y Fr(and)27 b(let)f Fo(X)786 1757 y Fn(2)853 1742 y Fr(=)h Fo(X)1037 1757 y Fn(1)1077 1742 y Fr(.)41 b(The)28 b(con)m(tact)f(v)m(ector)h(\014eld)f(is)f(rev)m (ersed,)k(i.e.)41 b Fo(V)2888 1757 y Fn(2)2955 1742 y Fr(=)28 b Fp(\000)p Fo(V)3193 1757 y Fn(1)3233 1742 y Fr(,)f(so)386 1858 y(it)g(p)s(oin)m(ts)h(out)m(w)m(ards)i(along)c Fo(@)1488 1873 y Fn(+)1548 1858 y Fo(h)1604 1873 y Fn(2)1671 1858 y Fr(=)i Fo(@)1826 1873 y Fk(\000)1885 1858 y Fo(h)1941 1873 y Fn(1)2009 1858 y Fr(and)g(in)m(w)m(ards)h(along)e Fo(@)2855 1873 y Fk(\000)2915 1858 y Fo(h)2971 1873 y Fn(2)3038 1858 y Fr(=)g Fo(@)3192 1873 y Fn(+)3252 1858 y Fo(h)3308 1873 y Fn(1)3348 1858 y Fr(.)386 1974 y(The)33 b(c)m(haracteristic)g(surface)g Fo(S)1574 1989 y Fn(2)1646 1974 y Fr(of)f Fo(V)1814 1989 y Fn(2)1886 1974 y Fr(satis\014es)h Fo(S)2305 1989 y Fn(2)2373 1974 y Fr(=)27 b Fo(S)2536 1989 y Fn(1)2576 1974 y Fr(.)486 2090 y(The)33 b(singular)f(foliation)d (on)k Fo(@)1625 2105 y Fk(\000)1684 2090 y Fo(h)1740 2105 y Fn(1)1812 2090 y Fr(consists)h(of)e(parallel)e(straigh)m(t)j (segmen)m(ts)386 2207 y(and)g(Figure)e(8)h(sho)m(ws)j(the)e(singular)e (foliation)e(on)j Fo(@)2350 2222 y Fn(+)2410 2207 y Fo(h)2466 2222 y Fn(1)2533 2207 y Fr(=)c Fo(@)2688 2222 y Fk(\000)2747 2207 y Fo(h)2803 2222 y Fn(2)2843 2207 y Fr(.)1109 2864 y @beginspecial 0 @llx 0 @lly 185 @urx 59 @ury 1850 @rwi @setspecial %%BeginDocument: artchar.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: charaom.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Thu Aug 19 15:39:36 2004 %%For: tvogel@math81 (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 185 59 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -4.0 60.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 1984 m -1000 -1000 l 4147 -1000 l 4147 1984 l cp clip 0.06000 0.06000 sc % Polyline 7.500 slw n 1164 962 m 1237 886 l gs col0 s gr % Polyline n 1011 962 m 1237 731 l gs col0 s gr % Polyline n 857 962 m 1237 582 l gs col0 s gr % Polyline n 715 962 m 1237 426 l gs col0 s gr % Polyline n 550 972 m 1237 275 l gs col0 s gr % Polyline n 408 962 m 1237 124 l gs col0 s gr % Polyline n 245 962 m 1157 46 l gs col0 s gr % Polyline n 94 960 m 1008 46 l gs col0 s gr % Polyline n 2697 962 m 2757 886 l gs col0 s gr % Polyline n 2522 962 m 2757 731 l gs col0 s gr % Polyline n 2369 962 m 2757 582 l gs col0 s gr % Polyline n 2215 962 m 2757 426 l gs col0 s gr % Polyline n 2073 962 m 2757 275 l gs col0 s gr % Polyline n 1930 962 m 2757 124 l gs col0 s gr % Polyline n 1777 962 m 2679 46 l gs col0 s gr % Polyline n 1635 962 m 2526 46 l gs col0 s gr % Polyline n 1482 950 m 2372 46 l gs col0 s gr % Polyline n 2993 950 m 3135 808 l gs col0 s gr % Polyline 15.000 slw [15 45] 45 sd n 1312 950 m 1312 46 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 2832 950 m 2832 54 l gs col0 s gr [] 0 sd % Polyline 7.500 slw n 94 962 m 94 46 l 3135 46 l 3135 962 l cp gs col0 s gr % Polyline n 94 808 m 855 46 l gs col0 s gr % Polyline n 94 658 m 700 46 l gs col0 s gr % Polyline n 94 501 m 549 46 l gs col0 s gr % Polyline n 94 351 m 395 46 l gs col0 s gr % Polyline n 94 203 m 246 46 l gs col0 s gr % Polyline n 1387 886 m 2222 46 l gs col0 s gr % Polyline n 1387 731 m 2070 46 l gs col0 s gr % Polyline n 1387 582 m 1918 46 l gs col0 s gr % Polyline n 1387 426 m 1765 46 l gs col0 s gr % Polyline n 1387 275 m 1616 46 l gs col0 s gr % Polyline n 1387 124 m 1462 46 l gs col0 s gr % Polyline n 2906 886 m 3135 658 l gs col0 s gr % Polyline n 2906 731 m 3135 501 l gs col0 s gr % Polyline n 2906 582 m 3135 351 l gs col0 s gr % Polyline n 2906 426 m 3135 203 l gs col0 s gr % Polyline n 2906 275 m 3135 46 l gs col0 s gr % Polyline n 2906 124 m 2979 46 l gs col0 s gr $F2psEnd rs %%EndDocument @endspecial 1623 3080 a Fs(Figure)37 b(8.)486 3319 y Fr(On)27 b Fo(h)699 3334 y Fn(3)766 3319 y Fr(=)g Fo(D)953 3282 y Fn(3)1020 3319 y Fr(consider)g(the)h(standard)f(con)m(tact)h (structure)g(de\014ned)h(b)m(y)f Fo(\013)3232 3334 y Fn(3)3299 3319 y Fr(=)386 3435 y Fo(dz)h Fr(+)c Fo(x)17 b(dy)39 b Fr(and)d(the)h(con)m(tact)g(v)m(ector)g(\014eld)f Fo(V)2103 3450 y Fn(3)2176 3435 y Fr(=)e Fp(\000)p Fo(x@)2469 3450 y Fl(x)2539 3435 y Fp(\000)25 b Fr(2)p Fo(y)t(@)2793 3450 y Fl(y)2859 3435 y Fp(\000)g Fr(3)p Fo(z)t(@)3110 3450 y Fl(z)3151 3435 y Fr(.)54 b(Let)386 3551 y Fo(X)467 3566 y Fn(3)534 3551 y Fr(=)28 b Fo(@)689 3566 y Fl(x)746 3551 y Fr(+)13 b Fo(x@)941 3566 y Fl(z)995 3551 y Fp(\000)g Fo(@)1136 3566 y Fl(y)1178 3551 y Fr(.)42 b(Again)27 b(a)h(simple)f(calculation)e(sho)m(ws)30 b(that)e([)p Fo(V)2959 3566 y Fn(3)2998 3551 y Fo(;)17 b(X)3123 3566 y Fn(3)3162 3551 y Fr(])29 b(and)386 3667 y Fo(X)467 3682 y Fn(3)541 3667 y Fr(are)34 b(linearly)f(indep)s(enden)m(t)j(ev)m (erywhere.)52 b(The)35 b(c)m(haracteristic)g(surface)g(of)386 3783 y Fo(V)443 3798 y Fn(3)515 3783 y Fr(is)1091 3919 y Fo(S)1151 3934 y Fn(3)1218 3919 y Fr(=)1322 3839 y Fj(\010)1380 3919 y Fr(\()p Fo(x;)17 b(y)t(;)g(z)t Fr(\))27 b Fp(2)h Fo(D)1902 3934 y Fn(3)1942 3835 y Fj(\014)1942 3894 y(\014)2007 3919 y Fo(z)33 b Fr(=)27 b Fp(\000)p Fr(3)p Fo(=)p Fr(2)17 b Fo(xy)2536 3839 y Fj(\011)2643 3919 y Fo(:)386 4082 y Fr(In)33 b(particular,)e(the)i(dividing)d(set)k (\000)e(on)h Fo(S)1977 4045 y Fn(2)2048 4082 y Fr(is)f(connected.)486 4198 y(No)m(w)i(w)m(e)i(com)m(bine)d(the)i(handles)g(with)f(the)g(con)m (tact)h(v)m(ector)g(\014elds)g(and)f(the)386 4314 y(Legendrian)25 b(v)m(ector)h(\014elds)f(describ)s(ed)h(ab)s(o)m(v)m(e)f(in)g(order)g (to)f(construct)i(a)f(con)m(tact)386 4430 y(structure)f(on)g Fo(D)1005 4394 y Fn(3)1067 4430 y Fr(together)g(with)e(a)h(con)m(tact)h (v)m(ector)h(\014eld)e Fo(V)44 b Fr(and)24 b(a)f(Legendrian)386 4546 y(v)m(ector)34 b(\014eld)e Fo(C)960 4561 y Fn(1)1032 4546 y Fr(as)g(in)g(Prop)s(osition)f(4.5.)486 4663 y(W)-8 b(e)28 b(orien)m(t)f Fo(@)974 4678 y Fk(\006)1034 4663 y Fo(h)1090 4678 y Fl(i)1146 4663 y Fr(suc)m(h)j(that)d(the)i(orien)m (tation)d(of)i Fo(@)2381 4678 y Fk(\006)2440 4663 y Fo(h)2496 4678 y Fl(i)2552 4663 y Fr(follo)m(w)m(ed)f(b)m(y)i Fo(V)3112 4678 y Fl(i)3168 4663 y Fr(giv)m(es)386 4779 y(the)34 b(orien)m(tation)d(of)i(the)h(con)m(tact)g(structure.)47 b(Moreo)m(v)m(er,)35 b(w)m(e)f(can)g(asso)s(ciate)f(a)386 4895 y(sign)23 b(to)g(eac)m(h)i(comp)s(onen)m(t)e(of)g Fo(@)1541 4910 y Fk(\006)1601 4895 y Fo(h)1657 4910 y Fl(i)1689 4895 y Fp(n)t Fr(\()p Fo(S)1841 4910 y Fl(i)1873 4895 y Fp(\\)t Fo(@)1994 4910 y Fk(\006)2053 4895 y Fo(h)2109 4910 y Fl(i)2138 4895 y Fr(\))g(as)h(follo)m(ws:)38 b(If)23 b(the)h(orien)m(tation)386 5011 y(of)40 b Fo(X)586 5026 y Fl(i)614 5011 y Fo(;)17 b Fr([)p Fo(V)742 5026 y Fl(i)770 5011 y Fo(;)g(X)895 5026 y Fl(i)923 5011 y Fr(])p Fo(;)g(V)1051 5026 y Fl(i)1119 5011 y Fr(at)40 b Fo(p)h Fr(is)f(the)h(con)m(tact)g (orien)m(tation,)g(then)g(the)g(comp)s(onen)m(t)386 5128 y(con)m(taining)30 b Fo(p)g Fr(is)h(p)s(ositiv)m(e.)42 b(Otherwise)32 b(this)f(region)f(is)g(negativ)m(e.)43 b(Notice)31 b(that)386 5244 y(along)g Fo(S)706 5259 y Fl(i)734 5244 y Fr(,)i(the)g(v)m(ector)g(\014elds)g Fo(X)1585 5259 y Fl(i)1613 5244 y Fo(;)17 b Fr([)p Fo(V)1741 5259 y Fl(i)1769 5244 y Fo(;)g(X)1894 5259 y Fl(i)1922 5244 y Fr(])p Fo(;)g(V)2050 5259 y Fl(i)2111 5244 y Fr(are)32 b(not)h(linearly)d(indep)s(enden)m(t.)386 5390 y Fy(Prop)s(osition)g (4.14.)39 b Fm(Ther)-5 b(e)30 b(is)h(a)f(c)-5 b(ontact)31 b(structur)-5 b(e)32 b Fp(C)37 b Fm(on)30 b Fo(D)2762 5354 y Fn(3)2832 5390 y Fm(to)-5 b(gether)31 b(with)386 5506 y(a)42 b(c)-5 b(ontact)43 b(ve)-5 b(ctor)43 b(\014eld)f Fo(V)64 b Fm(and)42 b(a)g(L)-5 b(e)g(gendrian)42 b(ve)-5 b(ctor)43 b(\014eld)f Fo(X)50 b Fm(without)43 b(ze-)386 5623 y(r)-5 b(o)g(es)41 b(such)g(that)h Fr([)p Fo(V)5 b(;)17 b(X)8 b Fr(])41 b Fm(and)g Fo(X)49 b Fm(ar)-5 b(e)41 b(line)-5 b(arly)41 b(indep)-5 b(endent)40 b(everywher)-5 b(e)40 b(and)p eop %%Page: 35 35 35 34 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(35)386 459 y Fo(V)61 b Fm(is)39 b(tr)-5 b(ansverse)38 b(to)i Fo(@)5 b(D)1342 423 y Fn(3)1382 459 y Fm(.)58 b(Mor)-5 b(e)g(over,)40 b(the)f(dividing)f(set)i(on)f Fo(@)5 b(D)2914 423 y Fn(3)2993 459 y Fm(has)39 b(thr)-5 b(e)g(e)386 575 y(c)g(onne)g(cte)g(d)34 b(c)-5 b(omp)g(onents.)386 783 y(Pr)g(o)g(of.)41 b Fr(Consider)e(the)g(o)m(v)m(ert)m(wisted)h(con) m(tact)f(structure)g(on)f Fh(R)2772 747 y Fn(3)2856 783 y Fr(whic)m(h)h(is)f(de-)386 899 y(\014ned)f(b)m(y)f Fo(\013)e Fr(=)g(cos\()p Fo(r)1187 863 y Fn(2)1226 899 y Fr(\))17 b Fo(dz)29 b Fr(+)24 b(sin\()p Fo(r)1711 863 y Fn(2)1750 899 y Fr(\))17 b Fo(d')35 b Fr(\(in)g(p)s(olar)f(co)s (ordinates\).)53 b(By)36 b(Theo-)386 1026 y(rem)27 b(3.11)h(the)g (sphere)i Fo(S)6 b Fr(\()p Fo(r)1389 1041 y Fn(0)1428 1026 y Fr(\))28 b(with)g(radius)f Fo(r)2044 1041 y Fn(0)2111 1026 y Fr(=)2215 940 y Fj(p)p 2315 940 206 4 v 86 x Fr(3)p Fo(\031)t(=)p Fr(2)g(around)h(the)g(origin)e(is)386 1142 y(con)m(v)m(ex,)j(i.e.)41 b(there)27 b(is)e(a)g(con)m(tact)i(v)m(ector) f(\014eld)g Fo(V)47 b Fr(transv)m(erse)28 b(to)e Fo(S)6 b Fr(\()p Fo(r)2949 1157 y Fn(0)2988 1142 y Fr(\).)41 b(W)-8 b(e)26 b(as-)386 1258 y(sume)j(that)f Fo(V)50 b Fr(p)s(oin)m(ts)29 b(in)m(w)m(ards.)43 b(Consider)29 b(a)f(collar)f Fo(U)38 b Fp(')28 b Fo(S)6 b Fr(\()p Fo(r)2730 1273 y Fn(0)2769 1258 y Fr(\))14 b Fp(\002)g Fr([0)p Fo(;)j(")p Fr(])p Fo(;)g(")27 b(>)h Fr(0)386 1374 y(of)23 b Fo(S)6 b Fr(\()p Fo(r)636 1389 y Fn(0)675 1374 y Fr(\))23 b(suc)m(h)h(that)f Fo(V)45 b Fr(corresp)s(onds)24 b(to)f(the)h (standard)f(v)m(ector)h(\014eld)f Fo(@)2977 1389 y Fl(\034)3044 1374 y Fr(induced)386 1491 y(b)m(y)33 b(the)g(second)h(factor.)486 1607 y(Clearly)-8 b(,)32 b(one)h(can)g(c)m(ho)s(ose)h(a)f(no)m(where)h (v)-5 b(anishing)32 b(section)h(of)g(k)m(er)q(\()p Fo(\013)q Fr(\))f(along)386 1723 y Fo(S)6 b Fr(\()p Fo(r)534 1738 y Fn(0)573 1723 y Fr(\))31 b(and)g(extend)i(it)d(to)h(a)g(Legendrian)f (v)m(ector)j(\014eld)e Fo(X)2526 1738 y Fl(U)2616 1723 y Fr(on)g Fo(U)41 b Fr(with)31 b(the)h(de-)386 1839 y(sired)h(prop)s (ert)m(y)-8 b(.)46 b(Since)33 b Fo(S)6 b Fr(\()p Fo(r)1453 1854 y Fn(0)1492 1839 y Fr(\))33 b(is)g(simply)e(connected,)k(all)c(no) m(where)k(v)-5 b(anishing)386 1955 y(sections)40 b(of)g Fp(C)46 b Fr(along)38 b Fo(S)6 b Fr(\()p Fo(r)1389 1970 y Fn(0)1428 1955 y Fr(\))40 b(are)g(homotopic.)64 b(W)-8 b(e)40 b(orien)m(t)g(the)g(b)s(oundary)g(of)386 2072 y Fo(U)49 b Fr(suc)m(h)40 b(that)e(the)g(orien)m(tation)f(of)h(the)g(b) s(oundary)h(follo)m(w)m(ed)f(b)m(y)h Fo(V)60 b Fr(giv)m(es)38 b(the)386 2188 y(orien)m(tation)31 b(of)h(the)h(con)m(tact)g (structure.)486 2304 y(In)46 b(order)h(to)f(pro)m(v)m(e)h(the)g(prop)s (osition)e(w)m(e)i(w)m(an)m(t)g(to)f(extend)i(the)f(con)m(tact)386 2420 y(structure,)33 b Fo(X)911 2435 y Fl(U)1002 2420 y Fr(and)f Fo(@)1242 2435 y Fl(\034)1318 2420 y Fr(to)f(the)i(in)m (terior)d(of)h(the)i(sphere.)45 b(T)-8 b(o)32 b(do)f(so)h(w)m(e)h (attac)m(h)386 2537 y(the)g(handles)g Fo(h)963 2552 y Fn(1)1002 2537 y Fo(;)17 b(h)1102 2552 y Fn(2)1142 2537 y Fo(;)g(h)1242 2552 y Fn(3)1281 2537 y Fr(.)486 2653 y(The)24 b(dividing)e(set)j(\000)1250 2668 y Fl(U)1333 2653 y Fr(on)e Fo(@)1510 2668 y Fn(+)1570 2653 y Fo(U)38 b Fr(=)28 b Fo(S)6 b Fr(\()p Fo(r)1926 2668 y Fn(0)1965 2653 y Fr(\))t Fp(\002)t(f)p Fo(")p Fp(g)24 b Fr(consists)h(of)e(three) i(connected)386 2769 y(comp)s(onen)m(ts)36 b Fo(\015)977 2784 y Fn(0)1016 2769 y Fo(;)17 b(\015)1111 2784 y Fn(+)1170 2769 y Fo(;)g(\015)1265 2784 y Fk(\000)1323 2769 y Fr(,)37 b(as)f(sho)m(wn)i(in)d(the)h(left)f(part)h(of)g(Figure)f(9.)53 b(Regions)386 2885 y(of)29 b Fo(@)545 2900 y Fn(+)605 2885 y Fo(U)40 b Fr(where)31 b Fo(X)r(;)17 b Fr([)p Fo(V)5 b(;)17 b(X)8 b Fr(])p Fo(;)17 b(V)51 b Fr(is)29 b(the)i(con)m(tact)f (orien)m(tation)e(are)i(called)e(p)s(ositiv)m(e)386 3002 y(and)33 b(if)e([)p Fo(X)r(;)17 b Fr([)p Fo(V)5 b(;)17 b(X)8 b Fr(])p Fo(;)17 b(V)54 b Fr(is)33 b(he)g(opp)s(osite)f(orien)m (tation)f(they)j(are)f(called)e(negativ)m(e.)446 4634 y @beginspecial 0 @llx 0 @lly 327 @urx 151 @ury 3443 @rwi @setspecial %%BeginDocument: surg.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: surg.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Wed Oct 13 10:25:39 2004 %%For: tvogel@math2h (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 327 151 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -8.0 159.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /endash 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 3641 m -1000 -1000 l 6568 -1000 l 6568 3641 l cp clip 0.06000 0.06000 sc % Arc 15.000 slw gs n 1348.5 -765.6 2435.6 119.4 60.4 arcn gs col0 s gr gr % Arc gs n 1347.2 -1211.1 1684.1 112.3 67.3 arcn gs col0 s gr gr % Arc gs n 1346.0 810.0 1678.0 112.3 67.2 arcn gs col0 s gr gr 7.500 slw % Ellipse n 1356 1353 1197 1196 0 360 DrawEllipse gs col0 s gr % Polyline 15.000 slw [60] 0 sd n 157 1363 m 160 1361 l 167 1358 l 177 1352 l 192 1344 l 209 1334 l 226 1325 l 244 1315 l 260 1307 l 274 1299 l 287 1292 l 300 1286 l 313 1280 l 325 1274 l 338 1268 l 350 1262 l 363 1257 l 376 1251 l 388 1245 l 401 1240 l 414 1234 l 428 1228 l 442 1222 l 457 1216 l 470 1211 l 484 1206 l 498 1200 l 513 1195 l 527 1189 l 542 1184 l 558 1178 l 573 1172 l 588 1167 l 603 1161 l 618 1156 l 634 1151 l 649 1146 l 664 1141 l 679 1136 l 695 1132 l 710 1127 l 726 1123 l 742 1119 l 757 1115 l 773 1110 l 789 1106 l 805 1102 l 821 1098 l 836 1094 l 853 1091 l 869 1087 l 885 1084 l 902 1081 l 918 1078 l 936 1076 l 953 1074 l 971 1071 l 990 1069 l 1008 1067 l 1026 1065 l 1044 1063 l 1062 1062 l 1078 1060 l 1094 1058 l 1109 1056 l 1123 1055 l 1141 1053 l 1156 1052 l 1170 1051 l 1183 1050 l 1195 1049 l 1206 1049 l 1218 1048 l 1230 1047 l 1243 1047 l 1257 1046 l 1271 1045 l 1287 1045 l 1305 1044 l 1325 1044 l 1347 1043 l 1367 1043 l 1383 1042 l 1394 1042 l 1398 1042 l 1399 1042 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1407 1044 m 1408 1044 l 1411 1044 l 1419 1045 l 1430 1046 l 1445 1047 l 1461 1049 l 1477 1050 l 1492 1051 l 1505 1053 l 1519 1054 l 1531 1055 l 1544 1056 l 1557 1058 l 1572 1059 l 1587 1060 l 1602 1062 l 1617 1063 l 1632 1065 l 1646 1066 l 1660 1067 l 1673 1069 l 1685 1070 l 1699 1072 l 1712 1073 l 1724 1074 l 1735 1076 l 1746 1077 l 1757 1078 l 1768 1080 l 1779 1082 l 1792 1084 l 1805 1086 l 1816 1088 l 1828 1091 l 1841 1093 l 1853 1096 l 1866 1099 l 1879 1103 l 1892 1106 l 1905 1109 l 1918 1112 l 1930 1115 l 1942 1118 l 1954 1121 l 1968 1124 l 1981 1128 l 1993 1131 l 2006 1135 l 2018 1138 l 2030 1141 l 2042 1145 l 2055 1148 l 2068 1152 l 2082 1156 l 2094 1160 l 2106 1163 l 2119 1167 l 2132 1171 l 2145 1175 l 2158 1179 l 2172 1184 l 2185 1188 l 2198 1192 l 2210 1196 l 2222 1201 l 2233 1205 l 2246 1210 l 2257 1216 l 2268 1222 l 2279 1228 l 2289 1234 l 2299 1241 l 2310 1247 l 2320 1252 l 2330 1258 l 2341 1263 l 2351 1268 l 2362 1272 l 2372 1276 l 2382 1280 l 2393 1283 l 2403 1287 l 2413 1291 l 2423 1294 l 2433 1299 l 2443 1303 l 2453 1308 l 2463 1313 l 2475 1319 l 2489 1327 l 2502 1334 l 2515 1341 l 2526 1347 l 2533 1351 l 2536 1353 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1992 347 m 1991 347 l 1988 345 l 1980 342 l 1968 336 l 1953 330 l 1938 323 l 1923 317 l 1910 311 l 1898 307 l 1888 303 l 1876 299 l 1864 295 l 1852 292 l 1840 289 l 1828 287 l 1817 284 l 1805 281 l 1792 277 l 1782 274 l 1771 271 l 1760 267 l 1749 264 l 1737 260 l 1726 257 l 1714 253 l 1704 250 l 1694 247 l 1685 245 l 1675 243 l 1666 242 l 1659 241 l 1652 240 l 1645 240 l 1638 239 l 1630 238 l 1620 237 l 1609 235 l 1598 233 l 1586 231 l 1574 228 l 1562 226 l 1549 224 l 1536 222 l 1523 220 l 1512 219 l 1501 218 l 1490 217 l 1478 217 l 1467 216 l 1455 215 l 1443 215 l 1431 215 l 1417 214 l 1403 214 l 1390 214 l 1376 214 l 1361 214 l 1346 214 l 1330 215 l 1314 215 l 1297 215 l 1281 216 l 1266 216 l 1251 217 l 1236 217 l 1223 218 l 1208 219 l 1194 220 l 1180 222 l 1168 223 l 1155 224 l 1143 226 l 1131 228 l 1118 229 l 1104 232 l 1090 234 l 1078 236 l 1066 239 l 1054 242 l 1042 244 l 1030 247 l 1018 250 l 1005 253 l 993 256 l 980 260 l 966 263 l 951 267 l 936 272 l 922 276 l 906 282 l 888 288 l 867 295 l 844 303 l 820 311 l 794 320 l 770 329 l 748 336 l 730 343 l 717 347 l 710 350 l 706 351 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 708 2331 m 711 2330 l 718 2327 l 729 2322 l 744 2317 l 761 2310 l 779 2302 l 797 2295 l 814 2289 l 829 2283 l 843 2278 l 857 2274 l 871 2270 l 885 2266 l 900 2261 l 916 2257 l 932 2253 l 948 2249 l 964 2245 l 979 2241 l 995 2237 l 1010 2233 l 1024 2230 l 1038 2227 l 1051 2224 l 1064 2222 l 1077 2220 l 1089 2219 l 1101 2217 l 1113 2216 l 1124 2215 l 1136 2213 l 1148 2212 l 1160 2211 l 1172 2209 l 1185 2208 l 1198 2207 l 1211 2206 l 1224 2204 l 1238 2203 l 1251 2202 l 1265 2201 l 1279 2200 l 1292 2199 l 1305 2198 l 1319 2197 l 1331 2197 l 1344 2196 l 1359 2196 l 1373 2195 l 1387 2195 l 1401 2195 l 1415 2195 l 1428 2195 l 1442 2196 l 1456 2196 l 1469 2196 l 1483 2197 l 1496 2198 l 1510 2199 l 1522 2200 l 1535 2200 l 1547 2201 l 1559 2202 l 1572 2204 l 1586 2205 l 1600 2207 l 1616 2210 l 1628 2212 l 1640 2215 l 1653 2218 l 1667 2220 l 1681 2224 l 1695 2227 l 1709 2230 l 1724 2234 l 1738 2237 l 1752 2241 l 1767 2244 l 1780 2248 l 1794 2251 l 1808 2255 l 1822 2259 l 1836 2263 l 1852 2268 l 1870 2274 l 1888 2280 l 1909 2287 l 1929 2293 l 1949 2300 l 1966 2306 l 1980 2311 l 1990 2314 l 1996 2316 l 1999 2317 l gs col0 s gr [] 0 sd /Times-Roman-iso ff 195.00 scf sf 701 252 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 165 789 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 195.00 scf sf 165 2014 m gs 1 -1 sc (+) col0 sh gr /Symbol ff 210.00 scf sf 1773 711 m gs 1 -1 sc (g) col0 sh gr /Symbol ff 210.00 scf sf 1849 789 m gs 1 -1 sc (+) col0 sh gr /Symbol ff 210.00 scf sf 701 1783 m gs 1 -1 sc (g) col0 sh gr /Symbol ff 210.00 scf sf 778 1936 m gs 1 -1 sc (0) col0 sh gr /Symbol ff 210.00 scf sf 2079 2518 m gs 1 -1 sc (g) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 808 2595 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 2170 2641 m gs 1 -1 sc (-) col0 sh gr % Arc gs n 4295.5 -728.0 2397.0 119.4 60.4 arcn gs col0 s gr gr % Arc 7.500 slw gs n 4241.8 1255.9 194.6 3.6 115.8 arc gs col0 s gr gr % Arc 15.000 slw gs n 4293.0 828.0 1647.0 112.4 67.2 arcn gs col0 s gr gr % Arc gs n 4294.2 -1162.1 1653.1 112.3 67.3 arcn gs col0 s gr gr 7.500 slw % Ellipse n 4299 1357 1178 1178 0 360 DrawEllipse gs col0 s gr % Polyline n 4334 1720 m 4336 1723 l 4341 1731 l 4348 1742 l 4357 1758 l 4369 1776 l 4380 1795 l 4391 1813 l 4401 1829 l 4410 1843 l 4418 1855 l 4425 1866 l 4433 1878 l 4440 1889 l 4446 1898 l 4453 1907 l 4459 1916 l 4465 1924 l 4471 1933 l 4478 1943 l 4485 1954 l 4493 1966 l 4500 1977 l 4508 1989 l 4516 2001 l 4524 2014 l 4532 2027 l 4541 2040 l 4550 2054 l 4558 2067 l 4567 2081 l 4575 2094 l 4583 2107 l 4590 2120 l 4597 2133 l 4604 2146 l 4611 2160 l 4619 2175 l 4626 2189 l 4633 2204 l 4640 2218 l 4646 2232 l 4653 2245 l 4658 2258 l 4664 2269 l 4668 2278 l 4674 2293 l 4678 2302 l 4680 2309 l 4682 2315 l 4684 2322 l 4685 2330 l 4685 2339 l 4685 2347 l 4684 2356 l 4683 2365 l 4681 2373 l 4680 2380 l 4678 2386 l 4675 2390 l 4673 2394 l 4670 2397 l 4668 2401 l 4666 2405 l 4664 2409 l 4662 2413 l 4661 2415 l 4661 2416 l gs col0 s gr % Polyline n 4621 2459 m 4620 2460 l 4616 2464 l 4609 2471 l 4601 2479 l 4594 2486 l 4587 2491 l 4579 2495 l 4570 2499 l 4561 2503 l 4551 2507 l 4542 2510 l 4534 2512 l 4526 2513 l 4518 2513 l 4509 2513 l 4500 2512 l 4496 2512 l 4495 2512 l gs col0 s gr % Polyline n 4117 193 m 4116 194 l 4109 196 l 4096 201 l 4081 207 l 4069 212 l 4058 217 l 4050 221 l 4042 224 l 4033 228 l 4025 231 l 4016 235 l 4008 239 l 3999 244 l 3991 250 l 3983 257 l 3975 265 l 3966 273 l 3957 283 l 3948 292 l 3940 301 l 3934 310 l 3929 320 l 3926 330 l 3925 341 l 3925 353 l 3926 366 l 3927 378 l 3928 389 l 3929 399 l 3929 406 l 3929 409 l 3930 411 l 3930 413 l 3931 414 l 3931 415 l 3932 416 l 3933 416 l 3934 416 l 3935 416 l gs col0 s gr % Polyline n 4700 1683 m 4700 1684 l 4702 1687 l 4707 1694 l 4714 1704 l 4722 1718 l 4732 1732 l 4740 1745 l 4747 1756 l 4754 1766 l 4759 1774 l 4765 1784 l 4771 1792 l 4776 1801 l 4781 1809 l 4786 1816 l 4791 1824 l 4796 1833 l 4801 1841 l 4806 1849 l 4812 1858 l 4817 1866 l 4822 1874 l 4827 1883 l 4833 1891 l 4839 1900 l 4845 1909 l 4850 1917 l 4856 1925 l 4862 1933 l 4868 1942 l 4875 1951 l 4881 1960 l 4887 1968 l 4893 1977 l 4899 1985 l 4905 1992 l 4912 2001 l 4918 2009 l 4924 2017 l 4930 2025 l 4936 2033 l 4942 2040 l 4949 2047 l 4956 2054 l 4964 2060 l 4971 2066 l 4980 2072 l 4988 2077 l 4996 2083 l 5005 2087 l 5014 2092 l 5023 2095 l 5033 2097 l 5044 2099 l 5055 2100 l 5066 2100 l 5078 2101 l 5089 2100 l 5100 2100 l 5111 2099 l 5122 2097 l 5133 2094 l 5144 2091 l 5154 2087 l 5165 2083 l 5176 2079 l 5186 2075 l 5196 2071 l 5205 2067 l 5216 2061 l 5227 2055 l 5239 2048 l 5251 2041 l 5259 2037 l 5262 2034 l 5263 2034 l gs col0 s gr % Polyline n 3977 497 m 3977 498 l 3979 502 l 3982 510 l 3987 524 l 3994 541 l 4001 559 l 4008 577 l 4015 592 l 4022 606 l 4028 618 l 4034 628 l 4040 638 l 4047 648 l 4053 659 l 4061 669 l 4068 679 l 4075 689 l 4083 699 l 4090 710 l 4097 720 l 4103 730 l 4110 740 l 4118 752 l 4125 764 l 4132 776 l 4140 788 l 4147 799 l 4154 811 l 4161 823 l 4168 835 l 4176 847 l 4184 860 l 4191 871 l 4198 882 l 4206 894 l 4214 906 l 4221 918 l 4229 930 l 4237 942 l 4245 954 l 4253 966 l 4261 977 l 4268 988 l 4276 999 l 4283 1009 l 4291 1019 l 4298 1029 l 4304 1037 l 4311 1045 l 4317 1053 l 4324 1061 l 4331 1070 l 4338 1080 l 4346 1090 l 4354 1102 l 4363 1116 l 4370 1127 l 4378 1139 l 4386 1153 l 4395 1167 l 4404 1182 l 4414 1197 l 4423 1213 l 4433 1229 l 4443 1246 l 4452 1262 l 4462 1277 l 4471 1292 l 4480 1307 l 4488 1320 l 4496 1333 l 4503 1345 l 4512 1360 l 4521 1374 l 4528 1387 l 4536 1399 l 4543 1410 l 4550 1421 l 4556 1432 l 4563 1443 l 4570 1454 l 4576 1465 l 4583 1476 l 4590 1487 l 4598 1499 l 4606 1512 l 4614 1524 l 4622 1537 l 4631 1550 l 4639 1562 l 4646 1573 l 4653 1583 l 4659 1592 l 4664 1599 l 4665 1601 l 4666 1602 l 4667 1603 l 4668 1604 l 4669 1605 l 4669 1606 l 4670 1607 l 4671 1607 l 4672 1608 l 4672 1607 l 4673 1607 l 4673 1606 l 4673 1605 l 4673 1604 l 4673 1603 l 4673 1602 l gs col0 s gr % Polyline n 3282 747 m 3285 744 l 3290 738 l 3299 728 l 3311 715 l 3326 699 l 3341 684 l 3356 670 l 3371 658 l 3385 648 l 3399 642 l 3413 638 l 3426 636 l 3440 636 l 3455 637 l 3470 638 l 3487 640 l 3504 642 l 3521 645 l 3538 648 l 3555 652 l 3572 656 l 3588 661 l 3603 667 l 3618 674 l 3632 682 l 3644 691 l 3655 700 l 3667 711 l 3677 723 l 3688 735 l 3698 748 l 3708 762 l 3718 775 l 3728 789 l 3738 804 l 3748 818 l 3759 832 l 3769 847 l 3779 862 l 3790 877 l 3801 892 l 3811 906 l 3821 921 l 3830 936 l 3840 952 l 3850 968 l 3860 984 l 3870 1001 l 3880 1017 l 3891 1034 l 3901 1051 l 3911 1068 l 3921 1085 l 3931 1102 l 3941 1118 l 3951 1135 l 3961 1151 l 3971 1167 l 3981 1183 l 3991 1199 l 4001 1215 l 4011 1231 l 4021 1247 l 4032 1263 l 4042 1279 l 4053 1296 l 4063 1312 l 4074 1329 l 4084 1345 l 4095 1361 l 4105 1377 l 4115 1393 l 4125 1408 l 4134 1422 l 4143 1436 l 4152 1449 l 4160 1462 l 4172 1480 l 4183 1497 l 4195 1514 l 4208 1532 l 4221 1551 l 4234 1570 l 4246 1587 l 4256 1602 l 4264 1613 l 4269 1620 l 4271 1623 l gs col0 s gr % Polyline 15.000 slw [60] 0 sd n 3666 2353 m 3669 2352 l 3676 2349 l 3686 2345 l 3701 2339 l 3717 2332 l 3735 2325 l 3752 2319 l 3769 2313 l 3784 2307 l 3798 2302 l 3811 2298 l 3825 2294 l 3839 2290 l 3854 2286 l 3869 2282 l 3885 2277 l 3901 2273 l 3917 2269 l 3933 2266 l 3948 2262 l 3963 2258 l 3977 2255 l 3991 2252 l 4004 2249 l 4017 2247 l 4029 2245 l 4041 2243 l 4052 2241 l 4064 2240 l 4075 2238 l 4086 2237 l 4098 2235 l 4110 2234 l 4122 2232 l 4134 2231 l 4147 2230 l 4160 2229 l 4173 2227 l 4187 2226 l 4201 2225 l 4215 2224 l 4229 2223 l 4242 2222 l 4255 2222 l 4268 2221 l 4280 2220 l 4291 2220 l 4306 2220 l 4319 2220 l 4331 2221 l 4342 2222 l 4353 2223 l 4365 2224 l 4377 2225 l 4390 2226 l 4401 2227 l 4413 2228 l 4424 2228 l 4437 2229 l 4449 2230 l 4462 2231 l 4474 2232 l 4486 2233 l 4497 2234 l 4507 2235 l 4519 2237 l 4530 2239 l 4541 2241 l 4551 2244 l 4561 2246 l 4571 2248 l 4579 2250 l 4587 2252 l 4597 2254 l 4605 2255 l 4613 2256 l 4617 2257 l 4618 2257 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 4353 1053 m 4354 1053 l 4357 1053 l 4364 1054 l 4376 1055 l 4391 1056 l 4407 1058 l 4423 1059 l 4437 1060 l 4451 1062 l 4464 1063 l 4476 1064 l 4488 1065 l 4502 1067 l 4516 1068 l 4530 1069 l 4545 1071 l 4559 1072 l 4574 1074 l 4588 1075 l 4601 1076 l 4614 1078 l 4626 1079 l 4640 1081 l 4652 1082 l 4664 1084 l 4675 1085 l 4686 1086 l 4696 1088 l 4707 1089 l 4719 1091 l 4731 1093 l 4744 1095 l 4755 1097 l 4767 1100 l 4779 1102 l 4792 1105 l 4805 1108 l 4818 1111 l 4831 1114 l 4844 1117 l 4856 1120 l 4869 1123 l 4880 1126 l 4892 1129 l 4905 1133 l 4918 1136 l 4931 1140 l 4943 1143 l 4954 1146 l 4966 1150 l 4978 1153 l 4991 1157 l 5004 1161 l 5017 1165 l 5029 1168 l 5041 1172 l 5053 1176 l 5066 1179 l 5079 1183 l 5092 1187 l 5106 1191 l 5119 1195 l 5131 1199 l 5143 1203 l 5155 1207 l 5166 1211 l 5178 1216 l 5190 1222 l 5201 1228 l 5212 1234 l 5222 1240 l 5232 1246 l 5242 1252 l 5252 1258 l 5263 1263 l 5273 1268 l 5283 1273 l 5293 1277 l 5303 1281 l 5313 1285 l 5323 1289 l 5333 1293 l 5343 1296 l 5353 1300 l 5362 1305 l 5372 1309 l 5382 1314 l 5392 1319 l 5404 1325 l 5417 1332 l 5430 1339 l 5443 1346 l 5453 1352 l 5460 1355 l 5463 1357 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 4928 367 m 4927 367 l 4924 365 l 4917 362 l 4905 357 l 4890 350 l 4875 344 l 4860 337 l 4847 332 l 4836 328 l 4826 324 l 4814 320 l 4802 317 l 4791 314 l 4779 311 l 4768 308 l 4756 306 l 4744 302 l 4732 299 l 4722 296 l 4711 293 l 4700 289 l 4689 286 l 4678 282 l 4666 279 l 4655 275 l 4645 272 l 4635 269 l 4626 267 l 4616 265 l 4608 264 l 4600 263 l 4593 263 l 4587 263 l 4580 262 l 4571 261 l 4562 260 l 4552 258 l 4541 256 l 4529 253 l 4518 250 l 4506 248 l 4493 245 l 4481 243 l 4468 241 l 4457 240 l 4446 239 l 4434 238 l 4421 238 l 4409 238 l 4396 237 l 4384 237 l 4372 237 l 4360 236 l 4350 236 l 4338 236 l 4327 236 l 4316 236 l 4306 236 l 4296 236 l 4286 236 l 4276 236 l 4265 236 l 4254 236 l 4242 236 l 4231 236 l 4219 235 l 4207 235 l 4195 235 l 4184 235 l 4174 236 l 4164 237 l 4155 238 l 4147 239 l 4139 241 l 4130 242 l 4122 244 l 4113 246 l 4103 247 l 4092 249 l 4079 250 l 4064 253 l 4046 255 l 4030 257 l 4019 259 l 4013 260 l 4012 260 l gs col0 s gr [] 0 sd % Polyline n 3676 375 m 3676 376 l 3677 381 l 3680 391 l 3685 408 l 3691 429 l 3697 450 l 3703 470 l 3708 488 l 3713 504 l 3718 518 l 3723 531 l 3728 543 l 3733 554 l 3739 566 l 3745 578 l 3752 591 l 3759 604 l 3767 617 l 3774 630 l 3783 643 l 3791 656 l 3799 668 l 3807 681 l 3816 693 l 3821 702 l 3827 710 l 3833 720 l 3840 730 l 3847 741 l 3855 753 l 3863 766 l 3873 780 l 3882 795 l 3892 810 l 3903 827 l 3914 845 l 3926 863 l 3938 882 l 3951 902 l 3964 923 l 3977 945 l 3991 967 l 4001 984 l 4012 1001 l 4024 1020 l 4036 1040 l 4049 1061 l 4063 1084 l 4078 1109 l 4095 1135 l 4112 1164 l 4131 1196 l 4152 1229 l 4174 1265 l 4196 1302 l 4220 1341 l 4244 1380 l 4268 1419 l 4290 1456 l 4311 1491 l 4330 1522 l 4346 1548 l 4359 1569 l 4368 1584 l 4374 1594 l 4377 1600 l 4379 1602 l gs col0 s gr % Polyline n 4468 1733 m 4470 1736 l 4475 1744 l 4482 1756 l 4493 1773 l 4506 1793 l 4519 1815 l 4533 1837 l 4547 1858 l 4559 1878 l 4570 1896 l 4581 1912 l 4590 1926 l 4598 1940 l 4606 1953 l 4614 1965 l 4622 1978 l 4631 1992 l 4640 2006 l 4648 2019 l 4657 2034 l 4666 2048 l 4676 2062 l 4685 2076 l 4693 2090 l 4702 2103 l 4711 2116 l 4719 2128 l 4727 2139 l 4735 2151 l 4744 2164 l 4754 2178 l 4764 2192 l 4774 2206 l 4785 2221 l 4795 2235 l 4806 2249 l 4816 2261 l 4825 2273 l 4834 2283 l 4841 2293 l 4848 2301 l 4858 2311 l 4867 2320 l 4876 2327 l 4885 2335 l 4895 2341 l 4902 2346 l 4906 2349 l 4907 2349 l gs col0 s gr % Polyline [60] 0 sd n 3808 463 m 3808 464 l 3808 469 l 3809 480 l 3809 493 l 3810 505 l 3812 516 l 3813 526 l 3816 536 l 3817 542 l 3820 550 l 3823 559 l 3827 569 l 3832 581 l 3839 594 l 3846 609 l 3855 626 l 3866 645 l 3877 665 l 3890 687 l 3904 711 l 3913 725 l 3922 740 l 3932 756 l 3942 773 l 3954 791 l 3966 810 l 3978 831 l 3992 852 l 4005 874 l 4020 896 l 4035 919 l 4050 942 l 4065 966 l 4080 989 l 4095 1013 l 4110 1036 l 4125 1058 l 4139 1080 l 4153 1102 l 4167 1123 l 4180 1144 l 4194 1164 l 4208 1186 l 4223 1208 l 4237 1230 l 4252 1252 l 4267 1274 l 4282 1296 l 4297 1318 l 4311 1340 l 4326 1361 l 4340 1382 l 4354 1402 l 4367 1421 l 4379 1440 l 4391 1457 l 4402 1473 l 4413 1488 l 4422 1502 l 4431 1514 l 4439 1526 l 4446 1536 l 4459 1554 l 4470 1569 l 4479 1582 l 4487 1593 l 4494 1603 l 4501 1611 l 4506 1617 l 4509 1621 l 4511 1623 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 4554 1713 m 4554 1714 l 4557 1717 l 4562 1726 l 4571 1739 l 4581 1754 l 4592 1771 l 4603 1787 l 4612 1802 l 4621 1815 l 4628 1827 l 4635 1838 l 4642 1849 l 4649 1860 l 4656 1871 l 4663 1883 l 4671 1896 l 4679 1909 l 4688 1922 l 4696 1936 l 4705 1950 l 4713 1963 l 4722 1976 l 4730 1989 l 4738 2002 l 4747 2014 l 4756 2027 l 4765 2041 l 4775 2055 l 4785 2069 l 4795 2084 l 4805 2098 l 4815 2112 l 4823 2125 l 4831 2137 l 4838 2148 l 4844 2158 l 4852 2173 l 4859 2187 l 4865 2200 l 4869 2212 l 4873 2223 l 4875 2233 l 4876 2240 l 4876 2246 l 4876 2249 l 4876 2251 l 4875 2253 l 4874 2255 l 4872 2256 l 4871 2258 l 4869 2258 l 4866 2259 l 4865 2260 l 4864 2260 l 4863 2260 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 4818 2305 m 4817 2305 l 4812 2305 l 4802 2305 l 4790 2304 l 4778 2304 l 4769 2303 l 4760 2303 l 4753 2302 l 4745 2300 l 4737 2298 l 4727 2296 l 4715 2293 l 4703 2289 l 4692 2286 l 4687 2284 l 4686 2284 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 3896 287 m 3895 287 l 3890 289 l 3880 293 l 3869 298 l 3858 303 l 3849 308 l 3843 312 l 3837 317 l 3833 322 l 3828 329 l 3824 337 l 3819 348 l 3814 359 l 3810 369 l 3808 374 l 3808 375 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 3124 1366 m 3127 1364 l 3134 1361 l 3144 1355 l 3158 1347 l 3175 1338 l 3192 1328 l 3209 1319 l 3225 1310 l 3239 1303 l 3252 1296 l 3264 1290 l 3276 1284 l 3289 1278 l 3301 1273 l 3313 1267 l 3326 1261 l 3338 1256 l 3350 1250 l 3363 1245 l 3376 1239 l 3389 1234 l 3403 1228 l 3418 1222 l 3431 1217 l 3445 1212 l 3459 1206 l 3474 1201 l 3490 1195 l 3506 1190 l 3522 1184 l 3538 1178 l 3553 1173 l 3569 1167 l 3584 1162 l 3598 1157 l 3611 1153 l 3623 1149 l 3639 1144 l 3654 1140 l 3667 1136 l 3680 1133 l 3693 1130 l 3705 1127 l 3717 1124 l 3729 1122 l 3740 1119 l 3752 1116 l 3763 1113 l 3775 1111 l 3788 1108 l 3801 1105 l 3816 1102 l 3829 1099 l 3840 1097 l 3846 1096 l 3850 1095 l gs col0 s gr [] 0 sd /Symbol ff 210.00 scf sf 5065 1685 m gs 1 -1 sc (g) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 5131 1830 m gs 1 -1 sc (0) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3436 315 m gs 1 -1 sc (p) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3526 435 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 195.00 scf sf 5461 1980 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 195.00 scf sf 4906 255 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 5431 765 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 3856 2625 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 5002 2508 m gs 1 -1 sc (p) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5101 2640 m gs 1 -1 sc (-) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 1623 4850 a Fs(Figure)37 b(9.)486 5158 y Fr(W)-8 b(e)37 b(attac)m(h)h Fo(h)1020 5173 y Fn(1)1097 5158 y Fr(to)f Fo(U)48 b Fr(using)37 b(an)h(orien)m(tation)e (preserving)i(em)m(b)s(edding)f Fo( )3272 5173 y Fn(1)3348 5158 y Fr(:)386 5274 y Fo(@)437 5289 y Fk(\000)496 5274 y Fo(h)552 5289 y Fn(1)626 5274 y Fp(\000)-17 b(!)34 b Fo(@)871 5289 y Fn(+)930 5274 y Fo(U)47 b Fr(suc)m(h)37 b(that)f Fo( )1544 5289 y Fn(1)1584 5274 y Fr(\()p Fo(p)1671 5289 y Fk(\006)1730 5274 y Fr(\))d Fp(2)h Fo(\015)1952 5289 y Fk(\006)2047 5274 y Fr(and)i(suc)m(h)i(that)e Fo( )2742 5289 y Fn(1)2817 5274 y Fr(preserv)m(es)j(the)386 5390 y(singular)k(foliation.)77 b(Moreo)m(v)m(er,)50 b(w)m(e)c(assume)f(that)g Fo( )2487 5405 y Fn(1)2571 5390 y Fr(maps)g(the)g(p)s(ositiv)m(e,)386 5506 y(resp)s(ectiv)m(ely)31 b(negativ)m(e,)g(regions)f(of)f Fo(@)1813 5521 y Fk(\000)1873 5506 y Fo(h)1929 5521 y Fn(1)1998 5506 y Fr(to)h(the)h(p)s(ositiv)m(e,) f(resp)s(ectiv)m(ely)h(neg-)386 5623 y(ativ)m(e,)38 b(regions)e(of)g Fo(@)1158 5638 y Fn(+)1218 5623 y Fo(U)10 b Fr(.)56 b(Suc)m(h)38 b(a)f(map)f(exists)h(b)s(ecause)h(the)f(dividing)e(set)j(on)p eop %%Page: 36 36 36 35 bop 386 259 a Fq(36)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fo(@)437 474 y Fk(\000)496 459 y Fo(U)44 b Fr(is)32 b(transv)m(erse)j(to)d(the)i(singular)d(foliation.)40 b(It)33 b(maps)g(p)s(oin)m(ts)f(where)i Fo(V)3237 474 y Fn(1)3309 459 y Fr(is)386 575 y(tangen)m(t)f(to)f(the)h(con)m(tact)g (structure)h(to)e(\000)1970 590 y Fl(U)2029 575 y Fr(.)486 691 y(F)-8 b(ollo)m(wing)44 b([Gi1)o(])k(one)f(obtains)h(a)f(smo)s(oth) g(con)m(tact)h(structure)h Fp(C)3050 655 y Fk(0)3121 691 y Fr(and)f(a)386 807 y(smo)s(oth)35 b(con)m(tact)i(v)m(ector)g (\014eld)f(on)g Fo(U)1808 771 y Fk(0)1866 807 y Fr(=)e Fo(U)h Fp([)2143 822 y Fl( )2189 831 y Fg(1)2253 807 y Fo(h)2309 822 y Fn(1)2385 807 y Fr(and)h(w)m(e)h(can)g(mo)s(dify)d (the)386 924 y(b)s(oundary)23 b(of)e Fo(U)33 b Fr(b)s(efore)23 b(w)m(e)g(attac)m(h)g Fo(h)1777 939 y Fn(1)1839 924 y Fr(suc)m(h)g(that)f(w)m(e)i(obtain)d(a)h(smo)s(oth)g(section)386 1040 y Fo(X)475 1004 y Fk(0)536 1040 y Fr(of)39 b Fp(C)712 1004 y Fk(0)774 1040 y Fr(with)f(the)h(prop)s(ert)m(y)g(that)f Fo(X)1886 1004 y Fk(0)1909 1040 y Fo(;)17 b Fr([)p Fo(V)2059 1004 y Fk(0)2082 1040 y Fo(;)g(X)2215 1004 y Fk(0)2238 1040 y Fr(])38 b(are)h(linearly)d(indep)s(enden)m(t.)386 1156 y(Here)d(w)m(e)h(use)f(that)g Fo( )1203 1171 y Fn(1)1275 1156 y Fr(preserv)m(es)i(p)s(ositiv)m(e)d(and)h(negativ)m(e)g(regions.) 486 1272 y(The)f(dividing)e(set)i(\000)1273 1236 y Fk(0)1327 1272 y Fr(on)g Fo(@)1513 1287 y Fn(+)1572 1272 y Fo(U)1648 1236 y Fk(0)1704 1272 y Fr(con)m(tains)f Fo(\015)2136 1287 y Fn(0)2175 1272 y Fr(.)43 b(The)33 b(other)e(comp)s(onen)m(t)h (con-)386 1389 y(sists)i(of)g Fo(S)28 b Fp(\\)c Fo(@)942 1404 y Fn(+)1001 1389 y Fo(h)1057 1404 y Fn(1)1131 1389 y Fr(together)34 b(with)f(the)i(parts)f(of)f Fo(\015)2324 1404 y Fn(+)2383 1389 y Fo(;)17 b(\015)2478 1404 y Fk(\000)2570 1389 y Fr(whic)m(h)35 b(do)e(not)h(lie)e(in)386 1505 y(the)42 b(attac)m(hing)e(region)h(of)f Fo(h)1479 1520 y Fn(1)1519 1505 y Fr(.)70 b(The)42 b(righ)m(t)e(part)h(of)g(Figure)g (9)g(sho)m(ws)h(\000)3165 1469 y Fk(0)3189 1505 y Fr(.)69 b(In)386 1621 y(particular)37 b(\000)903 1585 y Fk(0)965 1621 y Fr(has)j(t)m(w)m(o)f(connected)i(comp)s(onen)m(ts.)63 b(The)40 b(homotop)m(y)f(class)g(of)386 1737 y Fo(X)475 1701 y Fk(0)523 1737 y Fr(as)26 b(a)e(no)m(where)j(v)-5 b(anishing)24 b(section)h(of)g Fp(C)1988 1701 y Fk(0)2037 1737 y Fr(along)f(the)h(torus)h Fo(@)2744 1752 y Fn(+)2803 1737 y Fo(U)2879 1701 y Fk(0)2928 1737 y Fr(is)f(not)g(fully)386 1853 y(determined)36 b(b)m(y)i(this)e(construction.)56 b(The)37 b(restriction)f(of)g Fo(X)2721 1817 y Fk(0)2780 1853 y Fr(to)h Fo(\015)2955 1868 y Fn(0)3030 1853 y Fr(dep)s(ends)386 1970 y(only)h(on)g Fo(X)828 1985 y Fl(U)926 1970 y Fr(and)g(it)g(is)g (therefore)h(uniquely)g(determined)f(up)h(to)f(homotop)m(y)-8 b(.)386 2086 y(On)26 b(the)g(other)g(hand,)i(the)e(homotop)m(y)f(class) h(of)g Fo(X)2242 2050 y Fk(0)2291 2086 y Fr(along)e(a)i(curv)m(e)h (whic)m(h)f(in)m(ter-)386 2202 y(sects)32 b(the)e(b)s(elt)f(sphere)j (of)d Fo(h)1444 2217 y Fn(1)1514 2202 y Fr(in)g(exactly)i(one)f(p)s (oin)m(t)f(dep)s(ends)j(on)e(the)g(v)m(ertical)386 2318 y(mo)s(di\014cation)g(of)i(the)h(b)s(oundary)-8 b(,)33 b(cf.)43 b(Remark)33 b(2.21.)486 2435 y(W)-8 b(e)34 b(will)e(use)k (this)e(\015exibilit)m(y)e(when)k(w)m(e)f(attac)m(h)f Fo(h)2396 2450 y Fn(2)2436 2435 y Fr(.)49 b(Using)34 b(Theorem)g(3.11)386 2551 y(w)m(e)e(\014rst)g(deform)e(the)i(torus)f Fo(@)1520 2566 y Fn(+)1579 2551 y Fo(U)1655 2515 y Fk(0)1710 2551 y Fr(b)m(y)h(an)f(admissible)f(isotop)m(y)h(suc)m(h)h(that)f(it)f (is)386 2667 y(in)i(standard)h(form.)43 b(The)34 b(deformed)e(space)i (is)f(still)d(denoted)k(b)m(y)g Fo(U)2933 2631 y Fk(0)2957 2667 y Fr(.)44 b(W)-8 b(e)33 b(ma)m(y)386 2783 y(v)-5 b(ary)25 b(the)h(slop)s(e)f(of)g(the)h(Legendrian)f(ruling)e(suc)m(h)k (that)e(there)h(is)f(an)g(orien)m(tation)386 2900 y(preserving)45 b(em)m(b)s(edding)f Fo( )1435 2915 y Fn(2)1523 2900 y Fr(:)49 b Fo(@)1650 2915 y Fk(\000)1709 2900 y Fo(h)1765 2915 y Fn(2)1853 2900 y Fp(\000)-16 b(!)48 b Fo(@)2113 2915 y Fn(+)2172 2900 y Fo(U)2248 2863 y Fk(0)2317 2900 y Fr(whic)m(h)d(preserv)m(es)i(singular)386 3016 y(foliations)42 b(and)i(whic)m(h)i(maps)e(p)s(ositiv)m(e,)j(resp)s(ectiv)m(ely)f (negativ)m(e,)i(regions)c(of)386 3132 y Fo(@)437 3147 y Fk(\000)496 3132 y Fo(h)552 3147 y Fn(2)625 3132 y Fr(to)32 b(p)s(ositiv)m(e,)g(resp)s(ectiv)m(ely)h(negativ)m(e,)g (regions)f(of)h(the)g(b)s(oundary)-8 b(.)486 3248 y(No)m(w)35 b(w)m(e)g(can)g(extend)h(the)f(con)m(tact)g(structure)h Fp(C)2349 3212 y Fk(0)2407 3248 y Fr(and)f(the)f(con)m(tact)i(v)m (ector)386 3364 y(\014eld)j Fo(V)683 3328 y Fk(0)746 3364 y Fr(from)f Fo(U)1059 3328 y Fk(0)1123 3364 y Fr(to)h Fo(U)1325 3328 y Fk(00)1408 3364 y Fr(=)h Fo(U)1600 3328 y Fk(0)1651 3364 y Fp([)1717 3379 y Fl( )1763 3388 y Fg(2)1829 3364 y Fo(h)1885 3379 y Fn(2)1964 3364 y Fr(using)f Fp(C)2278 3379 y Fn(2)2358 3364 y Fr(and)h Fo(V)2612 3379 y Fn(2)2651 3364 y Fr(.)65 b(W)-8 b(e)40 b(denote)g(the)386 3481 y(extensions)32 b(b)m(y)g Fp(C)1046 3445 y Fk(00)1119 3481 y Fr(and)f Fo(V)1385 3445 y Fk(00)1428 3481 y Fr(.)43 b(When)31 b(w)m(e)h(w)m(an)m(t)g(to)e(extend)i Fo(X)2680 3445 y Fk(0)2734 3481 y Fr(to)e Fo(U)2927 3445 y Fk(00)3001 3481 y Fr(using)g Fo(X)3335 3496 y Fn(2)386 3597 y Fr(w)m(e)h(ha)m(v)m (e)g(to)f(ensure)i(that)d Fo( )1439 3612 y Fn(2)p Fk(\003)1514 3597 y Fr(\()p Fo(X)1633 3612 y Fn(2)1673 3597 y Fr(\))h(is)f (homotopic)g(to)g Fo(X)2516 3561 y Fk(0)2569 3597 y Fr(and)i(that)e (the)i(orien-)386 3713 y(tations)j(of)h(the)h(con)m(tact)g(structures)h (giv)m(en)e(b)m(y)i Fo(X)2290 3677 y Fk(0)2313 3713 y Fo(;)17 b Fr([)p Fo(V)2462 3677 y Fk(0)2485 3713 y Fo(;)g(X)2618 3677 y Fk(0)2641 3713 y Fr(])36 b(and)f Fo(X)2977 3728 y Fn(2)3016 3713 y Fo(;)17 b Fr([)p Fo(V)3144 3728 y Fn(2)3184 3713 y Fo(;)g(X)3309 3728 y Fn(2)3348 3713 y Fr(])386 3829 y(are)33 b(coheren)m(t.)486 3946 y(The)43 b(latter)f(requiremen)m(t)h(is)g(satis\014ed)g(since)g Fo( )2330 3961 y Fn(2)2413 3946 y Fr(maps)f(p)s(ositiv)m(e,)j(resp)s (ec-)386 4062 y(tiv)m(ely)39 b(negativ)m(e,)i(regions)d(of)h Fo(@)1587 4077 y Fk(\000)1646 4062 y Fo(h)1702 4077 y Fn(2)1781 4062 y Fr(to)g(p)s(ositiv)m(e,)h(resp)s(ectiv)m(ely)g (negativ)m(e,)h(re-)386 4178 y(gions)32 b(of)g Fo(@)798 4193 y Fn(+)857 4178 y Fo(U)933 4142 y Fk(0)957 4178 y Fr(.)486 4294 y(In)38 b(order)g(to)f(ensure)i(that)f Fo( )1586 4309 y Fn(2)1626 4294 y Fr(\()p Fo(X)1745 4309 y Fn(2)1784 4294 y Fr(\))g(is)f(homotopic)f(to)h Fo(X)2658 4258 y Fk(0)2719 4294 y Fr(w)m(e)i(use)g(the)f(fact)386 4411 y(that)32 b(the)g(homotop)m(y)g(class)h(of)e Fo(X)1647 4374 y Fk(0)1702 4411 y Fr(along)g Fo(@)2013 4426 y Fn(+)2073 4411 y Fo(U)2149 4374 y Fk(0)2205 4411 y Fr(is)g(not)h(uniquely)h (determined.)386 4527 y(Cho)s(osing)k(di\013eren)m(t)h(v)-5 b(alues)37 b(for)g Fo(k)k Fr(in)36 b(Remark)h(2.21)g(on)h(the)g(t)m(w)m (o)g(connected)386 4643 y(comp)s(onen)m(ts)i(of)g(the)g(attac)m(hing)f (region)g(of)g Fo(h)2136 4658 y Fn(1)2216 4643 y Fr(w)m(e)i(can)f(v)-5 b(ary)40 b(the)g(homotop)m(y)386 4759 y(class)i(of)f Fo(X)833 4723 y Fk(0)898 4759 y Fr(along)f(closed)i(curv)m(es)h(whic)m (h)f(in)m(tersect)h(the)f(b)s(elt)f(sphere)i(of)e Fo(h)3335 4774 y Fn(1)386 4875 y Fr(exactly)26 b(once.)42 b(In)26 b(this)f(w)m(a)m(y)i(w)m(e)g(can)f(ensure)h(the)f Fo( )2292 4890 y Fn(2)p Fk(\003)2367 4875 y Fr(\()p Fo(X)2486 4890 y Fn(2)2526 4875 y Fr(\))f(is)g(homotopic)f(to)i Fo(X)3352 4839 y Fk(0)386 4992 y Fr(along)i(the)h(attac)m(hing)g(curv)m(e)i(of)e Fo(h)1656 5007 y Fn(2)1695 4992 y Fr(.)42 b(No)m(w)30 b(w)m(e)h(can)e(use)h(v)m(ertical)f(mo)s(di\014cations)386 5108 y(of)42 b(the)i(b)s(oundary)f(and)g(w)m(e)h(obtain)e(a)h (Legendrian)g(v)m(ector)h(\014eld)f Fo(X)3025 5072 y Fk(00)3110 5108 y Fr(on)g Fo(U)3332 5072 y Fk(00)386 5224 y Fr(suc)m(h)34 b(that)e Fo(X)906 5188 y Fk(00)981 5224 y Fr(and)h([)p Fo(V)1276 5188 y Fk(00)1319 5224 y Fo(;)17 b(X)1452 5188 y Fk(00)1494 5224 y Fr(])32 b(are)h(linearly)d (indep)s(enden)m(t.)486 5340 y(The)41 b(attac)m(hing)e(curv)m(e)j(of)e Fo(h)1575 5355 y Fn(2)1655 5340 y Fr(in)m(tersects)h(eac)m(h)h(comp)s (onen)m(t)e(of)g(\000)3008 5304 y Fk(0)3071 5340 y Fr(exactly)386 5457 y(once.)i(The)26 b(dividing)e(set)j(\000)1414 5420 y Fk(00)1482 5457 y Fr(of)e Fo(@)1637 5472 y Fn(+)1697 5457 y Fo(U)1773 5420 y Fk(00)1841 5457 y Fr(con)m(tains)h(b)s(oth)g (comp)s(onen)m(ts)g(of)f(\000)3136 5420 y Fk(0)3185 5457 y Fr(with)386 5573 y(the)33 b(segmen)m(ts)h(con)m(tained)f(in)f(the)i (image)d(of)h Fo( )2151 5588 y Fn(2)2224 5573 y Fr(remo)m(v)m(ed.)45 b(The)34 b(endp)s(oin)m(ts)f(of)p eop %%Page: 37 37 37 36 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(37)386 459 y Fr(the)46 b(remaining)d(curv)m(es)48 b(are)e(connected)h (b)m(y)g(the)f(t)m(w)m(o)g(segmen)m(ts)h Fo(S)37 b Fp(\\)31 b Fo(@)3192 474 y Fn(+)3252 459 y Fo(h)3308 474 y Fn(2)3348 459 y Fr(.)386 575 y(Hence)48 b(\000)751 539 y Fk(00)839 575 y Fr(is)e(connected)j(and)d Fo(@)1675 590 y Fn(+)1735 575 y Fo(U)1811 539 y Fk(00)1900 575 y Fr(is)g(di\013eomorphic)f(to)h (a)g(sphere.)86 b(All)386 691 y(no)m(where)30 b(v)-5 b(anishing)28 b(Legendrian)h(v)m(ector)h(\014elds)f(along)f Fo(@)2537 706 y Fn(+)2596 691 y Fo(U)2672 655 y Fk(00)2744 691 y Fr(are)h(homotopic.)486 807 y(Using)f(Theorem)g(3.11)g(again,)g (w)m(e)i(c)m(hange)f(the)g(singular)e(foliation)e(on)j Fo(@)3196 822 y Fn(+)3256 807 y Fo(U)3332 771 y Fk(00)386 924 y Fr(suc)m(h)36 b(that)e(it)f(b)s(ecomes)i(equiv)-5 b(alen)m(t)34 b(to)g(the)h(singular)d(foliation)f(on)j Fo(@)3008 939 y Fk(\000)3068 924 y Fo(h)3124 939 y Fn(3)3164 924 y Fr(.)48 b(W)-8 b(e)386 1040 y(attac)m(h)41 b Fo(h)751 1055 y Fn(3)832 1040 y Fr(using)f(an)h(em)m(b)s(edding)g(of)f Fo(@)1911 1055 y Fk(\000)1971 1040 y Fo(h)2027 1055 y Fn(3)2107 1040 y Fr(in)m(to)g Fo(@)2364 1055 y Fn(+)2424 1040 y Fo(U)2500 1004 y Fk(00)2584 1040 y Fr(whic)m(h)i(satis\014es)f (the)386 1156 y(same)25 b(conditions)g(on)g(the)h(orien)m(tations)e(as) i Fo( )2075 1171 y Fn(1)2140 1156 y Fr(and)g Fo( )2386 1171 y Fn(2)2426 1156 y Fr(.)41 b(After)25 b(a)g(v)m(ertical)g(mo)s(d-) 386 1272 y(i\014cation)31 b(of)h(the)i(b)s(oundary)f(w)m(e)h(can)f (attac)m(h)g Fo(h)2159 1287 y Fn(3)2231 1272 y Fr(to)g Fo(U)2427 1236 y Fk(00)2503 1272 y Fr(suc)m(h)h(that)e Fp(C)2992 1236 y Fk(00)3035 1272 y Fo(;)17 b(V)3157 1236 y Fk(00)3200 1272 y Fo(;)g(X)3333 1236 y Fk(00)386 1389 y Fr(extend)33 b(to)e(a)h(con)m(tact)g(structure)h Fp(C)38 b Fr(on)31 b(a)h(ball,)e(a)h(con)m(tact)h(v)m(ector)h(\014eld)e Fo(V)54 b Fr(and)386 1505 y(a)32 b(no)m(where)i(v)-5 b(anishing)32 b(section)g Fo(X)41 b Fr(of)32 b Fp(C)39 b Fr(with)32 b(the)h(desired)g(prop)s(erties.)193 b Fi(\003)386 1699 y Fr(4.6.2.)48 b Fm(Mo)-5 b(del)30 b(Engel)g(structur)-5 b(es)31 b(on)e Fo(R)1864 1714 y Fn(3)1904 1699 y Fm(.)49 b Fr(W)-8 b(e)28 b(apply)f(Prop)s(osition)e(4.5)i(to)g Fp(C)6 b Fo(;)17 b(V)386 1815 y Fr(and)35 b Fo(C)648 1830 y Fn(1)718 1815 y Fr(=)c Fo(X)42 b Fr(constructed)36 b(in)e(the)h(pro)s(of)f(of)g(Prop)s(osition)f(4.14.)48 b(This)35 b(yields)386 1932 y(mo)s(del)20 b(Engel)i(structures)h Fp(D)1449 1947 y Fl(k)1492 1932 y Fo(;)17 b(k)30 b Fp(2)e Fh(Z)p Fr(,)22 b(on)f Fo(R)2027 1947 y Fn(3)2089 1932 y Fr(suc)m(h)i(that)f(the)g(con)m(tact)g(structure)386 2048 y(on)32 b Fo(@)572 2063 y Fk(\000)632 2048 y Fo(R)706 2063 y Fn(3)773 2048 y Fr(=)c Fo(@)928 2063 y Fk(\000)987 2048 y Fo(U)33 b Fp(\002)23 b Fo(S)1252 2012 y Fn(1)1324 2048 y Fr(is)32 b(o)m(v)m(ert)m(wisted)i(b)m(y)g(Theorem)e(3.19.)486 2164 y(Because)39 b Fo(S)6 b Fr(\()p Fo(r)1012 2179 y Fn(0)1052 2164 y Fr(\))38 b(and)g(the)h(con)m(tact)g(form)e Fo(\013)i Fr(used)h(in)d(the)i(pro)s(of)f(of)f(Prop)s(o-)386 2280 y(sition)45 b(4.14)h(are)g(in)m(v)-5 b(arian)m(t)45 b(under)i(rotations)e(around)i(the)g Fo(z)t Fr({axis)f(w)m(e)i(ma)m(y) 386 2396 y(assume)33 b(that)f(the)h(same)g(is)f(true)h(for)f Fo(V)54 b Fr(and)33 b Fo(\013)q Fr(\()p Fo(V)21 b Fr(\).)43 b(F)-8 b(or)32 b Fo(s)c Fp(2)g Fr([0)p Fo(;)17 b Fr(1])32 b(let)705 2564 y Fo(\014)760 2579 y Fl(s)824 2564 y Fr(=)c Fo(sr)1018 2579 y Fn(0)1074 2564 y Fr(cos\()p Fo(r)1289 2523 y Fn(2)1286 2589 y(0)1345 2564 y Fr(sin)1465 2523 y Fn(2)1504 2564 y Fr(\()p Fo(\022)s Fr(\)\))17 b Fo(d\022)25 b Fr(+)d(sin\()p Fo(r)2107 2523 y Fn(2)2104 2589 y(0)2162 2564 y Fr(sin)2282 2523 y Fn(2)2321 2564 y Fr(\()p Fo(\022)s Fr(\)\))17 b Fo(d')22 b Fp(\000)g Fo(\013)q Fr(\()p Fo(V)f Fr(\))c Fo(dt)386 2731 y Fr(on)30 b Fo(@)570 2746 y Fk(\000)629 2731 y Fo(R)703 2746 y Fn(3)771 2731 y Fr(=)d Fo(S)940 2695 y Fn(2)997 2731 y Fp(\002)17 b Fo(S)1157 2695 y Fn(1)1226 2731 y Fr(in)30 b(spherical)f(co)s(ordinates)h(\()p Fo(\022)s(;)17 b(')p Fr(\))29 b(on)h Fo(S)2718 2695 y Fn(2)2758 2731 y Fr(.)42 b(According)30 b(to)386 2847 y(\(18\))o(,)f(the)f(con)m(tact)g(structure)h(on)f Fo(@)1712 2862 y Fk(\000)1771 2847 y Fo(R)1845 2862 y Fn(3)1913 2847 y Fr(is)f(de\014ned)i(b)m(y)f Fo(\014)2522 2862 y Fn(1)2589 2847 y Fr(and)g Fo(\014)2829 2862 y Fl(s)2894 2847 y Fr(is)f(a)g(con)m(tact)386 2964 y(form)d(for)h(all)e Fo(s)p Fr(.)41 b(By)26 b(Theorem)g(2.2)f(the)h(con)m(tact)f(structures) j(de\014ned)e(b)m(y)h Fo(\014)3153 2979 y Fn(0)3218 2964 y Fr(and)386 3080 y Fo(\014)441 3095 y Fn(1)512 3080 y Fr(are)k(isotopic.)42 b(The)32 b(self)f(di\013eomorphism)e Fo(f)42 b Fr(of)31 b Fo(S)2389 3044 y Fn(2)2447 3080 y Fp(\002)20 b Fo(S)2610 3044 y Fn(1)2681 3080 y Fr(with)31 b Fo(f)11 b Fr(\()p Fo(\022)s(;)17 b(';)g(t)p Fr(\))27 b(=)386 3196 y(\()p Fo(\022)s(;)17 b Fp(\000)p Fo(';)g Fp(\000)p Fo(t)p Fr(\))36 b(preserv)m(es)i Fo(\014)1363 3211 y Fn(0)1403 3196 y Fr(.)52 b(Moreo)m(v)m(er)38 b Fo(f)46 b Fr(preserv)m(es)38 b(the)e(Legendrian)g(curv)m(es)386 3312 y Fp(f)p Fo(p)p Fp(g)22 b(\002)h Fo(S)723 3276 y Fn(1)796 3312 y Fr(when)34 b Fo(p)c Fp(2)f(f)p Fo(')g Fr(=)g(0)p Fp(g)k Fr(and)g Fo(\013)q Fr(\()p Fo(V)21 b Fr(\))34 b(v)-5 b(anishes)34 b(at)f Fo(p)p Fr(.)46 b(The)34 b(orien)m(tations)386 3429 y(are)f(rev)m(ersed)h(b)m(y)g Fo(f)11 b Fr(.)486 3554 y(Hence)41 b Fo(f)51 b Fr(extends)42 b(to)e(a)g(di\013eomorphism)2173 3528 y Fj(e)2155 3554 y Fo(f)51 b Fr(of)40 b Fo(R)2447 3569 y Fn(3)2526 3554 y Fr(whic)m(h)h(preserv)m(es)i(the)386 3671 y(con)m(tact)35 b(structure)h(on)f Fo(@)1341 3686 y Fk(\000)1401 3671 y Fo(R)1475 3686 y Fn(3)1549 3671 y Fr(but)g(rev)m(erses)j(its)c(orien) m(tation)f(and)i(whic)m(h)h(pre-)386 3787 y(serv)m(es)42 b(a)d(Legendrian)h(curv)m(e)h(on)f Fo(@)1738 3802 y Fk(\000)1797 3787 y Fo(R)1871 3802 y Fn(3)1951 3787 y Fr(but)g(rev)m(erses)i(its)d (orien)m(tation.)64 b(The)386 3903 y(rotation)34 b(n)m(um)m(b)s(er)i (of)f(the)h(in)m(tersection)f(line)g(\014eld)g(on)g Fo(@)2523 3918 y Fk(\000)2583 3903 y Fo(R)2657 3918 y Fn(3)2732 3903 y Fr(along)f Fp(f)p Fo(p)p Fp(g)24 b(\002)g Fo(S)3335 3867 y Fn(1)386 4019 y Fr(is)35 b Fp(\000j)p Fo(k)s Fp(j)h Fr(for)g(the)g(mo)s(del)e(Engel)i(structures)i(w)m(e)f(ha)m(v)m(e)g (obtained)e(so)i(far.)53 b(If)36 b(w)m(e)386 4151 y(push)c(forw)m(ard)f (the)g(mo)s(del)e(Engel)h(structure)i(w)m(e)g(obtained)f(so)f(far)h (using)3192 4125 y Fj(e)3174 4151 y Fo(f)42 b Fr(w)m(e)386 4267 y(obtain)35 b(mo)s(del)f(Engel)i(structures)i(whic)m(h)f(represen) m(t)h(the)f(missing)d(homotop)m(y)386 4383 y(classes)42 b(of)f(in)m(tersection)g(line)f(\014elds)h(on)g Fo(@)2003 4398 y Fk(\000)2063 4383 y Fo(R)2137 4398 y Fn(3)2176 4383 y Fr(.)69 b(Th)m(us)43 b(w)m(e)f(ha)m(v)m(e)h(pro)m(v)m(ed)f(the) 386 4500 y(follo)m(wing)30 b(theorem.)386 4641 y Fy(Theorem)53 b(4.15.)48 b Fm(Ther)-5 b(e)47 b(ar)-5 b(e)47 b(mo)-5 b(del)46 b(Engel)h(structur)-5 b(es)48 b(on)f Fo(R)2889 4656 y Fn(3)2976 4641 y Fm(such)g(that)386 4757 y(the)38 b(induc)-5 b(e)g(d)38 b(c)-5 b(ontact)39 b(structur)-5 b(e)39 b(on)g Fo(@)1858 4772 y Fk(\000)1917 4757 y Fo(R)1991 4772 y Fn(3)2069 4757 y Fm(is)g Fp(C)h Fr(=)35 b(k)m(er)q(\()p Fo(\014)2604 4772 y Fn(1)2643 4757 y Fr(\))k Fm(and)f(al)5 b(l)38 b(p)-5 b(ossible)386 4874 y(orientations)41 b(of)h Fp(C)49 b Fm(and)41 b(homotopy)h(classes)f(of)h(interse)-5 b(ction)41 b(line)h(\014elds)f(ar)-5 b(e)386 4990 y(r)g(e)g(alize)g(d.) 927 5216 y Fr(5.)48 b Fs(A)-9 b(tt)i(a)n(ching)38 b(maps)g(f)n(or)g(r)n (ound)g(handles)486 5390 y Fr(In)33 b(the)h(follo)m(wing)c(sections)k (w)m(e)g(explain)e(ho)m(w)i(to)f(extend)h(Engel)f(structures)386 5506 y(from)26 b Fo(M)37 b Fr(to)27 b Fo(M)21 b Fp([)1037 5521 y Fl( )1101 5506 y Fo(R)1175 5521 y Fl(l)1228 5506 y Fr(using)27 b(a)f(suitable)g(mo)s(del)g(Engel)g(structure)j(on)d Fo(R)3078 5521 y Fl(l)3131 5506 y Fr(if)g Fo( )31 b Fr(is)386 5623 y(an)c(attac)m(hing)f(map)h(for)f(a)h(round)g(handle)g(of)g(index) g Fo(l)j Fr(=)e(1)p Fo(;)17 b Fr(2)p Fo(;)g Fr(3.)40 b(F)-8 b(or)26 b(this)h Fo( )k Fr(has)p eop %%Page: 38 38 38 37 bop 386 259 a Fq(38)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fr(to)32 b(ha)m(v)m(e)h(certain)e(prop)s(erties)h(explained)g(in) f(Section)h(5.1)g(and)g(w)m(e)h(will)c(isotop)s(e)386 575 y Fo( )45 b Fr(and)d(apply)f(v)m(ertical)g(mo)s(di\014cation)e(of)i (the)h(b)s(oundary)g(to)f Fo(M)53 b Fr(to)41 b(bring)g Fo( )386 691 y Fr(in)m(to)32 b(the)h(desired)g(form.)42 b(This)33 b(is)f(discussed)i(in)e(Section)h(5.2)f(to)g(Section)g(5.4.) 386 870 y(5.1.)48 b Fy(Extending)26 b(Engel)f(structures.)49 b Fr(Let)23 b Fo(M)34 b Fr(b)s(e)23 b(an)f(Engel)h(manifold)d(with)386 986 y(orien)m(ted)47 b(c)m(haracteristic)f(foliation)d(and)k(transv)m (erse)h(b)s(oundaries)f(and)f Fo(R)3253 1001 y Fl(l)3326 986 y Fr(a)386 1102 y(round)33 b(handle)f(of)g(index)h Fo(l)d Fp(2)e(f)p Fr(1)p Fo(;)17 b Fr(2)p Fo(;)g Fr(3)p Fp(g)31 b Fr(with)h(a)g(mo)s(del)f(Engel)i(structure.)386 1240 y Fy(Prop)s(osition)k(5.1.)42 b Fm(Assume)36 b(that)g(an)f(emb)-5 b(e)g(dding)34 b Fo( )f Fr(:)d Fo(@)2587 1255 y Fk(\000)2646 1240 y Fo(R)2720 1255 y Fl(l)2776 1240 y Fp(!)e Fo(@)2955 1255 y Fn(+)3015 1240 y Fo(M)47 b Fm(maps)386 1356 y(the)28 b(oriente)-5 b(d)28 b(c)-5 b(ontact)29 b(structur)-5 b(e)29 b Fp(C)1701 1371 y Fl(R)1788 1356 y Fm(on)f Fo(@)1972 1371 y Fk(\000)2031 1356 y Fo(R)2105 1371 y Fl(l)2160 1356 y Fm(to)h(the)f(oriente)-5 b(d)28 b(c)-5 b(ontact)28 b(struc-)386 1472 y(tur)-5 b(e)35 b Fp(C)638 1487 y Fl(M)753 1472 y Fm(on)f Fo(@)943 1487 y Fn(+)1003 1472 y Fo(M)45 b Fm(and)35 b(pr)-5 b(eserves)34 b(interse)-5 b(ction)34 b(line)g(\014elds.)486 1588 y(Then)48 b(the)h(Engel)f(structur)-5 b(e)50 b(extends)e(fr)-5 b(om)49 b Fo(M)59 b Fm(to)50 b Fo(M)2633 1552 y Fk(0)2710 1588 y Fr(=)k Fo(M)43 b Fp([)3043 1603 y Fl( )3129 1588 y Fo(M)60 b Fm(by)386 1705 y(the)37 b(mo)-5 b(del)36 b(Engel)h(structur)-5 b(e)38 b(on)f Fo(R)1733 1720 y Fl(l)1759 1705 y Fm(.)51 b(If)37 b(the)g(Engel)f(structur)-5 b(es)39 b(ar)-5 b(e)36 b(oriente)-5 b(d,)386 1821 y(then)35 b(one)g(obtains)g(an)g(oriente)-5 b(d)35 b(Engel)g(structur)-5 b(e)36 b(on)f Fo(M)2569 1785 y Fk(0)2629 1821 y Fm(if)g Fo( )k Fm(pr)-5 b(eserves)35 b(the)386 1937 y(orientation)f(of)h(the)g(interse)-5 b(ction)34 b(line)g(\014elds.)386 2106 y(Pr)-5 b(o)g(of.)41 b Fr(Using)f(the)h(normal)e(form)g(theorem)i(\(Theorem)g(2.16\))f(for)g (neigh)m(b)s(or-)386 2222 y(ho)s(o)s(ds)j(of)g(transv)m(erse)j(h)m(yp)s (ersurfaces)g(in)d(Engel)g(manifolds)e(w)m(e)k(can)e(em)m(b)s(ed)386 2338 y(collars)f Fo(U)767 2353 y Fl(R)869 2338 y Fr(of)i Fo(@)1043 2353 y Fk(\000)1103 2338 y Fo(R)1177 2353 y Fl(l)1203 2338 y Fr(,)j(resp)s(ectiv)m(ely)e Fo(U)1886 2353 y Fl(M)2009 2338 y Fr(of)f Fo(@)2183 2353 y Fn(+)2242 2338 y Fo(M)10 b Fr(,)48 b(in)m(to)c Fh(P)p Fp(C)2742 2353 y Fl(R)2802 2338 y Fr(,)j(resp)s(ectiv)m(ely)386 2454 y Fh(P)p Fp(C)497 2469 y Fl(M)578 2454 y Fr(.)d(W)-8 b(e)33 b(iden)m(tify)f Fo(U)1238 2469 y Fl(R)1328 2454 y Fr(and)h Fo(U)1584 2469 y Fl(M)1696 2454 y Fr(with)f(their)g(image.) 486 2580 y(By)h(the)g(second)h(part)f(of)f(Prop)s(osition)f(2.17)h(w)m (e)i(obtain)e(an)g(em)m(b)s(edding)3269 2554 y Fj(e)3253 2580 y Fo( )g Fr(:)386 2706 y Fo(U)452 2721 y Fl(R)538 2706 y Fp(\000)-17 b(!)28 b Fh(P)p Fp(C)837 2721 y Fl(M)948 2706 y Fr(from)i Fo( )t Fr(.)42 b(Since)31 b Fo(')f Fr(preserv)m(es)j (in)m(tersection)d(line)f(\014elds,)3071 2680 y Fj(e)3055 2706 y Fo( )34 b Fr(maps)386 2823 y Fo(@)437 2838 y Fk(\000)496 2823 y Fo(R)570 2838 y Fl(l)641 2823 y Fp(\032)45 b Fo(U)829 2838 y Fl(R)929 2823 y Fr(to)d Fo(@)1109 2838 y Fn(+)1168 2823 y Fo(M)55 b Fp(\032)45 b Fo(U)1505 2838 y Fl(M)1626 2823 y Fr(and)e(b)s(ecause)g Fo( )k Fr(preserv)m(es)e(the)d(orien)m (tation)386 2951 y(of)g(the)h(con)m(tact)f(structure,)1512 2925 y Fj(e)1496 2951 y Fo( )t Fr(\()p Fo(U)1667 2966 y Fl(R)1724 2951 y Fr(\))h(and)f Fo(U)2070 2966 y Fl(M)2192 2951 y Fr(lie)e(on)j(opp)s(osite)e(sides)i(of)f(the)386 3067 y(h)m(yp)s(ersurface)35 b Fo(@)1007 3082 y Fn(+)1066 3067 y Fo(M)j Fp(\032)29 b Fh(P)p Fp(C)1415 3082 y Fl(M)1496 3067 y Fr(.)43 b(This)33 b(pro)m(v)m(es)i(the)e(claim.)781 b Fi(\003)486 3236 y Fr(Strictly)37 b(sp)s(eaking)h Fo(M)1352 3200 y Fk(0)1414 3236 y Fr(is)g(a)g(manifold)d(with)j(corners.)61 b(W)-8 b(e)39 b(smo)s(oth)e(these)386 3352 y(corners)45 b(b)m(y)h(cutting)d(out)i(a)f(piece)g(of)g Fo(R)1960 3367 y Fl(l)2031 3352 y Fr(suc)m(h)i(that)e(the)h(b)s(oundary)f(of)g (the)386 3468 y(resulting)34 b(manifold)d(is)k(still)d(transv)m(erse.) 52 b(Prop)s(osition)33 b(5.1)h(can)h(of)f(course)h(b)s(e)386 3584 y(form)m(ulated)c(with)h(an)h(Engel)f(manifold)e(instead)i(of)g (the)h(round)g(handle)g Fo(R)3199 3599 y Fl(l)3225 3584 y Fr(.)486 3701 y(The)47 b(follo)m(wing)d(three)j(sections)g(describ)s (e)g(ho)m(w)g(to)f(c)m(ho)s(ose)h(mo)s(del)e(Engel)386 3817 y(structures)29 b(on)e Fo(R)1041 3832 y Fn(1)1080 3817 y Fo(;)17 b(R)1198 3832 y Fn(2)1238 3817 y Fo(;)g(R)1356 3832 y Fn(3)1422 3817 y Fr(and)27 b(ho)m(w)h(to)e(isotop)s(e)g(the)i (attac)m(hing)e(map)g(in)h(order)386 3933 y(to)h(satisfy)h(the)g (conditions)e(in)h(Prop)s(osition)f(5.1.)42 b(It)28 b(is)h(enough)g(to) f(ensure)i(that)386 4049 y(the)j(attac)m(hing)e(map)h(preserv)m(es)j (to)d(homotop)m(y)g(class)g(of)g(the)h(in)m(tersection)f(line)386 4166 y(\014eld)g(since)h(w)m(e)h(can)f(then)g(apply)f(v)m(ertical)g(mo) s(di\014cations)e(of)j(the)g(b)s(oundary)-8 b(.)486 4282 y(Before)25 b(w)m(e)i(con)m(tin)m(ue)f(let)f(us)h(remark)f(that)h(w)m (e)g(will)e(alw)m(a)m(ys)i(assume)g(that)f(the)386 4398 y(attac)m(hing)36 b(map)h Fo( )i Fr(:)c Fo(@)1258 4413 y Fk(\000)1318 4398 y Fo(R)1392 4413 y Fl(l)1454 4398 y Fp(\000)-17 b(!)35 b Fo(@)1700 4413 y Fn(+)1760 4398 y Fo(M)48 b Fr(preserv)m(es)40 b(the)d(con)m(tact)h(orien)m(tations.) 386 4514 y(If)33 b(this)g(is)g(not)g(the)h(case)g(w)m(e)g(can)g (replace)f Fo( )38 b Fr(b)m(y)c Fo( )26 b Fp(\016)d Fo(f)44 b Fr(where)34 b Fo(f)44 b Fr(is)33 b(the)h(di\013eo-)386 4631 y(morphism)g(of)i Fo(@)1011 4646 y Fk(\000)1071 4631 y Fo(R)1145 4646 y Fl(l)1207 4631 y Fr(induced)h(b)m(y)g(complex)f (conjugation)f(of)g Fo(S)2811 4594 y Fn(1)2851 4631 y Fr(.)54 b(W)-8 b(e)37 b(obtain)386 4747 y(di\013eomorphic)31 b(manifolds)f(when)k(w)m(e)f(attac)m(h)g Fo(R)2211 4762 y Fl(l)2270 4747 y Fr(to)f Fo(M)43 b Fr(using)32 b Fo( )37 b Fr(or)32 b Fo( )26 b Fp(\016)c Fo(f)11 b Fr(.)386 4925 y(5.2.)48 b Fy(A)m(ttac)m(hing)43 b(maps)i(for)g(round)g(handles)g(of)g (index)f Fr(1)p Fy(.)49 b Fr(The)39 b(mo)s(del)386 5042 y(Engel)28 b(structures)i(on)e(round)h(handles)g(of)f(index)g(1)g (induce)h(con)m(tact)g(structures)386 5158 y(on)42 b Fo(@)582 5173 y Fk(\000)642 5158 y Fo(R)716 5173 y Fn(1)798 5158 y Fr(suc)m(h)i(that)f(the)g(attac)m(hing)e(curv)m(es)k Fo(\015)2228 5173 y Fk(\006)2331 5158 y Fr(=)g Fp(f\006)p Fr(1)p Fp(g)29 b(\002)g(f)p Fr(0)p Fp(g)f(\002)i Fo(S)3163 5122 y Fn(1)3245 5158 y Fr(are)386 5274 y(Legendrian.)56 b(Let)37 b Fo( )i Fr(:)c Fo(@)1336 5289 y Fk(\000)1395 5274 y Fo(R)1469 5289 y Fn(1)1544 5274 y Fp(\000)-16 b(!)34 b Fo(@)1790 5289 y Fn(+)1850 5274 y Fo(M)47 b Fr(b)s(e)38 b(an)e(em)m(b)s(edding)h(preserving)g(the)386 5390 y(con)m(tact)48 b(orien)m(tation.)88 b(W)-8 b(e)48 b(w)m(an)m(t)h(to)e(isotop)s(e)h Fo( )j Fr(suc)m(h)f(that)d(the)i (resulting)386 5506 y(map)38 b(preserv)m(es)k(orien)m(ted)d(con)m(tact) g(structures)h(and)f(the)h(homotop)m(y)e(class)h(of)386 5623 y(in)m(tersection)i(line)f(\014elds)i(for)f(a)g(suitable)g(c)m (hoice)g(of)g(mo)s(del)f(Engel)h(structure)p eop %%Page: 39 39 39 38 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(39)386 459 y Fp(D)463 474 y Fl(k)r(;m)587 459 y Fr(.)60 b(After)38 b(a)g(v)m(ertical)f(mo)s(di\014cation)e(of)j Fo(@)2107 474 y Fn(+)2166 459 y Fo(M)49 b Fr(w)m(e)39 b(then)f(obtain)f(an)h(Engel)386 575 y(structure)c(on)e Fo(M)44 b Fr(with)32 b Fo(R)1373 590 y Fn(1)1445 575 y Fr(attac)m(hed.)486 691 y(It)22 b(is)f(a)h(standard)h(fact)f(in)f (con)m(tact)i(top)s(ology)e(that)g(ev)m(ery)j(em)m(b)s(edded)g(curv)m (e)f(in)386 807 y(a)j(con)m(tact)h(manifold)c(can)k(b)m(y)g(isotop)s (ed)f(to)g(a)g(Legendrian)g(curv)m(e)h(suc)m(h)h(that)e(the)386 924 y(isotop)m(y)38 b(nev)m(er)h(lea)m(v)m(es)g(an)f(arbitrarily)d (small)g(tubular)i(neigh)m(b)s(orho)s(o)s(d)g(of)g(the)386 1040 y(original)23 b(curv)m(e,)29 b([Aeb].)42 b(Hence)27 b(w)m(e)g(ma)m(y)f(assume)h(that)e Fo( )t Fr(\()p Fo(\015)2623 1055 y Fk(\006)2682 1040 y Fr(\))h(are)g(Legendrian)386 1156 y(curv)m(es.)486 1272 y(Next)43 b(w)m(e)h(w)m(an)m(t)g(to)e (isotop)s(e)g Fo( )47 b Fr(suc)m(h)d(that)f(the)g(resulting)f(map)g (preserv)m(es)386 1389 y(orien)m(ted)31 b(con)m(tact)g(structures)i (and)d(the)i(homotop)m(y)e(class)h(of)f(in)m(tersection)h(line)386 1505 y(\014elds)37 b(along)e Fo(\015)955 1520 y Fk(\006)1050 1505 y Fr(for)h(a)g(suitable)f(c)m(hoice)i(of)f(mo)s(del)f(Engel)h (structures)j(on)d Fo(R)3308 1520 y Fn(1)3348 1505 y Fr(.)386 1621 y(F)-8 b(or)32 b(this)h(w)m(e)h(stabilize)e(the)h(curv)m (es)j Fo( )t Fr(\()p Fo(\015)1901 1636 y Fk(\006)1959 1621 y Fr(\).)45 b(W)-8 b(e)34 b(ma)m(y)f(assume)h(that)f(all)e(stabi-) 386 1737 y(lizations)g(of)i Fo( )t Fr(\()p Fo(\015)1040 1752 y Fk(\006)1099 1737 y Fr(\))g(are)g(carried)g(out)h(in)e(disjoin)m (t)h(tubular)f(neigh)m(b)s(orho)s(o)s(ds)h(of)386 1853 y(the)i(original)c(curv)m(es.)51 b(Then)36 b(one)e(can)h(extend)h(the)f (isotop)m(y)f(of)g Fo( )39 b Fr(from)33 b Fo(\015)3195 1868 y Fk(\006)3288 1853 y Fr(to)386 1970 y Fo(@)437 1985 y Fk(\000)496 1970 y Fo(R)570 1985 y Fn(1)643 1970 y Fr(and)f(one)h(obtains)f(a)g(stabilized)g(attac)m(hing)f(map.)486 2086 y(Let)37 b Fo( )728 2101 y Fn(+)825 2086 y Fr(and)g Fo( )1082 2101 y Fk(\000)1179 2086 y Fr(b)s(e)h(the)g(restrictions)e (of)h Fo( )42 b Fr(to)37 b Fp(f)p Fo(x)f Fr(=)g(1)p Fp(g)25 b(\002)h Fo(D)2908 2050 y Fn(1)2972 2086 y Fp(\002)g Fo(S)3141 2050 y Fn(1)3218 2086 y Fr(and)386 2202 y Fp(f)p Fo(x)i Fr(=)g Fp(\000)p Fr(1)p Fp(g)22 b(\002)h Fo(D)1005 2166 y Fn(2)1066 2202 y Fp(\002)g Fo(S)1232 2166 y Fn(1)1271 2202 y Fr(.)44 b(By)33 b(fr\()p Fo(\015)1652 2217 y Fk(\006)1710 2202 y Fo(;)17 b(m)p Fr(\))33 b(w)m(e)g(denote)h(a)e(con)m(tact)h (framing)e(of)h Fo(\015)3316 2217 y Fk(\006)386 2318 y Fr(if)f Fo(R)549 2333 y Fn(1)622 2318 y Fr(carries)h(the)h(mo)s(del)e (Engel)h(structure)i Fp(D)2156 2333 y Fl(k)r(;m)2281 2318 y Fr(.)486 2435 y(Since)47 b Fo( )k Fr(is)c(orien)m(tation)f (preserving)i(there)g(exist)f(in)m(tegers)h Fo(n)2920 2450 y Fn(+)2979 2435 y Fo(;)17 b(n)3081 2450 y Fk(\000)3187 2435 y Fr(suc)m(h)386 2551 y(that)46 b Fo( )51 b Fr(maps)46 b(a)h(con)m(tact)g(framing)e(along)g Fo(\015)2148 2566 y Fk(\006)2253 2551 y Fr(to)i(a)f(framing)f(represen)m(ting)386 2667 y Fo(n)444 2682 y Fk(\006)525 2667 y Fp(\001)22 b Fr(fr\()p Fo( )744 2682 y Fk(\006)803 2667 y Fr(\()p Fo(\015)892 2682 y Fk(\006)951 2667 y Fr(\)\))32 b(when)i Fo(R)1388 2682 y Fn(1)1460 2667 y Fr(carries)f(the)g(mo)s(del)e(Engel)h (structure)i Fp(D)2995 2682 y Fl(k)r(;)p Fn(0)3092 2667 y Fr(.)386 2830 y Fy(Prop)s(osition)j(5.2.)43 b Fm(We)36 b(c)-5 b(an)36 b(cho)-5 b(ose)35 b(a)h(mo)-5 b(del)35 b(Engel)h(structur)-5 b(e)38 b(on)d Fo(R)3144 2845 y Fn(1)3220 2830 y Fm(and)386 2947 y(stabilize)f Fo( )820 2962 y Fk(\006)914 2947 y Fm(such)h(that)g(the)g(stabilize)-5 b(d)34 b(maps)556 3110 y Fr(\(i\))40 b Fm(send)i(c)-5 b(ontact)42 b(fr)-5 b(amings)41 b(of)h Fo(\015)1858 3125 y Fk(\006)1959 3110 y Fm(to)g(fr)-5 b(amings)42 b(of)f Fo( )2675 3125 y Fk(\006)2735 3110 y Fr(\()p Fo(\015)2824 3125 y Fk(\006)2882 3110 y Fr(\))i Fm(which)e(ar)-5 b(e)700 3226 y(homotopic)34 b(to)h(a)g(c)-5 b(ontact)35 b(fr)-5 b(aming,)33 b(and)529 3342 y Fr(\(ii\))39 b Fm(the)27 b(r)-5 b(otation)26 b(numb)-5 b(ers)25 b(of)h Fp(D)j Fm(along)c(the)h(stabilize)-5 b(d)25 b(L)-5 b(e)g(gendrian)25 b(curves)700 3458 y(obtaine)-5 b(d)35 b(fr)-5 b(om)34 b Fo( )1380 3473 y Fn(+)1439 3458 y Fr(\()p Fo(\015)1528 3473 y Fn(+)1587 3458 y Fr(\))h Fm(and)f Fo( )1912 3473 y Fk(\000)1972 3458 y Fr(\()p Fo(\015)2061 3473 y Fk(\000)2119 3458 y Fr(\))h Fm(ar)-5 b(e)35 b(b)-5 b(oth)34 b(e)-5 b(qual,)386 3622 y(if)35 b(and)f(only)h(if)386 3832 y Fr(\(26\))293 b Fo(n)911 3847 y Fn(+)992 3832 y Fr(+)22 b(rot\()p Fo( )1316 3847 y Fn(+)1375 3832 y Fr(\()p Fo(\015)1464 3847 y Fn(+)1523 3832 y Fr(\)\))27 b Fp(\021)h Fo(n)1789 3847 y Fk(\000)1871 3832 y Fr(+)22 b(rot\()p Fo( )2195 3847 y Fk(\000)2254 3832 y Fr(\()p Fo(\015)2343 3847 y Fk(\000)2401 3832 y Fr(\)\))100 b(mo)s(d)32 b(2)j Fo(:)386 4063 y Fm(Pr)-5 b(o)g(of.)41 b Fr(W)-8 b(e)39 b(equip)g Fo(R)1201 4078 y Fn(1)1279 4063 y Fr(with)g(the)g(mo)s(del)e(Engel)h (structure)i Fp(D)2757 4078 y Fl(k)r(;)p Fn(0)2854 4063 y Fr(.)62 b(Let)39 b(\()p Fo(S;)17 b(T)d Fr(\))386 4179 y(b)s(e)33 b(a)f(framing)e(of)i Fo(\015)5 b Fr(\).)44 b(Because)34 b Fo( )j Fr(is)32 b(orien)m(tation)e(preserving,)k(the)f (framings)994 4391 y Fo(m)23 b Fp(\001)1151 4310 y Fj(\000)1197 4391 y Fo( )1260 4406 y Fk(\006\003)1355 4391 y Fr(\()p Fo(S;)17 b(T)d Fr(\))1606 4310 y Fj(\001)1802 4391 y Fr(and)151 b Fo( )2173 4406 y Fk(\006\003)2267 4310 y Fj(\000)2313 4391 y Fo(m)22 b Fp(\001)g Fr(\()p Fo(S;)17 b(T)d Fr(\))2721 4310 y Fj(\001)386 4601 y Fr(are)41 b(homotopic.)65 b(If)41 b(w)m(e)h(use)f(the)g(mo)s(del)e(Engel)i (structure)g Fp(D)2786 4616 y Fl(k)r(;m)2952 4601 y Fr(instead)f(of)386 4717 y Fp(D)463 4732 y Fl(k)r(;)p Fn(0)593 4717 y Fr(on)32 b Fo(R)802 4732 y Fn(1)875 4717 y Fr(this)g(implies)1000 4932 y Fo( )1063 4947 y Fk(\006\003)1158 4851 y Fj(\000)1220 4932 y Fr(fr\()p Fo(\015)1377 4947 y Fk(\006)1435 4932 y Fo(;)17 b(m)p Fr(\))1602 4851 y Fj(\001)1675 4932 y Fr(=)28 b(\()p Fo(m)22 b Fr(+)g Fo(n)2080 4947 y Fk(\006)2140 4932 y Fr(\))g Fp(\001)g Fr(fr)o(\()p Fo( )2418 4947 y Fk(\006)2478 4932 y Fr(\()p Fo(\015)2567 4947 y Fk(\006)2625 4932 y Fr(\)\))33 b Fo(:)386 5148 y Fr(In)39 b(\(11\))g(w)m(e)h(ha)m(v) m(e)g(determined)f(the)h(e\013ect)g(of)f(p)s(ositiv)m(e)f(and)h (negativ)m(e)h(stabi-)386 5264 y(lization)28 b(on)j(con)m(tact)h (framings.)41 b(Since)31 b(w)m(e)i(w)m(an)m(t)e(the)h(stabilized)e(em)m (b)s(eddings)403 5364 y Fj(e)386 5390 y Fo( )449 5405 y Fk(\006)551 5390 y Fr(to)43 b(map)f(con)m(tact)h(framings)e(of)h Fo(\015)1843 5405 y Fk(\006)1945 5390 y Fr(to)g(framings)f(of)2621 5364 y Fj(e)2605 5390 y Fo( )2668 5405 y Fk(\006)2727 5390 y Fr(\()p Fo(\015)2816 5405 y Fk(\006)2875 5390 y Fr(\))h(whic)m(h)i(are)386 5506 y(homotopic)34 b(to)i(con)m(tact)g (framings,)f(w)m(e)i(ha)m(v)m(e)g(to)f(apply)g(p)s(ositiv)m(e)f(or)g (negativ)m(e)386 5623 y(stabilization)d(\()p Fo(n)1038 5638 y Fk(\006)1121 5623 y Fr(+)24 b Fo(m)p Fr(\){times.)51 b(Hence)37 b Fo(n)2051 5638 y Fk(\006)2134 5623 y Fr(+)24 b Fo(m)36 b Fr(has)g(to)f(b)s(e)g(non{negativ)m(e.)p eop %%Page: 40 40 40 39 bop 386 259 a Fq(40)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fr(If)32 b Fo(n)541 417 y Fn(+)541 480 y(+)601 459 y Fo(;)17 b(n)703 417 y Fk(\000)703 480 y Fn(+)762 459 y Fo(;)g(n)864 417 y Fn(+)864 480 y Fk(\000)923 459 y Fo(;)g(n)1025 417 y Fk(\000)1025 480 y(\000)1111 459 y Fp(2)28 b Fh(N)1271 474 y Fn(0)1349 459 y Fr(satisfy)1355 622 y Fo(n)1413 637 y Fn(+)1495 622 y Fr(+)22 b Fo(m)28 b Fr(=)f Fo(n)1867 581 y Fn(+)1867 647 y(+)1949 622 y Fr(+)22 b Fo(n)2105 581 y Fk(\000)2105 647 y Fn(+)2192 622 y Fp(\025)28 b Fr(0)1355 780 y Fo(n)1413 795 y Fk(\000)1495 780 y Fr(+)22 b Fo(m)28 b Fr(=)f Fo(n)1867 738 y Fn(+)1867 804 y Fk(\000)1949 780 y Fr(+)22 b Fo(n)2105 738 y Fk(\000)2105 804 y(\000)2192 780 y Fp(\025)28 b Fr(0)k Fo(;)386 704 y Fr(\(27\))386 947 y(then)41 b(it)e(follo)m(ws)g(from)g(\(10\))h(that) g(the)h(rotation)d(n)m(um)m(b)s(ers)k(of)d(the)i(stabilized)386 1063 y(Legendrian)32 b(curv)m(es)j(are)d(giv)m(en)h(b)m(y)652 1250 y(rot)794 1139 y Fj(\020\020)913 1250 y Fr(\()p Fo(\033)1006 1265 y Fn(+)1065 1250 y Fr(\))1103 1209 y Fl(n)1146 1179 y Fg(+)1146 1230 y(+)1201 1250 y Fr(\()p Fo(\033)1294 1265 y Fk(\000)1353 1250 y Fr(\))1391 1209 y Fl(n)1434 1179 y Fe(\000)1434 1230 y Fg(+)1490 1250 y Fo( )1553 1265 y Fn(+)1613 1139 y Fj(\021)1689 1250 y Fr(\()p Fo(\015)1778 1265 y Fn(+)1836 1250 y Fr(\))1874 1139 y Fj(\021)1961 1250 y Fr(=)28 b(rot\()p Fo( )2291 1265 y Fn(+)2350 1250 y Fr(\()p Fo(\015)2439 1265 y Fn(+)2498 1250 y Fr(\)\))22 b(+)g Fo(n)2752 1209 y Fn(+)2752 1275 y(+)2833 1250 y Fp(\000)h Fo(n)2991 1209 y Fk(\000)2991 1275 y Fn(+)651 1467 y Fr(rot)792 1356 y Fj(\020)q(\020)911 1467 y Fr(\()p Fo(\033)1004 1482 y Fn(+)1064 1467 y Fr(\))1102 1426 y Fl(n)1145 1396 y Fg(+)1145 1446 y Fe(\000)1201 1467 y Fr(\()p Fo(\033)1294 1482 y Fk(\000)1353 1467 y Fr(\))1391 1426 y Fl(n)1434 1396 y Fe(\000)1434 1446 y(\000)1490 1467 y Fo( )1553 1482 y Fk(\000)1613 1356 y Fj(\021)1689 1467 y Fr(\()p Fo(\015)1778 1482 y Fk(\000)1836 1467 y Fr(\))1874 1356 y Fj(\021)1961 1467 y Fr(=)28 b(rot\()p Fo( )2291 1482 y Fk(\000)2350 1467 y Fr(\()p Fo(\015)2439 1482 y Fk(\000)2498 1467 y Fr(\)\))22 b(+)g Fo(n)2752 1426 y Fn(+)2752 1491 y Fk(\000)2833 1467 y Fp(\000)h Fo(n)2991 1426 y Fk(\000)2991 1491 y(\000)3082 1467 y Fo(:)386 1654 y Fr(After)f(su\016cien)m(tly)h(man)m(y)g (stabilizations)c(w)m(e)24 b(also)d(w)m(an)m(t)i(the)g(rotation)e(n)m (um)m(b)s(ers)386 1770 y(along)29 b(the)i(image)e(of)h Fo(\015)1250 1785 y Fn(+)1340 1770 y Fr(and)g Fo(\015)1578 1785 y Fk(\000)1668 1770 y Fr(to)g(b)s(e)h(equal.)42 b(This)31 b(can)g(b)s(e)g(ac)m(hiev)m(ed)h(if)e(and)386 1886 y(only)i(if)f(w)m(e)j(can)f(solv)m(e)g(\(27\))f(and)829 2048 y Fo(n)887 2006 y Fn(+)887 2072 y Fk(\000)968 2048 y Fp(\000)23 b Fo(n)1126 2006 y Fk(\000)1126 2072 y(\000)1207 2048 y Fp(\000)g Fo(n)1365 2006 y Fn(+)1365 2072 y(+)1446 2048 y Fr(+)f Fo(n)1602 2006 y Fk(\000)1602 2072 y Fn(+)1689 2048 y Fr(=)28 b(rot\()p Fo( )2019 2063 y Fn(+)2078 2048 y Fr(\()p Fo(\015)2167 2063 y Fn(+)2226 2048 y Fr(\)\))22 b Fp(\000)g Fr(rot\()p Fo( )2649 2063 y Fk(\000)2708 2048 y Fr(\()p Fo(\015)2797 2063 y Fk(\000)2856 2048 y Fr(\)\))-2546 b(\(28\))386 2221 y(with)25 b(nonnegativ)m(e)h(in)m (tegers)g Fo(n)1548 2180 y Fn(+)1548 2243 y(+)1607 2221 y Fo(;)17 b(n)1709 2180 y Fk(\000)1709 2243 y Fn(+)1768 2221 y Fo(;)g(n)1870 2180 y Fn(+)1870 2243 y Fk(\000)1929 2221 y Fo(;)g(n)2031 2180 y Fk(\000)2031 2243 y(\000)2116 2221 y Fr(and)25 b Fo(m)j Fp(2)g Fh(Z)p Fr(.)39 b(Then)26 b(w)m(e)h(can)f(tak)m(e)707 2384 y Fo(k)31 b Fr(=)d(rot)o(\()p Fo( )1118 2399 y Fn(+)1178 2384 y Fr(\()p Fo(\015)1267 2399 y Fn(+)1325 2384 y Fr(\)\))22 b(+)g Fo(n)1579 2343 y Fn(+)1579 2409 y(+)1661 2384 y Fp(\000)g Fo(n)1818 2343 y Fk(\000)1818 2409 y Fn(+)1905 2384 y Fr(=)28 b(rot)o(\()p Fo( )2234 2399 y Fk(\000)2294 2384 y Fr(\()p Fo(\015)2383 2399 y Fk(\000)2441 2384 y Fr(\)\))22 b(+)g Fo(n)2695 2343 y Fn(+)2695 2409 y Fk(\000)2777 2384 y Fp(\000)g Fo(n)2934 2343 y Fk(\000)2934 2409 y(\000)3026 2384 y Fo(:)386 2546 y Fr(F)-8 b(rom)34 b(\(27\))g(and)h(\(28\))g(w)m(e)h (obtain)e(the)i(condition)d(\(26\).)51 b(Therefore)36 b(this)f(con-)386 2662 y(dition)c(is)h(necessary)-8 b(.)486 2778 y(Con)m(v)m(ersely)g(,)32 b(if)k(\(26\))30 b(is)g(satis\014ed,)h (then)g(the)f(under{determined)h(system)g(of)386 2894 y(equations)40 b(\(10\))f(and)g(\(28\))g(admits)f(solutions)h(in)g Fh(Z)p Fr(.)61 b(If)40 b(w)m(e)h(c)m(ho)s(ose)f Fo(m)g Fr(large)386 3011 y(enough)33 b(w)m(e)h(can)e(ac)m(hiev)m(e)i Fo(n)1441 2969 y Fn(+)1441 3032 y(+)1500 3011 y Fo(;)17 b(n)1602 2969 y Fk(\000)1602 3032 y Fn(+)1661 3011 y Fo(;)g(n)1763 2969 y Fn(+)1763 3032 y Fk(\000)1822 3011 y Fo(;)g(n)1924 2969 y Fk(\000)1924 3032 y(\000)2011 3011 y Fp(2)28 b Fh(N)2171 3026 y Fn(0)2216 3011 y Fr(.)1054 b Fi(\003)486 3182 y Fr(Note)34 b(that)g(in)g(this)g(pro)s(of)f(w)m(e)j (used)f(only)f(stabilization)d(to)j(mo)s(dify)f(Legen-)386 3298 y(drian)g(curv)m(es.)48 b(A)m(ttac)m(hing)34 b(a)f(b)m(ypass)j(to) d(a)g(con)m(v)m(ex)j(surface)f(along)d(a)h(segmen)m(t)386 3414 y(of)g(a)g(Legendrian)h(knot)g(con)m(tained)g(in)e(the)i(surface)h (can)f(b)s(e)g(view)m(ed)h(as)e(an)h(in-)386 3531 y(v)m(erse)25 b(pro)s(cedure)e(of)g(stabilization.)37 b(Under)24 b(the)f(assumption)f (that)h(the)g(con)m(tact)386 3647 y(structure)32 b(on)f Fo(@)987 3662 y Fn(+)1046 3647 y Fo(M)42 b Fr(is)30 b(o)m(v)m(ert)m (wisted)i(it)e(is)g(p)s(ossible)h(to)f(pro)m(v)m(e)i(Prop)s(osition)d (5.2)386 3763 y(for)36 b(mo)s(del)f(Engel)i(structures)h(with)f(a)f (\014xed)i(con)m(tact)g(framing,)d(i.e.)56 b(without)386 3879 y(c)m(ho)s(osing)32 b Fo(m)h Fr(big)f(enough)h(as)f(w)m(e)i(did)e (ab)s(o)m(v)m(e.)486 3995 y(Let)g(us)h(explain)f(the)h(meaning)e(of)39 b(\(26\))32 b(in)g(more)g(top)s(ological)c(terms.)44 b(Con-)386 4112 y(sider)36 b(an)h(orien)m(tation)e(preserving)i(attac)m (hing)f(map)g Fo( )i Fr(:)c Fo(@)2605 4127 y Fk(\000)2665 4112 y Fo(R)2739 4127 y Fn(1)2813 4112 y Fp(!)g Fo(@)2998 4127 y Fn(+)3058 4112 y Fo(M)10 b Fr(.)56 b(On)386 4228 y Fo(T)14 b(M)33 b Fr(w)m(e)23 b(ha)m(v)m(e)h(the)f(Engel)f(framing)e (and)j(w)m(e)g(ma)m(y)f(assume)h(that)f(its)g(comp)s(onen)m(ts)386 4344 y(except)41 b(of)f(the)g(comp)s(onen)m(t)g(spanning)f Fp(W)49 b Fr(form)38 b(a)i(framing)e(of)h(the)h(tangen)m(t)386 4460 y(bundle)35 b(of)f Fo(@)872 4475 y Fn(+)932 4460 y Fo(M)10 b Fr(.)51 b(Using)34 b Fo( )39 b Fr(w)m(e)d(pull)d(bac)m(k)j (this)f(trivialization)30 b(to)35 b Fo(@)3010 4475 y Fk(\000)3069 4460 y Fo(R)3143 4475 y Fn(1)3218 4460 y Fr(and)386 4577 y(add)c(a)g(v)m(ector)h(\014eld)f(whic)m(h)h(p)s(oin)m (ts)f(in)m(w)m(ards)h(and)f(is)g(transv)m(erse)i(to)e Fo(@)2996 4592 y Fk(\000)3056 4577 y Fo(R)3130 4592 y Fn(1)3169 4577 y Fr(.)43 b(W)-8 b(e)386 4693 y(obtain)31 b(a)i(framing)d(of)i Fo(T)14 b(R)1387 4708 y Fn(1)1459 4693 y Fr(along)31 b Fo(@)1770 4708 y Fk(\000)1830 4693 y Fo(R)1904 4708 y Fn(1)1943 4693 y Fr(.)486 4809 y(It)22 b(is)g(p)s(ossible)g(to)g(isotop)s(e)g Fo( )k Fr(suc)m(h)e(that)f(w)m (e)g(can)g(extend)h(the)f(Engel)f(structure)386 4925 y(from)h Fo(M)35 b Fr(to)25 b Fo(M)35 b Fr(with)24 b Fo(R)1266 4940 y Fn(1)1330 4925 y Fr(attac)m(hed)h(only)f(if)f(w)m(e)j (can)f(extend)h(the)f(Engel)f(framing)386 5042 y(from)h Fo(M)36 b Fr(to)26 b Fo(M)19 b Fp([)1032 5057 y Fl( )1094 5042 y Fo(R)1168 5057 y Fn(1)1207 5042 y Fr(.)42 b(This)26 b(condition)e(dep)s(ends)k(only)d(on)h(the)g(Engel)g(framing)386 5158 y(and)32 b(the)g(isotop)m(y)f(class)h(of)f Fo( )t Fr(.)43 b(If)32 b Fo( )t Fr(\()p Fo(\015)1805 5173 y Fk(\006)1864 5158 y Fr(\))f(are)h(Legendrian)f(one)h(can)g(sho)m(w)h (that)386 5274 y(suc)m(h)h(an)e(extension)i(exists)f(if)f(and)g(only)g (if)39 b(\(26\))32 b(is)g(satis\014ed.)486 5390 y(On)41 b(the)i(other)e(hand)i(extending)f(a)f(framing)f(from)h Fo(M)52 b Fr(to)42 b Fo(M)d Fp([)2982 5405 y Fl( )3063 5390 y Fo(R)3137 5405 y Fn(1)3219 5390 y Fr(is)i(a)386 5506 y(purely)f(top)s(ological)c(problem.)65 b(Th)m(us)41 b(if)e(w)m(e)i(start)f(with)f(an)h(attac)m(hing)f(map)386 5623 y Fo( )449 5638 y Fk(\006)551 5623 y Fr(and)k(end)g(up)h(with)e(a) g(map)g(violating)g(\(26\))g(then)h(it)f(is)g(not)h(p)s(ossible)f(to)p eop %%Page: 41 41 41 40 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(41)386 459 y Fr(construct)33 b(an)e(Engel)g(structure)i(on)e Fo(M)g Fp([)1961 474 y Fl( )2034 459 y Fo(R)2108 474 y Fn(1)2179 459 y Fr(suc)m(h)i(that)f(the)g(Engel)f(framing)386 575 y(on)h Fo(M)44 b Fr(is)32 b(homotopic)f(to)h(the)h(Engel)f(framing) f(induced)i(b)m(y)g Fp(D)s Fr(.)486 691 y(No)m(w)28 b(if)e Fo( )32 b Fr(:)27 b Fo(@)987 706 y Fk(\000)1047 691 y Fo(R)1121 706 y Fn(1)1188 691 y Fp(\000)-16 b(!)27 b Fo(@)1427 706 y Fn(+)1487 691 y Fo(M)38 b Fr(is)27 b(an)h(em)m(b)s (edding)f(suc)m(h)h(that)g Fo( )t Fr(\()p Fo(\015)2908 706 y Fk(\006)2966 691 y Fr(\))g(are)f(Leg-)386 807 y(endrian)h(curv)m (es)j(and)e Fo(')g Fr(preserv)m(es)i(con)m(tact)f(framings)d(and)i (rotation)e(n)m(um)m(b)s(ers)386 924 y(along)h Fo(\015)694 939 y Fk(\006)783 924 y Fr(then)i(b)m(y)h(Lemma)d(3.3)h(w)m(e)i(can)f (isotop)s(e)f Fo( )34 b Fr(relativ)m(e)29 b(to)g Fo(\015)2890 939 y Fk(\006)2979 924 y Fr(suc)m(h)i(that)386 1040 y(the)36 b(resulting)f(map)f(preserv)m(es)39 b(the)d(con)m(tact)g(structure)h (on)e(a)h(tubular)f(neigh-)386 1156 y(b)s(orho)s(o)s(d)f(of)i Fo(\015)942 1171 y Fk(\006)1000 1156 y Fr(.)53 b(Using)35 b(Remark)g(4.10)g(w)m(e)i(can)f(isotop)s(e)f(the)h(new)g(attac)m(hing) 386 1272 y(map)c(suc)m(h)i(that)e(its)g(image)f(lies)g(in)h(this)g (tubular)g(neigh)m(b)s(orho)s(o)s(d.)386 1410 y Fy(Theorem)42 b(5.3.)i Fm(L)-5 b(et)39 b Fo(M)50 b Fm(b)-5 b(e)39 b(an)f(oriente)-5 b(d)39 b(Engel)f(manifold)g(with)g(tr)-5 b(ansverse)386 1526 y(b)g(oundary)31 b(and)g(assume)g(that)h Fo( )g Fr(:)c Fo(@)1723 1541 y Fk(\000)1782 1526 y Fo(R)1856 1541 y Fn(1)1924 1526 y Fp(\000)-16 b(!)27 b Fo(@)2163 1541 y Fn(+)2223 1526 y Fo(M)42 b Fm(is)31 b(an)h(emb)-5 b(e)g(dding)30 b(and)h(the)386 1642 y(trivialization)i(of)h Fo(T)14 b(M)45 b Fm(induc)-5 b(e)g(d)33 b(by)h(the)g(Engel)g(structur) -5 b(e)35 b(c)-5 b(an)34 b(b)-5 b(e)33 b(extende)-5 b(d)34 b(to)386 1759 y Fo(M)f Fp([)579 1774 y Fl(')652 1759 y Fo(R)726 1774 y Fn(1)765 1759 y Fm(.)486 1875 y(Then)h(ther)-5 b(e)35 b(is)f(a)h(mo)-5 b(del)34 b(Engel)g(structur)-5 b(e)37 b(on)d Fo(R)2346 1890 y Fn(1)2421 1875 y Fm(such)g(that)i Fo( )j Fm(is)34 b(isotopic)386 2001 y(to)f(an)g(emb)-5 b(e)g(dding)1123 1974 y Fj(e)1106 2001 y Fo( )37 b Fm(which)c(pr)-5 b(eserves)32 b(oriente)-5 b(d)33 b(c)-5 b(ontact)33 b(structur)-5 b(es)34 b(and)f(the)386 2117 y(homotopy)h(class)g(of)h(the)g(interse)-5 b(ction)34 b(line)g(\014eld.)386 2295 y Fr(5.3.)48 b Fy(A)m(ttac)m(hing)28 b(maps)g(for)h(round)g(handles)g(of)g(index)g Fr(2)p Fy(.)48 b Fr(All)24 b(our)g(mo)s(del)386 2411 y(Engel)j(structures)i(on)e(round)h(2{handles)f(induce)g(the)h(same)f (singular)f(foliation)386 2527 y(on)37 b Fo(T)597 2491 y Fn(2)583 2552 y(0)673 2527 y Fr(=)e Fo(@)5 b(D)924 2491 y Fn(2)990 2527 y Fp(\002)26 b(f)p Fr(0)p Fp(g)f(\002)h Fo(S)1436 2491 y Fn(1)1511 2527 y Fp(\032)37 b Fo(@)1676 2542 y Fk(\000)1735 2527 y Fo(R)1809 2542 y Fn(2)1849 2527 y Fr(.)58 b(No)m(w)38 b(supp)s(ose)h(that)e Fo(M)48 b Fr(is)38 b(an)f(Engel)386 2644 y(manifold)e(with)j(transv)m(erse)i(b) s(oundary)f(and)f Fo( )j Fr(:)c Fo(@)2349 2659 y Fk(\000)2409 2644 y Fo(R)2483 2659 y Fn(2)2560 2644 y Fp(\000)-16 b(!)36 b Fo(@)2808 2659 y Fn(+)2868 2644 y Fo(M)49 b Fr(is)37 b(an)h(at-)386 2760 y(tac)m(hing)g(map)f(whic)m(h)i(preserv)m (es)h(the)f(orien)m(tations)e(induced)i(b)m(y)g(the)f(con)m(tact)386 2876 y(structures.)486 2992 y(If)c(w)m(e)h(w)m(an)m(t)h(to)e(attac)m(h) g Fo(R)1468 3007 y Fn(2)1542 2992 y Fr(to)h Fo(M)45 b Fr(and)34 b(extend)i(the)f(Engel)f(structure)i(from)386 3109 y Fo(M)54 b Fr(to)43 b Fo(M)e Fp([)30 b Fo(R)969 3124 y Fn(2)1009 3109 y Fr(,)46 b(w)m(e)f(ha)m(v)m(e)g(to)e(ensure)i (that)e(the)h(attac)m(hing)f(map)g(preserv)m(es)386 3225 y(con)m(tact)37 b(structures.)58 b(By)37 b(Theorem)h(3.11)e(and)h (Theorem)g(3.8)f(together)h(with)386 3341 y(Remark)i(4.13)f(it)g (su\016ces)j(to)e(mo)s(dify)e Fo( )43 b Fr(suc)m(h)d(that)f(after)g (the)g(deformation,)386 3457 y(the)30 b(image)e(of)h Fo(T)1009 3421 y Fn(2)995 3482 y(0)1077 3457 y Fr(is)g(con)m(v)m(ex)j (and)e(the)g(attac)m(hing)f(map)f(preserv)m(es)33 b(singular)28 b(fo-)386 3573 y(liation.)38 b(Recall)25 b(that)i(the)g(dividing)d(set) k(of)e Fo(T)2074 3537 y Fn(2)2060 3598 y(0)2139 3573 y Fr(consists)i(of)e(t)m(w)m(o)h(homotopically)386 3690 y(non{trivial)j(circles.)486 3806 y(Let)j Fo(N)40 b Fr(=)30 b Fo(@)936 3821 y Fn(+)995 3806 y Fo(M)45 b Fr(and)34 b Fo(T)1396 3770 y Fn(2)1464 3806 y Fr(=)c Fo( )t Fr(\()p Fo(T)1746 3770 y Fn(2)1732 3831 y(0)1785 3806 y Fr(\).)46 b(W)-8 b(e)34 b(assume)h(that)e(the)h(con)m(tact)g(struc-)386 3922 y(ture)k Fp(C)44 b Fr(on)38 b Fo(N)48 b Fr(is)37 b(o)m(v)m(ert)m(wisted.)61 b(Using)38 b(Lemma)e(3.21)h(and)h(Prop)s (osition)e(3.22)386 4038 y(w)m(e)k(can)g(bring)e Fo(T)1054 4002 y Fn(2)1132 4038 y Fr(in)m(to)h(the)g(desired)h(form.)62 b(It)39 b(is)g(clear)f(ho)m(w)i(to)f(obtain)f(the)386 4155 y(desired)33 b(isotop)m(y)g(of)f Fo( )k Fr(from)c(this.)386 4292 y Fy(Theorem)j(5.4.)40 b Fm(L)-5 b(et)33 b Fo(T)1310 4256 y Fn(2)1383 4292 y Fm(b)-5 b(e)32 b(an)h(emb)-5 b(e)g(dde)g(d)31 b(torus)j(in)f(an)f(overtwiste)-5 b(d)33 b(c)-5 b(ontact)386 4408 y(manifold)35 b Fr(\()p Fo(N)5 b(;)17 b Fp(C)6 b Fr(\))p Fm(.)50 b(Assume)37 b(that)g Fp(C)43 b Fm(is)36 b(orientable)g(and)g(that)h(the)g(Euler)f(class)386 4525 y(of)e(the)g(r)-5 b(estriction)33 b(of)h Fp(C)40 b Fm(to)34 b Fo(T)1523 4488 y Fn(2)1596 4525 y Fm(is)g(zer)-5 b(o.)44 b(Then)33 b(we)h(c)-5 b(an)33 b(isotop)-5 b(e)33 b Fo(T)2916 4488 y Fn(2)2990 4525 y Fm(such)g(that)386 4641 y(after)d(the)f(isotopy)h(the)f(singular)g(foliation)g(on)g(the)h (torus)g(is)f(in)g(standar)-5 b(d)29 b(form.)386 4757 y(Mor)-5 b(e)g(over)35 b(we)f(c)-5 b(an)34 b(pr)-5 b(escrib)g(e)34 b(the)h(slop)-5 b(e)34 b(of)h(the)g(dividing)e(curves.)486 4873 y(After)42 b(the)h(isotopy,)g(the)g(c)-5 b(omplement)40 b(of)i(a)g(tubular)h(neighb)-5 b(orho)g(o)g(d)41 b(of)h Fo(T)3336 4837 y Fn(2)386 4989 y Fm(c)-5 b(ontains)34 b(an)g(overtwiste)-5 b(d)35 b(disc.)386 5158 y(Pr)-5 b(o)g(of.)41 b Fr(It)28 b(su\016ces)i(to)e(\014nd)h(a)e(con)m(v)m(ex)k (torus)d(whic)m(h)h(is)e(isotopic)g(to)g(the)i(original)386 5274 y(one)d(suc)m(h)h(that)f(the)g(dividing)e(set)i(consists)h(of)e(t) m(w)m(o)i(homotopically)22 b(non{trivial)386 5390 y(comp)s(onen)m(ts)32 b(whic)m(h)h(ha)m(v)m(e)g(the)f(desired)h(slop)s(e.)43 b(Using)31 b(the)i(Giroux)d(\015exibilit)m(y)386 5506 y(theorem)36 b(\(Theorem)h(3.13\))f(one)h(can)g(arrange)g(the)g (singular)e(foliation)e(on)k Fo(T)3336 5470 y Fn(2)386 5623 y Fr(suc)m(h)25 b(that)e(it)f(has)i(standard)f(form.)39 b(W)-8 b(e)24 b(will)d(frequen)m(tly)j(use)h(Prop)s(osition)c(3.22.)p eop %%Page: 42 42 42 41 bop 386 259 a Fq(42)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fr(The)38 b(follo)m(wing)d(\014gures)k(represen)m(t)g(the)f (dividing)e(set)i(on)g(a)f(torus)h(b)s(efore)f(the)386 575 y(b)m(ypass)h(attac)m(hmen)m(t.)54 b(The)38 b(thic)m(k)m(ened)f (arc)f(represen)m(ts)j(the)e(attac)m(hing)e(curv)m(e)386 691 y Fo(\015)437 706 y Fn(1)508 691 y Fr(of)e(the)g(b)m(ypass.)486 807 y(1)535 771 y Fl(st)643 807 y Fr(Step:)71 b(Let)47 b Fo(D)1200 822 y Fl(ot)1310 807 y Fr(b)s(e)f(a)g(con)m(v)m(ex)j(o)m(v) m(ert)m(wisted)f(disc.)85 b(W)-8 b(e)47 b(p)s(erturb)f(the)386 924 y(em)m(b)s(edding)e(of)g Fo(T)1085 887 y Fn(2)1168 924 y Fr(suc)m(h)i(that)e(it)f(b)s(ecomes)h(transv)m(erse)j(to)d Fo(D)2817 939 y Fl(ot)2880 924 y Fr(.)79 b(Using)44 b(an)386 1040 y(extension)39 b(of)f(a)g(radial)e(v)m(ector)j(\014eld)f(on)g Fo(D)2046 1055 y Fl(ot)2148 1040 y Fr(w)m(e)i(can)e(isotop)s(e)g Fo(T)2893 1004 y Fn(2)2970 1040 y Fr(suc)m(h)i(that)386 1156 y(after)32 b(the)h(isotop)m(y)g Fo(T)1192 1120 y Fn(2)1253 1156 y Fp(\\)23 b Fo(D)1423 1171 y Fl(ot)1514 1156 y Fr(=)28 b Fp(;)k Fr(and)h(the)g(resulting)e(torus)i(is)f(con)m (v)m(ex.)486 1272 y(Since)48 b Fo(D)837 1287 y Fl(ot)948 1272 y Fr(is)g(con)m(v)m(ex)i(there)e(is)g(a)f(neigh)m(b)s(orho)s(o)s (d)g(whic)m(h)h(is)g(foliated)e(b)m(y)386 1389 y(o)m(v)m(ert)m(wisted)h (discs.)80 b(In)45 b(the)g(follo)m(wing)e(w)m(e)i(will)e(alw)m(a)m(ys)i (ensure)h(that)f(after)386 1505 y(eac)m(h)33 b(mo)s(di\014cation)c(of)j (the)g(em)m(b)s(edding)g(of)f Fo(T)2117 1469 y Fn(2)2156 1505 y Fr(,)i(there)f(is)g(an)g(o)m(v)m(ert)m(wisted)i(disc)386 1621 y(disjoin)m(t)25 b(from)g(the)h(deformed)g(torus:)41 b(If)26 b Fo(D)j Fr(is)c(a)h(b)m(ypass)i(for)d Fo(T)2705 1585 y Fn(2)2745 1621 y Fr(,)i(w)m(e)g(c)m(ho)s(ose)g(the)386 1737 y(neigh)m(b)s(orho)s(o)s(d)34 b(of)g Fo(T)1185 1701 y Fn(2)1248 1737 y Fp([)24 b Fo(D)37 b Fr(in)d(Lemma)g(3.21)g(so)h (small)e(that)h(its)h(complemen)m(t)386 1853 y(still)30 b(con)m(tains)j(o)m(v)m(ert)m(wisted)h(discs.)486 1970 y(2)535 1934 y Fl(nd)650 1970 y Fr(Step:)44 b(In)33 b(this)g(step)g(w)m (e)h(remo)m(v)m(e)f(all)d(homotopically)g(non{trivial)g(com-)386 2086 y(p)s(onen)m(ts)37 b(of)f(the)h(dividing)e(set.)56 b(If)36 b(there)i(are)e(no)g(suc)m(h)i(comp)s(onen)m(ts)f(w)m(e)h(con-) 386 2202 y(tin)m(ue)33 b(with)f(step)h(3.)486 2318 y(If)28 b(the)h(dividing)d(set)j(con)m(tains)f(more)g(than)g(t)m(w)m(o)h (homotopically)c(non{trivial)386 2435 y(comp)s(onen)m(ts,)39 b(then)e(w)m(e)h(reduce)h(the)e(n)m(um)m(b)s(er)g(of)g(its)f(comp)s (onen)m(ts)i(of)e(the)i(di-)386 2551 y(viding)f(set)i(using)f(the)h(b)m (ypass)h(attac)m(hmen)m(ts)f(in)f(the)g(left)g(part)g(of)g(Figure)f(10) 386 2667 y(often)g(enough.)57 b(W)-8 b(e)37 b(end)h(up)f(with)g(a)f (dividing)f(set)j(whic)m(h)g(con)m(tains)f(t)m(w)m(o)g(ho-)684 3509 y @beginspecial 0 @llx 0 @lly 287 @urx 79 @ury 2870 @rwi @setspecial %%BeginDocument: artweg.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: artweg.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Wed Oct 13 10:33:31 2004 %%For: tvogel@math2h (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 287 79 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -17.0 88.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 2457 m -1000 -1000 l 6052 -1000 l 6052 2457 l cp clip 0.06000 0.06000 sc % Polyline 7.500 slw n 300 165 m 2223 165 l 2223 1445 l 300 1445 l cp gs col0 s gr % Polyline n 300 552 m 2225 552 l gs col0 s gr % Polyline n 323 834 m 2225 834 l gs col0 s gr % Polyline n 323 1116 m 2225 1116 l gs col0 s gr % Polyline 15.000 slw n 1239 564 m 1239 1116 l gs col0 s gr /Symbol ff 180.00 scf sf 1485 1123 m gs 1 -1 sc (1) col0 sh gr /Symbol ff 210.00 scf sf 1395 973 m gs 1 -1 sc (g) col0 sh gr % Polyline 7.500 slw n 3120 166 m 5038 166 l 5038 1440 l 3120 1440 l cp gs col0 s gr % Polyline n 3120 833 m 5040 833 l gs col0 s gr % Polyline n 3120 483 m 5017 483 l gs col0 s gr % Polyline 15.000 slw n 4278 484 m 4278 487 l 4278 493 l 4278 504 l 4278 520 l 4278 540 l 4278 566 l 4278 594 l 4279 625 l 4279 656 l 4279 687 l 4279 716 l 4279 742 l 4279 766 l 4279 787 l 4279 805 l 4279 820 l 4279 833 l 4279 856 l 4279 872 l 4279 882 l 4279 889 l 4279 894 l 4279 901 l 4278 910 l 4276 921 l 4274 931 l 4271 941 l 4267 951 l 4263 962 l 4259 974 l 4255 985 l 4251 995 l 4247 1006 l 4242 1015 l 4238 1024 l 4232 1034 l 4226 1043 l 4220 1052 l 4214 1060 l 4207 1068 l 4200 1076 l 4193 1083 l 4184 1090 l 4174 1096 l 4163 1100 l 4151 1105 l 4139 1109 l 4127 1113 l 4114 1116 l 4102 1119 l 4091 1121 l 4080 1122 l 4069 1122 l 4058 1122 l 4047 1121 l 4036 1119 l 4025 1118 l 4014 1116 l 4004 1114 l 3994 1111 l 3984 1107 l 3975 1103 l 3965 1099 l 3956 1094 l 3947 1089 l 3938 1084 l 3929 1079 l 3921 1073 l 3913 1068 l 3905 1061 l 3898 1055 l 3890 1049 l 3883 1042 l 3876 1036 l 3870 1029 l 3864 1022 l 3859 1016 l 3854 1009 l 3850 1002 l 3845 995 l 3841 988 l 3837 981 l 3833 973 l 3830 965 l 3827 957 l 3825 948 l 3822 940 l 3820 931 l 3818 922 l 3817 914 l 3815 905 l 3814 896 l 3813 886 l 3813 873 l 3812 860 l 3812 848 l 3812 839 l 3812 835 l 3812 834 l gs col0 s gr /Symbol ff 210.00 scf sf 4305 1140 m gs 1 -1 sc (g) col0 sh gr /Symbol ff 180.00 scf sf 4395 1245 m gs 1 -1 sc (1) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 1595 3725 a Fs(Figure)h(10.)386 3987 y Fr(motopically)27 b(non{trivial)g(curv)m(es.)45 b(W)-8 b(e)31 b(remo)m(v)m(e)f(these)i (comp)s(onen)m(ts)e(with)g(the)386 4103 y(b)m(ypass)k(attac)m(hmen)m(t) f(in)f(the)h(righ)m(t)f(part)g(of)g(Figure)g(10)486 4219 y(3)535 4183 y Fl(r)r(d)643 4219 y Fr(Step:)48 b(Using)34 b(the)h(b)m(ypass)h(attac)m(hmen)m(t)f(in)f(Figure)f(11,)i(w)m(e)h (obtain)d(t)m(w)m(o)386 4336 y(new)41 b(comp)s(onen)m(ts)f(of)f(the)h (dividing)e(set.)65 b(Their)40 b(slop)s(e)f(dep)s(ends)j(on)d(the)h (at-)386 4452 y(tac)m(hing)25 b(curv)m(e)i(of)e(the)h(b)m(ypass.)43 b(F)-8 b(or)25 b(a)g(giv)m(en)h(iden)m(ti\014cation)e Fo(T)2734 4416 y Fn(2)2801 4452 y Fp(')k Fo(S)2972 4416 y Fn(1)3019 4452 y Fp(\002)8 b Fo(S)3170 4416 y Fn(1)3210 4452 y Fr(,)27 b(w)m(e)386 4568 y(can)38 b(ac)m(hiev)m(e)h(that)f(the)g (new)h(comp)s(onen)m(ts)f(of)f(the)i(dividing)d(set)i(are)g(isotopic) 386 4684 y(to)32 b Fp(f)p Fo(p)p Fp(g)22 b(\002)g Fo(S)841 4648 y Fn(1)913 4684 y Fr(for)32 b Fo(p)c Fp(2)g Fo(S)1299 4648 y Fn(1)1338 4684 y Fr(.)44 b(The)33 b(dashed)h(curv)m(e)g (represen)m(ts)h(this)d(circle.)486 4800 y(4)535 4764 y Fl(th)632 4800 y Fr(Step:)41 b(W)-8 b(e)27 b(are)g(left)g(with)f(a)h (con)m(v)m(ex)i(torus)f(whose)g(dividing)d(set)j(con)m(tains)386 4917 y(exactly)38 b(t)m(w)m(o)h(homotopically)34 b(non{trivial)i (dividing)g(curv)m(es)j Fo(\033)2795 4932 y Fn(1)2835 4917 y Fo(;)17 b(\033)2934 4932 y Fn(2)3012 4917 y Fr(with)37 b(the)386 5033 y(desired)d(slop)s(e.)45 b(If)33 b(this)g(is)f(the)i(en) m(tire)f(dividing)f(set)i(w)m(e)g(are)f(done.)46 b(Otherwise)386 5149 y(w)m(e)34 b(consider)e(the)h(t)m(w)m(o)h(ann)m(uli)d Fo(T)1628 5113 y Fn(2)1689 5149 y Fp(n)22 b Fr(\()p Fo(\033)1854 5164 y Fn(1)1916 5149 y Fp([)h Fo(\033)2060 5164 y Fn(2)2100 5149 y Fr(\).)486 5265 y(W)-8 b(e)30 b(claim)d(that)j(if)e(only)h(one)i (of)e(these)i(ann)m(uli)d(con)m(tains)i(other)g(comp)s(onen)m(ts)386 5382 y(of)j(the)h(dividing)e(set)i(\000,)g(then)g(there)g(is)g(at)f (least)g(one)h(comp)s(onen)m(t)g(of)f(\000)g(whic)m(h)386 5506 y(b)s(ounds)c(a)e(disc)1008 5481 y Fj(e)988 5506 y Fo(D)k Fr(con)m(taining)c(another)h(comp)s(onen)m(t)g(of)f(\000.)42 b(Assume)29 b(that)f(this)386 5623 y(is)40 b(not)g(true.)68 b(Then)41 b Fo(T)1274 5587 y Fn(2)1341 5623 y Fp(n)27 b Fr(\000)41 b(con)m(tains)f Fo(r)k(>)d Fr(0)f(discs,)j(one)e(ann)m (ulus)g(and)f(one)p eop %%Page: 43 43 43 42 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(43)593 1084 y @beginspecial 0 @llx 0 @lly 309 @urx 79 @ury 3090 @rwi @setspecial %%BeginDocument: neuecomp.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: neuecomp.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Thu Feb 19 17:49:59 2004 %%For: tvogel@math81 (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 309 79 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -15.0 94.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 2551 m -1000 -1000 l 6396 -1000 l 6396 2551 l cp clip 0.06000 0.06000 sc % Polyline 7.500 slw n 3480 1228 m 5377 1228 l gs col0 s gr % Polyline n 3480 1061 m 5377 1061 l gs col0 s gr % Polyline n 283 272 m 2188 272 l 2188 1539 l 283 1539 l cp gs col0 s gr % Polyline n 3471 272 m 5384 272 l 5384 1531 l 3471 1531 l cp gs col0 s gr % Polyline gs clippath 2998 879 m 3118 909 l 2998 939 l 3160 939 l 3160 879 l cp clip n 2515 909 m 3145 909 l gs col0 s gr gr % arrowhead n 2998 879 m 3118 909 l 2998 939 l 2998 909 l 2998 879 l cp gs 0.00 setgray ef gr col0 s % Polyline [60] 0 sd n 283 1446 m 2202 1446 l gs col0 s gr [] 0 sd % Polyline n 665 543 m 666 546 l 669 553 l 675 564 l 682 580 l 691 600 l 701 622 l 712 645 l 723 667 l 733 688 l 742 706 l 752 723 l 760 737 l 769 750 l 778 762 l 787 774 l 796 785 l 805 796 l 813 807 l 822 817 l 831 828 l 839 838 l 849 848 l 859 858 l 870 867 l 883 876 l 897 885 l 913 893 l 927 899 l 941 904 l 957 909 l 974 915 l 991 919 l 1010 924 l 1029 929 l 1048 934 l 1068 938 l 1088 943 l 1107 947 l 1127 951 l 1146 955 l 1166 958 l 1184 960 l 1203 963 l 1221 964 l 1239 965 l 1259 965 l 1279 964 l 1300 963 l 1321 961 l 1342 958 l 1363 955 l 1385 952 l 1406 948 l 1427 944 l 1448 940 l 1468 936 l 1488 932 l 1507 927 l 1525 921 l 1542 916 l 1558 909 l 1575 900 l 1590 891 l 1604 881 l 1617 869 l 1630 857 l 1642 845 l 1654 833 l 1665 820 l 1676 807 l 1687 794 l 1697 781 l 1707 768 l 1717 755 l 1726 742 l 1734 728 l 1742 713 l 1749 697 l 1756 678 l 1762 657 l 1769 635 l 1776 612 l 1782 590 l 1787 571 l 1792 555 l 1794 544 l 1796 537 l 1797 534 l gs col0 s gr % Polyline 15.000 slw n 920 893 m 923 891 l 930 887 l 942 880 l 957 871 l 975 860 l 994 850 l 1013 840 l 1030 831 l 1045 823 l 1059 817 l 1072 813 l 1085 809 l 1098 807 l 1111 805 l 1123 803 l 1136 802 l 1149 801 l 1161 800 l 1174 799 l 1187 799 l 1201 798 l 1216 798 l 1231 798 l 1245 799 l 1259 799 l 1274 800 l 1291 802 l 1307 803 l 1324 805 l 1342 806 l 1359 808 l 1376 810 l 1392 812 l 1407 814 l 1422 816 l 1435 819 l 1447 822 l 1462 827 l 1474 832 l 1485 838 l 1493 843 l 1501 849 l 1509 856 l 1517 862 l 1527 869 l 1538 877 l 1550 885 l 1561 893 l 1573 901 l 1586 910 l 1600 920 l 1613 930 l 1627 941 l 1641 952 l 1656 962 l 1669 972 l 1683 981 l 1697 989 l 1710 997 l 1723 1004 l 1737 1011 l 1750 1017 l 1763 1023 l 1777 1028 l 1790 1034 l 1803 1039 l 1816 1044 l 1830 1049 l 1843 1054 l 1856 1058 l 1869 1061 l 1882 1064 l 1895 1065 l 1907 1067 l 1920 1068 l 1932 1068 l 1945 1068 l 1957 1068 l 1969 1068 l 1982 1068 l 1995 1068 l 2008 1068 l 2021 1068 l 2035 1067 l 2049 1067 l 2066 1065 l 2085 1064 l 2105 1062 l 2126 1060 l 2146 1058 l 2163 1056 l 2176 1054 l 2184 1053 l 2187 1053 l 2188 1053 l gs col0 s gr % Polyline n 283 1061 m 285 1061 l 290 1061 l 299 1061 l 313 1061 l 332 1061 l 357 1061 l 386 1061 l 420 1061 l 458 1061 l 498 1061 l 540 1061 l 582 1061 l 625 1061 l 665 1061 l 704 1061 l 741 1061 l 774 1061 l 805 1061 l 832 1061 l 857 1061 l 878 1061 l 897 1061 l 913 1061 l 945 1061 l 967 1061 l 980 1061 l 986 1062 l 989 1062 l 991 1062 l 994 1063 l 1000 1062 l 1010 1062 l 1024 1061 l 1037 1060 l 1052 1059 l 1068 1057 l 1086 1056 l 1105 1054 l 1124 1052 l 1143 1051 l 1161 1049 l 1178 1046 l 1193 1044 l 1206 1041 l 1216 1037 l 1228 1030 l 1236 1020 l 1241 1008 l 1244 994 l 1246 980 l 1248 970 l 1248 966 l 1248 965 l gs col0 s gr % Polyline 7.500 slw n 3854 543 m 3855 546 l 3858 553 l 3863 565 l 3870 581 l 3879 602 l 3889 624 l 3899 647 l 3910 670 l 3920 690 l 3930 709 l 3939 725 l 3949 738 l 3958 750 l 3968 761 l 3978 771 l 3989 780 l 4000 790 l 4010 799 l 4021 808 l 4032 817 l 4043 825 l 4055 833 l 4067 840 l 4080 846 l 4094 850 l 4109 853 l 4125 854 l 4140 853 l 4156 850 l 4174 845 l 4192 838 l 4210 829 l 4229 820 l 4249 809 l 4269 798 l 4289 786 l 4309 776 l 4329 766 l 4349 757 l 4369 751 l 4389 746 l 4408 743 l 4428 742 l 4446 743 l 4463 745 l 4482 749 l 4500 754 l 4519 761 l 4539 769 l 4559 778 l 4578 788 l 4598 798 l 4618 808 l 4637 818 l 4657 827 l 4675 835 l 4693 842 l 4710 847 l 4726 851 l 4741 853 l 4755 854 l 4774 853 l 4791 848 l 4807 842 l 4821 833 l 4834 824 l 4846 813 l 4858 802 l 4870 790 l 4882 779 l 4893 767 l 4904 755 l 4914 742 l 4922 730 l 4929 716 l 4936 700 l 4943 682 l 4950 661 l 4957 638 l 4964 615 l 4970 593 l 4976 572 l 4980 556 l 4983 544 l 4985 537 l 4986 534 l gs col0 s gr /Symbol ff 165.00 scf sf 655 1236 m gs 1 -1 sc (g) col0 sh gr /Symbol ff 165.00 scf sf 725 1341 m gs 1 -1 sc (1) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 1595 1350 a Fs(Figure)38 b(11.)386 1571 y Fr(ann)m(ulus)c(with)g Fo(r)j Fr(holes.)48 b(The)36 b(Euler)e(n)m(um)m(b)s(er)g Fo(\037)p Fr(\()p Fp(C)6 b Fo(;)17 b(T)2434 1535 y Fn(2)2473 1571 y Fr(\))35 b(of)e(the)i (restriction)e(of)386 1687 y Fp(C)39 b Fr(to)32 b Fo(T)667 1651 y Fn(2)739 1687 y Fr(is)386 1852 y(\(29\))494 b Fo(\037)p Fr(\()p Fp(C)6 b Fo(;)17 b(T)1326 1811 y Fn(2)1365 1852 y Fr(\))28 b(=)g Fo(\037)p Fr(\()p Fo(T)1705 1811 y Fn(2)1691 1877 y(+)1750 1852 y Fr(\))22 b Fp(\000)g Fo(\037)p Fr(\()p Fo(T)2079 1811 y Fn(2)2065 1877 y Fk(\000)2125 1852 y Fr(\))27 b(=)h Fp(\006)p Fr(2)p Fo(r)j Fp(6)p Fr(=)c(0)32 b Fo(:)386 2020 y Fr(b)m(y)f(\(12\).)42 b(The)30 b(sign)f(dep)s(ends)h(on)f(the)h(orien)m(tations)e(of)g Fo(T)2536 1984 y Fn(2)2605 2020 y Fr(and)h(of)g(the)g(con)m(tact)386 2136 y(structure.)62 b(But)39 b(\(29\))e(con)m(tradicts)i(our)f (assumption)g(on)g(the)h(Euler)g(class)f(of)386 2252 y Fp(C)6 b Fr(.)68 b(In)41 b(order)g(to)f(reduce)i(the)f(n)m(um)m(b)s (er)g(of)f(connected)j(comp)s(onen)m(ts)e(of)f(\000)h(w)m(e)386 2369 y(p)s(erform)28 b(a)h(b)m(ypass)i(attac)m(hmen)m(t)e(as)h (indicated)e(in)g(the)i(left)e(part)h(of)g(Figure)f(12.)386 2485 y(Notice)i(that)h(this)g(do)s(es)g(not)g(a\013ect)g(the)h (homotopically)27 b(non{trivial)i(dividing)386 2601 y(curv)m(es.)684 3419 y @beginspecial 0 @llx 0 @lly 287 @urx 79 @ury 2870 @rwi @setspecial %%BeginDocument: artotweg.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: artotweg.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Wed Oct 13 10:49:39 2004 %%For: tvogel@math2h (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 287 79 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -3.0 85.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 2407 m -1000 -1000 l 5827 -1000 l 5827 2407 l cp clip 0.06000 0.06000 sc 7.500 slw % Ellipse n 1006 381 592 197 0 360 DrawEllipse gs col0 s gr % Ellipse n 1331 392 90 90 0 360 DrawEllipse gs col0 s gr % Ellipse n 1006 369 110 110 0 360 DrawEllipse gs col0 s gr % Ellipse n 703 381 90 90 0 360 DrawEllipse gs col0 s gr % Polyline n 75 113 m 1980 113 l 1980 1380 l 75 1380 l cp gs col0 s gr % Polyline n 75 1204 m 1980 1204 l gs col0 s gr % Polyline 15.000 slw n 1010 480 m 1010 960 l gs col0 s gr % Polyline 7.500 slw n 75 960 m 1980 960 l gs col0 s gr % Ellipse n 3491 431 116 115 0 360 DrawEllipse gs col0 s gr % Ellipse n 3491 953 116 115 0 360 DrawEllipse gs col0 s gr % Polyline n 2910 129 m 4813 129 l 4813 1395 l 2910 1395 l cp gs col0 s gr % Polyline n 2910 1186 m 4815 1186 l gs col0 s gr % Polyline n 2910 697 m 4814 697 l gs col0 s gr % Polyline 15.000 slw n 3491 547 m 3491 837 l gs col0 s gr /Symbol ff 210.00 scf sf 3570 825 m gs 1 -1 sc (g) col0 sh gr /Symbol ff 180.00 scf sf 3660 930 m gs 1 -1 sc (1) col0 sh gr /Symbol ff 210.00 scf sf 1050 735 m gs 1 -1 sc (g) col0 sh gr /Symbol ff 180.00 scf sf 1125 855 m gs 1 -1 sc (1) col0 sh gr /Symbol ff 210.00 scf sf 1050 735 m gs 1 -1 sc (g) col0 sh gr /Symbol ff 180.00 scf sf 1125 855 m gs 1 -1 sc (1) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 1595 3635 a Fs(Figure)38 b(12.)486 3911 y Fr(If)32 b(b)s(oth)h(ann)m(uli)e Fo(T)1180 3875 y Fn(2)1241 3911 y Fp(n)22 b Fr(\()p Fo(\033)1406 3926 y Fn(1)1468 3911 y Fp([)h Fo(\033)1612 3926 y Fn(2)1652 3911 y Fr(\))32 b(con)m(tain)g(connected)j(comp)s(onen)m(ts)e(of)f(\000)h(w)m(e)386 4027 y(reduce)h(the)g(n)m(um)m(b)s(er)g(of)e(comp)s(onen)m(ts)i(using)f (the)g(b)m(ypass)i(attac)m(hmen)m(t)f(in)e(the)386 4143 y(righ)m(t)39 b(part)h(of)g(Figure)f(12.)66 b(If)41 b(do)f(this)g (often)g(enough)h(w)m(e)g(end)g(up)f(with)g(the)386 4260 y(desired)33 b(con\014guration)f(of)g(dividing)f(curv)m(es)j(on)f Fo(T)2301 4223 y Fn(2)2340 4260 y Fr(.)930 b Fi(\003)486 4436 y Fr(A)42 b(b)m(ypass)h(attac)m(hmen)m(t)g(also)e(a\013ects)h (framings.)70 b(F)-8 b(or)41 b(our)h(purp)s(ose,)j(it)c(is)386 4552 y(enough)j(to)g(sho)m(w)i(this)d(for)h(a)g(particular)e(b)m(ypass) k(attac)m(hmen)m(t.)79 b(K.)44 b(Honda)386 4668 y(describ)s(ed)38 b(this)e(e\013ect)i(in)e(more)g(detail,)g(cf.)57 b(Prop)s(osition)35 b(4.7)h(in)h([Ho])f(where)386 4785 y(one)d(can)g(\014nd)g(a)f(pro)s(of) g(of)g(the)h(follo)m(wing)d(lemma.)386 4925 y Fy(Lemma)36 b(5.5.)42 b Fm(L)-5 b(et)34 b Fo(T)1236 4889 y Fn(2)1303 4925 y Fr(=)28 b Fo(S)1473 4889 y Fn(1)1533 4925 y Fp(\002)21 b Fo(S)1697 4889 y Fn(1)1771 4925 y Fm(b)-5 b(e)33 b(a)i(torus)f(in)g (standar)-5 b(d)34 b(form)g(c)-5 b(ontaine)g(d)386 5042 y(in)37 b(a)g(c)-5 b(ontact)37 b(manifold)f(such)h(that)h(the)f(L)-5 b(e)g(gendrian)36 b(divides)g(ar)-5 b(e)37 b(isotopic)g(to)386 5158 y Fp(f)p Fr(1)p Fp(g)16 b(\002)i Fo(S)712 5122 y Fn(1)751 5158 y Fm(.)44 b(L)-5 b(et)33 b Fo(X)40 b Fm(b)-5 b(e)32 b(a)h(nowher)-5 b(e)31 b(vanishing)g(L)-5 b(e)g(gendrian)31 b(ve)-5 b(ctor)33 b(\014eld)f(on)g(the)386 5274 y(c)-5 b(ontact)35 b(manifold.)486 5390 y(Assume)f(that)h(the)g(r)-5 b(otation)35 b(numb)-5 b(er)34 b(of)g Fo(X)43 b Fm(along)34 b(the)g(L)-5 b(e)g(gendrian)34 b(divides)386 5506 y(is)40 b(zer)-5 b(o)39 b(and)h(that)g(it)g(is)g(even)f(along)g Fo(S)1876 5470 y Fn(1)1941 5506 y Fp(\002)27 b(f)p Fr(0)p Fp(g)p Fm(.)59 b(We)40 b(attach)g(a)g(byp)-5 b(ass)39 b(as)h(in)386 5623 y(Figur)-5 b(e)32 b(13)g(to)h Fo(T)1008 5587 y Fn(2)1080 5623 y Fm(and)f(bring)g(the)h(char)-5 b(acteristic)32 b(foliation)g(in)g(standar)-5 b(d)32 b(form)p eop %%Page: 44 44 44 43 bop 386 259 a Fq(44)1096 b(THOMAS)25 b(V)n(OGEL)584 1063 y @beginspecial 0 @llx 0 @lly 311 @urx 80 @ury 3110 @rwi @setspecial %%BeginDocument: dehn.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: dehn.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Mon Sep 27 16:57:55 2004 %%For: tvogel@math81 (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 311 80 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -6.0 86.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 2425 m -1000 -1000 l 6269 -1000 l 6269 2425 l cp clip 0.06000 0.06000 sc % Polyline 7.500 slw n 120 125 m 2038 125 l 2038 1401 l 120 1401 l cp gs col0 s gr % Polyline n 3331 125 m 5257 125 l 5257 1393 l 3331 1393 l cp gs col0 s gr % Polyline gs clippath 2854 738 m 2974 768 l 2854 798 l 3016 798 l 3016 738 l cp clip n 2369 768 m 3001 768 l gs col0 s gr gr % arrowhead n 2854 738 m 2974 768 l 2854 798 l 2854 768 l 2854 738 l cp gs 0.00 setgray ef gr col0 s % Polyline n 120 512 m 2040 512 l gs col0 s gr % Polyline 15.000 slw n 1232 126 m 1380 513 l gs col0 s gr % Polyline 7.500 slw n 144 1074 m 2041 1074 l gs col0 s gr % Polyline 15.000 slw n 935 512 m 1232 1403 l gs col0 s gr % Polyline 7.500 slw n 4633 126 m 4633 127 l 4634 130 l 4636 139 l 4639 152 l 4643 171 l 4648 192 l 4654 215 l 4659 236 l 4664 256 l 4669 274 l 4674 289 l 4679 303 l 4684 316 l 4690 330 l 4696 343 l 4702 356 l 4708 369 l 4714 382 l 4720 395 l 4727 408 l 4734 420 l 4742 431 l 4750 442 l 4759 452 l 4769 461 l 4780 469 l 4791 476 l 4803 483 l 4815 489 l 4827 496 l 4840 502 l 4853 507 l 4866 513 l 4880 518 l 4894 522 l 4909 526 l 4923 529 l 4937 531 l 4953 532 l 4969 533 l 4986 533 l 5004 533 l 5022 533 l 5040 533 l 5057 533 l 5074 533 l 5090 533 l 5105 533 l 5119 534 l 5132 534 l 5148 534 l 5164 534 l 5179 535 l 5195 535 l 5211 534 l 5226 534 l 5238 534 l 5245 534 l 5248 534 l 5249 534 l gs col0 s gr % Polyline n 3334 522 m 3336 522 l 3341 522 l 3350 522 l 3363 522 l 3382 522 l 3405 522 l 3433 522 l 3465 523 l 3499 523 l 3535 523 l 3572 523 l 3609 523 l 3645 524 l 3679 524 l 3711 524 l 3741 524 l 3769 525 l 3795 525 l 3819 525 l 3841 526 l 3862 526 l 3887 526 l 3911 527 l 3934 528 l 3957 528 l 3980 529 l 4003 529 l 4025 529 l 4047 530 l 4069 530 l 4090 531 l 4111 531 l 4132 532 l 4152 533 l 4172 534 l 4191 535 l 4209 536 l 4225 537 l 4241 539 l 4262 542 l 4280 545 l 4297 548 l 4311 551 l 4324 553 l 4337 556 l 4348 560 l 4360 563 l 4371 568 l 4382 573 l 4392 580 l 4402 588 l 4410 598 l 4417 608 l 4423 619 l 4428 631 l 4431 643 l 4434 655 l 4437 668 l 4441 681 l 4445 696 l 4449 712 l 4455 730 l 4461 751 l 4466 766 l 4470 783 l 4476 800 l 4481 819 l 4487 839 l 4493 860 l 4499 882 l 4505 904 l 4511 927 l 4518 949 l 4524 972 l 4530 994 l 4536 1015 l 4542 1036 l 4547 1057 l 4553 1076 l 4557 1095 l 4562 1113 l 4567 1133 l 4572 1153 l 4576 1174 l 4581 1195 l 4586 1219 l 4592 1243 l 4597 1268 l 4602 1294 l 4607 1318 l 4612 1340 l 4615 1359 l 4618 1374 l 4620 1384 l 4621 1389 l 4622 1392 l gs col0 s gr % Polyline n 3334 1044 m 3337 1044 l 3343 1044 l 3353 1044 l 3369 1045 l 3389 1045 l 3414 1046 l 3442 1046 l 3473 1047 l 3504 1048 l 3536 1048 l 3566 1049 l 3595 1049 l 3621 1050 l 3646 1050 l 3669 1051 l 3691 1051 l 3712 1051 l 3733 1051 l 3754 1051 l 3775 1051 l 3796 1051 l 3818 1051 l 3840 1051 l 3861 1051 l 3883 1051 l 3905 1051 l 3927 1050 l 3949 1050 l 3970 1050 l 3991 1050 l 4011 1050 l 4031 1050 l 4050 1050 l 4068 1051 l 4086 1051 l 4108 1052 l 4128 1052 l 4148 1053 l 4167 1053 l 4186 1054 l 4205 1054 l 4223 1054 l 4241 1055 l 4258 1056 l 4275 1057 l 4291 1058 l 4306 1060 l 4319 1062 l 4331 1065 l 4348 1071 l 4360 1079 l 4368 1088 l 4374 1097 l 4379 1107 l 4384 1117 l 4390 1128 l 4397 1140 l 4402 1151 l 4406 1163 l 4410 1175 l 4414 1189 l 4417 1202 l 4421 1216 l 4424 1230 l 4427 1244 l 4431 1256 l 4434 1269 l 4437 1281 l 4440 1295 l 4444 1310 l 4448 1327 l 4452 1344 l 4455 1361 l 4458 1374 l 4460 1383 l 4461 1386 l 4461 1387 l gs col0 s gr % Polyline n 4461 132 m 4462 134 l 4463 140 l 4465 149 l 4468 163 l 4472 181 l 4477 203 l 4483 228 l 4489 254 l 4495 282 l 4501 309 l 4507 335 l 4513 359 l 4518 381 l 4523 402 l 4527 420 l 4531 437 l 4535 452 l 4540 471 l 4545 489 l 4549 506 l 4554 522 l 4559 537 l 4563 552 l 4568 567 l 4572 581 l 4577 596 l 4581 611 l 4586 626 l 4590 641 l 4595 657 l 4600 673 l 4605 689 l 4610 706 l 4615 723 l 4620 740 l 4625 758 l 4630 776 l 4636 794 l 4641 812 l 4646 829 l 4651 846 l 4656 862 l 4661 877 l 4666 891 l 4670 904 l 4676 920 l 4681 933 l 4686 945 l 4690 956 l 4694 965 l 4698 975 l 4703 984 l 4708 993 l 4715 1002 l 4723 1010 l 4732 1017 l 4743 1024 l 4754 1030 l 4765 1035 l 4777 1040 l 4789 1045 l 4802 1050 l 4815 1054 l 4830 1057 l 4846 1060 l 4859 1061 l 4872 1062 l 4887 1062 l 4902 1062 l 4918 1061 l 4934 1060 l 4951 1059 l 4968 1058 l 4984 1057 l 5001 1056 l 5017 1055 l 5032 1055 l 5047 1054 l 5062 1054 l 5079 1054 l 5096 1054 l 5115 1054 l 5136 1054 l 5159 1054 l 5182 1054 l 5204 1054 l 5222 1054 l 5236 1054 l 5245 1054 l 5248 1054 l 5249 1054 l gs col0 s gr /Symbol ff 210.00 scf sf 1095 780 m gs 1 -1 sc (g) col0 sh gr /Symbol ff 180.00 scf sf 1200 900 m gs 1 -1 sc (1) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 1595 1299 a Fs(Figure)38 b(13.)386 1552 y Fm(such)31 b(that)g(the)h(L)-5 b(e)g(gendrian)29 b(ruling)i(is)g (stil)5 b(l)31 b(tangent)g(to)h(the)f(foliation)f(fr)-5 b(om)31 b(the)386 1669 y(\014rst)k(factor)g(in)f Fo(S)1057 1632 y Fn(1)1119 1669 y Fp(\002)22 b Fo(S)1284 1632 y Fn(1)1323 1669 y Fm(.)486 1785 y(Then)41 b(the)h(r)-5 b(otation)43 b(numb)-5 b(er)42 b(along)f(the)h(L)-5 b(e)g(gendrian)42 b(divides)f(in)h(the)g(iso-)386 1901 y(top)-5 b(e)g(d)37 b(torus)g(is)g(o)-5 b(dd)37 b(\(and)f(ther)-5 b(efor)g(e)37 b(non{zer)-5 b(o\))36 b(while)g(the)h(r)-5 b(otation)37 b(numb)-5 b(er)386 2017 y(along)34 b(the)h(L)-5 b(e)g(gendrian)34 b(curve)g Fo(S)1636 1981 y Fn(1)1698 2017 y Fp(\002)22 b(f)p Fr(0)p Fp(g)35 b Fm(r)-5 b(emains)33 b(even.)386 2168 y Fy(Prop)s(osition)g(5.6.)40 b Fm(Assume)32 b(that)h Fo(M)44 b Fm(c)-5 b(arries)32 b(an)g(oriente)-5 b(d)32 b(Engel)h(structur)-5 b(e)386 2284 y(with)43 b(oriente)-5 b(d)42 b(char)-5 b(acteristic)43 b(foliation)f(such)h(that)g(the)g(b)-5 b(oundary)43 b(of)g Fo(M)54 b Fm(is)386 2400 y(tr)-5 b(ansverse.)75 b(Mor)-5 b(e)g(over,)47 b(assume)e(that)g(the)g(c)-5 b(ontact)45 b(structur)-5 b(e)47 b(on)e Fo(@)3096 2415 y Fn(+)3155 2400 y Fo(M)56 b Fm(is)386 2517 y(overtwiste)-5 b(d.)44 b(L)-5 b(et)35 b Fo( )d Fr(:)c Fo(@)1299 2532 y Fk(\000)1358 2517 y Fo(R)1432 2532 y Fn(2)1500 2517 y Fp(\000)-16 b(!)27 b Fo(@)1739 2532 y Fn(+)1798 2517 y Fo(M)46 b Fm(b)-5 b(e)35 b(an)f(attaching)g(map.)486 2633 y(We)d(c)-5 b(an)31 b(extend)h(the)f(Engel)g(structur)-5 b(e)33 b(fr)-5 b(om)31 b Fo(M)43 b Fm(to)31 b Fo(M)26 b Fp([)15 b Fo(R)2724 2648 y Fn(2)2796 2633 y Fm(using)31 b(a)h(mo)-5 b(del)386 2749 y(Engel)44 b(structur)-5 b(e)47 b(if)e(and)f(only)h(if)g(the)g(Engel)f(fr)-5 b(aming)45 b(on)f Fo(M)56 b Fm(extends)45 b(to)g(a)386 2865 y(trivialization)c(of) h Fo(M)c Fp([)28 b Fo(R)1389 2880 y Fn(2)1471 2865 y Fm(over)42 b(the)g Fr(2)p Fm({c)-5 b(el)5 b(l)41 b Fo(e)2188 2880 y Fn(2)2269 2865 y Fr(=)g Fo(D)2470 2829 y Fn(2)2537 2865 y Fp(\002)28 b(f)p Fr(0)p Fp(g)f(\002)h(f)p Fr(1)p Fp(g)40 b(\032)i Fo(R)3305 2880 y Fn(2)3345 2865 y Fm(.)386 2981 y(The)g(Engel)h(structur)-5 b(e)44 b(on)e Fo(M)d Fp([)28 b Fo(R)1739 2996 y Fn(2)1822 2981 y Fm(c)-5 b(an)42 b(b)-5 b(e)43 b(chosen)f(such)h(that)g(the)g(c)-5 b(ontact)386 3098 y(structur)g(e)36 b(on)e Fo(@)987 3113 y Fn(+)1047 3098 y Fr(\()p Fo(M)f Fp([)22 b Fo(R)1374 3113 y Fn(2)1414 3098 y Fr(\))35 b Fm(again)f(overtwiste)-5 b(d.)386 3298 y(Pr)g(o)g(of.)41 b Fr(The)34 b(2{cell)e Fo(e)1202 3313 y Fn(2)1275 3298 y Fr(is)h(of)g(course)h(attac)m(hed)g(using)f(the)h (restriction)e(of)h Fo( )k Fr(to)386 3414 y Fo(@)5 b(e)487 3429 y Fn(2)527 3414 y Fr(.)65 b(Clearly)-8 b(,)40 b(if)f(the)h(Engel)f (trivialization)c(can)40 b(not)f(b)s(e)h(extended)h(o)m(v)m(er)g Fo(e)3308 3429 y Fn(2)3348 3414 y Fr(,)386 3531 y(then)33 b(it)f(is)g(imp)s(ossible)e(to)i(extend)i(the)f(Engel)g(structure)g(to) g Fo(M)g Fp([)2886 3546 y Fl( )2932 3555 y Fg(2)2993 3531 y Fo(R)3067 3546 y Fn(2)3107 3531 y Fr(.)486 3647 y(The)28 b(con)m(tact)g(structure)h Fp(C)34 b Fr(on)27 b Fo(@)1700 3662 y Fn(+)1760 3647 y Fo(M)38 b Fr(is)27 b(orien)m(ted)h(and)f(the)h(in)m(tersection)g(line)386 3763 y(\014eld)i(yields)h(a)f(no)m(where)i(v)-5 b(anishing)30 b(Legendrian)g(v)m(ector)i(\014eld.)42 b(Therefore)32 b(the)386 3879 y(Euler)f(c)m(haracteristic)f(of)g Fp(C)37 b Fr(view)m(ed)32 b(as)f(a)f(bundle)h(is)f(zero.)44 b(W)-8 b(e)31 b(equip)g Fo(R)3115 3894 y Fn(2)3185 3879 y Fr(with)386 3995 y(one)37 b(of)f(the)i(mo)s(del)d(Engel)i(structures)h(from)e (Section)h(4.5.)56 b(By)37 b(Theorem)g(5.4)386 4112 y(and)g(Remark)f (4.13)g(w)m(e)i(can)f(assume)h(that)e Fo( )41 b Fr(preserv)m(es)e(con)m (tact)f(structures.)386 4228 y(In)33 b(particular,)e Fo( )t Fr(\()p Fo(@)5 b(e)1191 4243 y Fn(2)1231 4228 y Fr(\))33 b(is)f(Legendrian.)486 4344 y(W)-8 b(e)47 b(\014rst)h(sho)m(w)h(that)e(under)h(the)g(assumption)f(of)g(the)h (prop)s(osition,)h(the)386 4460 y(rotation)31 b(n)m(um)m(b)s(er)i (along)e Fo( )t Fr(\()p Fo(@)5 b(e)1581 4475 y Fn(2)1622 4460 y Fr(\))32 b(is)h(ev)m(en.)45 b(F)-8 b(or)32 b(this)g(w)m(e)i (homotop)e(the)h(Engel)386 4577 y(trivialization)j(on)j Fo(M)51 b Fr(suc)m(h)42 b(that)d(the)i(only)e(comp)s(onen)m(t)h Fo(W)54 b Fr(of)39 b(the)i(framing)386 4693 y(whic)m(h)31 b(is)g(not)f(tangen)m(t)i(to)e Fo(@)1453 4708 y Fn(+)1513 4693 y Fo(M)41 b Fr(spans)32 b(the)g(c)m(haracteristic)e(foliation.)40 b(In)31 b(this)386 4809 y(w)m(a)m(y)43 b(w)m(e)g(obtain)e(a)h(framing)e (of)i Fo(@)1692 4824 y Fn(+)1751 4809 y Fo(M)10 b Fr(.)73 b(Using)42 b(the)h(attac)m(hing)e(map)g(to)h(pull)386 4925 y(bac)m(k)e(this)e(framing,)g(w)m(e)i(obtain)d(a)h(framing)f(on)h Fo(@)2316 4940 y Fk(\000)2376 4925 y Fo(R)2450 4940 y Fn(2)2490 4925 y Fr(.)61 b(If)39 b(w)m(e)h(add)e(a)h(v)m(ector)386 5042 y(\014eld)27 b(along)f Fo(@)898 5057 y Fk(\000)957 5042 y Fo(R)1031 5057 y Fn(2)1098 5042 y Fr(whic)m(h)i(spans)g(and)g (orien)m(ts)f(the)h(c)m(haracteristic)f(foliation)d(w)m(e)386 5158 y(obtain)31 b(a)i(framing)d(of)i Fo(T)14 b(R)1387 5173 y Fn(2)1459 5158 y Fr(along)31 b Fo(@)1770 5173 y Fk(\000)1830 5158 y Fo(R)1904 5173 y Fn(2)1943 5158 y Fr(.)486 5274 y(Since)c Fo( )32 b Fr(preserv)m(es)f(con)m(tact)d (structures,)j(the)d(pull)e(bac)m(k)j(of)e(the)i(comp)s(onen)m(t)386 5390 y(of)37 b(the)i(Engel)f(framing)d(whic)m(h)k(is)f(orthogonal)e(to) h(the)i(con)m(tact)f(structure)i(on)386 5506 y Fo(@)437 5521 y Fn(+)496 5506 y Fo(M)47 b Fr(is)36 b(transv)m(erse)i(to)e(the)g (con)m(tact)h(structure)g(on)f Fo(@)2457 5521 y Fk(\000)2516 5506 y Fo(R)2590 5521 y Fn(2)2630 5506 y Fr(.)54 b(Without)35 b(loss)h(of)386 5623 y(generalit)m(y)-8 b(,)28 b(w)m(e)i(assume)f(that) g(these)g(comp)s(onen)m(ts)g(of)f(the)h(Engel)g(framings)e(are)p eop %%Page: 45 45 45 44 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(45)386 459 y Fr(preserv)m(ed)41 b(b)m(y)e(the)f(attac)m(hing)g(map.) 59 b(Then)40 b(the)e(pullbac)m(k)g(framing)e(and)i(the)386 575 y(Engel)28 b(framing)d(on)j Fo(R)1214 590 y Fn(2)1282 575 y Fr(ha)m(v)m(e)h(t)m(w)m(o)g(comp)s(onen)m(ts)f(in)f(common.)41 b(When)29 b(w)m(e)g(w)m(an)m(t)386 691 y(to)38 b(compare)h(the)g(pull)e (bac)m(k)j(framing)c(with)i(the)h(Engel)g(trivialization)34 b(along)386 807 y Fo(e)431 822 y Fn(2)503 807 y Fr(it)e(is)g(therefore) h(enough)g(to)f(consider)h(the)g(rotation)e(n)m(um)m(b)s(ers)i(along)f Fo(@)5 b(e)3240 822 y Fn(2)3280 807 y Fr(.)486 924 y(By)34 b(\(iii\))d(of)j(Prop)s(osition)e(4.12,)i(the)h(rotation)d(n)m(um)m(b)s (er)j(along)e Fo(@)5 b(e)2970 939 y Fn(2)3044 924 y Fr(induced)386 1040 y(b)m(y)33 b(the)f(mo)s(del)e(Engel)i(structure)h(on)f Fo(R)1875 1055 y Fn(2)1947 1040 y Fr(is)f(ev)m(en.)45 b(If)32 b(the)g(rotation)f(n)m(um)m(b)s(er)h(of)386 1156 y(the)d(pull)f(bac)m(k)i(framing)d(along)h Fo(@)5 b(e)1681 1171 y Fn(2)1750 1156 y Fr(is)29 b(o)s(dd,)h(then)f(the)h(pull)e(bac)m (k)i(framing)d(and)386 1272 y(the)g(Engel)g(framing)d(are)j(not)g (homotopic)e(along)g Fo(@)5 b(e)2320 1287 y Fn(2)2387 1272 y Fr(and)27 b(the)g(pull)f(bac)m(k)h(fram-)386 1389 y(ing)i(can)i(not)f(b)s(e)h(extended)h(o)m(v)m(er)f(the)g(disc)g Fo(D)2087 1352 y Fn(2)2143 1389 y Fp(\002)18 b(f)p Fr(0)p Fp(g)27 b(\032)i Fo(h)2576 1404 y Fn(2)2615 1389 y Fr(.)43 b(This)30 b(con)m(tradicts)386 1505 y(the)j(assumption.)43 b(Hence)34 b(the)f(rotation)e(n)m(um)m(b)s(er)i(along)e Fo( )t Fr(\()p Fo(@)5 b(e)2761 1520 y Fn(2)2801 1505 y Fr(\))33 b(is)f(ev)m(en.)486 1621 y(If)g(the)h(rotation)d(n)m(um)m(b) s(er)j(along)e(the)i(Legendrian)f(divides)g(of)g Fo( )t Fr(\()p Fo(T)3027 1585 y Fn(2)3013 1646 y(0)3066 1621 y Fr(\))g(is)g(not)386 1737 y(zero,)40 b(then)e(b)m(y)h(Prop)s(osition) d(4.12)i(w)m(e)h(can)f(c)m(ho)s(ose)h(a)f(suitable)f(mo)s(del)f(Engel) 386 1853 y(structure)26 b(suc)m(h)g(that)e(the)h(isotop)s(ed)f(attac)m (hing)f(map)h(preserv)m(es)j(the)e(homotop)m(y)386 1970 y(class)33 b(of)f(the)h(in)m(tersection)f(line)f(\014elds.)486 2086 y(If)26 b(the)i(rotation)d(n)m(um)m(b)s(er)i(along)f(the)h (Legendrian)g(divides)f(of)h Fo( )t Fr(\()p Fo(T)2978 2050 y Fn(2)2964 2111 y(0)3017 2086 y Fr(\))g(is)f(zero,)386 2202 y(then)37 b(w)m(e)g(attac)m(h)g(a)f(b)m(ypass)i(as)f(in)e(Figure)h (13)g(to)g(isotop)s(e)f Fo( )41 b Fr(once)c(again.)53 b(By)386 2318 y(Lemma)37 b(5.5,)i(the)g(rotation)e(n)m(um)m(b)s(er)i (along)d(the)j(Legendrian)f(rulings)f(of)h(the)386 2435 y(image)f(of)h Fo(T)862 2398 y Fn(0)848 2459 y(0)940 2435 y Fr(remains)g(ev)m(en)j(and)e(the)g(rotation)f(n)m(um)m(b)s(er)h (along)e(the)j(Legen-)386 2551 y(drian)32 b(divides)g(is)g(non{zero.) 486 2667 y(The)k(new)g(attac)m(hing)f(map)f(preserv)m(es)k(the)e (singular)e(foliation)d(on)36 b Fo(T)3103 2631 y Fn(2)3089 2692 y(0)3177 2667 y Fr(after)386 2783 y(one)f(pushes)i(forw)m(ard)e (the)g(mo)s(del)f(Engel)h(structures)h(from)e(Prop)s(osition)f(4.12)386 2900 y(with)f(the)h(di\013eomorphism)d(\002)1538 2915 y Fn(1)1610 2900 y Fr(from)h(\(22\).)486 3016 y(When)41 b(w)m(e)h(restrict)e(this)g(di\013eomorphism)f(to)h Fo(@)5 b(D)2434 2980 y Fn(2)2501 3016 y Fp(\002)29 b(f)p Fr(0)p Fp(g)e(\002)h Fo(S)2954 2980 y Fn(1)3033 3016 y Fr(then)42 b(w)m(e)386 3132 y(obtain)26 b(a)h(righ)m(t)f(handed)i(Dehn)f(t)m (wist.)41 b(No)m(w)28 b(w)m(e)g(can)f(c)m(ho)s(ose)h(the)g(mo)s(del)d (Engel)386 3248 y(structure)36 b(suc)m(h)g(that)e(the)h(new)g(attac)m (hing)f(map)f(preserv)m(es)38 b(orien)m(ted)c(con)m(tact)386 3364 y(structures)j(and)e(the)h(homotop)m(y)f(class)g(of)g(the)h(in)m (tersection)f(line)f(\014eld.)52 b(After)386 3481 y(a)29 b(v)m(ertical)f(mo)s(di\014cation)f(of)h(the)i(b)s(oundary)f(of)g Fo(M)10 b Fr(,)30 b(w)m(e)g(can)g(extend)g(the)g(Engel)386 3597 y(structure)k(o)m(v)m(er)f Fo(R)1086 3612 y Fn(2)1159 3597 y Fr(using)f(a)g(mo)s(del)f(Engel)h(structure)i(from)e(Section)g (4.5.)98 b Fi(\003)386 3854 y Fr(5.4.)48 b Fy(A)m(ttac)m(hing)60 b(maps)g(for)h(round)g(handles)h(of)e(index)h Fr(3)p Fy(.)49 b Fr(Let)k Fo( )66 b Fr(:)386 3970 y Fo(@)437 3985 y Fk(\000)496 3970 y Fo(R)570 3985 y Fn(3)638 3970 y Fp(\000)-16 b(!)27 b Fo(@)877 3985 y Fn(+)937 3970 y Fo(M)36 b Fr(b)s(e)26 b(an)g(orien)m(tation)e(preserving)j(attac)m (hing)e(map)g(for)h(a)f(round)386 4086 y(3{handle)d(carrying)g(a)h(mo)s (del)e(Engel)h(structure)j(from)c(Section)i(4.6.)40 b(The)23 b(mo)s(del)386 4203 y(Engel)29 b(structures)h(on)f Fo(R)1313 4218 y Fn(3)1382 4203 y Fr(induce)g(o)m(v)m(ert)m(wisted)h(con)m(tact)g (structures)h(on)d Fo(@)3174 4218 y Fk(\000)3234 4203 y Fo(R)3308 4218 y Fn(3)3348 4203 y Fr(.)386 4319 y(By)k(Theorem)f (3.18)g(w)m(e)h(can)g(isotop)s(e)e Fo( )36 b Fr(to)31 b(a)g(con)m(tact)h(di\013eomorphism)c(if)j(and)386 4435 y(only)i(if)g Fo( )38 b Fr(is)33 b(orien)m(tation)f(preserving)j(and)f (the)g(image)e(of)i(the)g(con)m(tact)g(struc-)386 4551 y(ture)29 b(on)g Fo(@)771 4566 y Fk(\000)831 4551 y Fo(R)905 4566 y Fn(3)974 4551 y Fr(is)f(homotopic)g(to)g(the)i(con)m(tact)f (structure)i(on)e Fo(@)2760 4566 y Fn(+)2819 4551 y Fo(M)40 b Fr(as)29 b(a)g(plane)386 4668 y(\014eld.)486 4784 y(The)43 b(con)m(tact)g(structure)h(on)e Fo(@)1674 4799 y Fk(\000)1733 4784 y Fo(R)1807 4799 y Fn(3)1889 4784 y Fr(is)g(orien)m(table)g(and)g (it)g(extends)i(to)e Fo(R)3335 4799 y Fn(3)386 4900 y Fr(as)36 b(a)f(plane)g(\014eld.)53 b(The)37 b(follo)m(wing)c(lemma)g (sho)m(ws)38 b(that)d(this)g(determines)h(the)386 5016 y(homotop)m(y)c(class)h(of)f(the)h(plane)f(\014eld)g(completely)-8 b(.)386 5174 y Fy(Lemma)53 b(5.7.)c Fm(Ther)-5 b(e)47 b(is)g(a)g(unique)h(homotopy)f(class)g(of)g(orientable)g(plane)386 5290 y(\014elds)34 b(on)g Fo(S)840 5254 y Fn(2)902 5290 y Fp(\002)22 b Fo(S)1067 5254 y Fn(1)1134 5290 y Fr(=)28 b Fo(@)5 b(D)1378 5254 y Fn(3)1440 5290 y Fp(\002)23 b Fo(S)1606 5254 y Fn(1)1680 5290 y Fm(which)34 b(extends)g(to)h Fo(D)2501 5254 y Fn(3)2562 5290 y Fp(\002)23 b Fo(S)2728 5254 y Fn(1)2767 5290 y Fm(.)386 5506 y(Pr)-5 b(o)g(of.)41 b Fr(Recall)36 b(from)g([HH)q(])h(that)h(the)g(Grassmannian)e(of)h (orien)m(ted)h(planes)f(in)386 5623 y Fh(R)452 5587 y Fn(3)497 5623 y Fr(,)31 b(resp)s(ectiv)m(ely)g Fh(R)1150 5587 y Fn(4)1196 5623 y Fr(,)g(is)e(Gr)1464 5638 y Fn(2)1503 5623 y Fr(\(3\))f Fp(')g Fo(S)1827 5587 y Fn(2)1866 5623 y Fr(,)j(resp)s(ectiv)m(ely)g(Gr)2568 5638 y Fn(2)2607 5623 y Fr(\(4\))c Fp(')i Fo(S)2931 5587 y Fn(2)2987 5623 y Fp(\002)18 b Fo(S)3148 5587 y Fn(2)3218 5623 y Fr(and)p eop %%Page: 46 46 46 45 bop 386 259 a Fq(46)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fr(that)32 b(the)h(inclusion)e Fh(R)1238 423 y Fn(3)1311 459 y Fp(\000)-16 b(!)28 b Fh(R)1565 423 y Fn(4)1643 459 y Fr(induces)34 b(the)f(diagonal)d(map)1031 646 y(\001)e(:)f(Gr)1309 661 y Fn(2)1348 646 y Fr(\(3\))h Fp(')g Fo(S)1672 605 y Fn(2)1739 646 y Fp(\000)-16 b(!)27 b Fo(S)1993 605 y Fn(2)2054 646 y Fp(\002)c Fo(S)2220 605 y Fn(2)2287 646 y Fp(')28 b Fr(Gr)2506 661 y Fn(2)2546 646 y Fr(\(4\))k Fo(:)386 834 y Fr(Let)c Fp(C)608 849 y Fn(0)676 834 y Fr(and)h Fp(C)914 849 y Fn(1)982 834 y Fr(b)s(e)f(t)m(w)m(o)h(plane)f(\014elds)g(on)g Fo(S)1988 798 y Fn(2)2041 834 y Fp(\002)13 b Fo(S)2197 798 y Fn(1)2265 834 y Fr(that)28 b(extend)i(to)d(the)i(in)m(terior)386 950 y(of)k Fo(D)582 914 y Fn(3)644 950 y Fp(\002)23 b Fo(S)810 914 y Fn(1)849 950 y Fr(.)46 b(W)-8 b(e)34 b(view)f Fp(C)1368 965 y Fn(0)1408 950 y Fo(;)17 b Fp(C)1504 965 y Fn(1)1577 950 y Fr(as)34 b(maps)f(from)f Fo(S)2251 914 y Fn(2)2313 950 y Fp(\002)23 b Fo(S)2479 914 y Fn(1)2552 950 y Fr(to)33 b(Gr)2786 965 y Fn(2)2826 950 y Fr(\(3\))g(and)g(their) 386 1066 y(extensions)h(as)f(maps)f(from)f Fo(D)1545 1030 y Fn(3)1606 1066 y Fp(\002)23 b Fo(S)1772 1030 y Fn(1)1844 1066 y Fr(to)32 b(Gr)2077 1081 y Fn(2)2117 1066 y Fr(\(4\).)486 1182 y(Because)23 b Fp(f)p Fr(0)p Fp(g\002)p Fo(S)1140 1146 y Fn(1)1201 1182 y Fr(is)e(a)h(strong)f (deformation)f(retract)i(of)g Fo(D)2674 1146 y Fn(3)2713 1182 y Fp(\002)p Fo(S)2856 1146 y Fn(1)2917 1182 y Fr(and)g(Gr)3211 1197 y Fn(2)3250 1182 y Fr(\(4\))386 1299 y(is)32 b(simply)e (connected,)k(there)f(is)f(a)f(homotop)m(y)h(b)s(et)m(w)m(een)i(the)f (extensions)g(of)f Fp(C)3335 1314 y Fn(0)386 1415 y Fr(and)42 b Fp(C)637 1430 y Fn(1)720 1415 y Fr(through)g(plane)g(\014elds)h(in)e Fo(T)14 b Fr(\()p Fo(D)1944 1379 y Fn(3)2012 1415 y Fp(\002)29 b Fo(S)2184 1379 y Fn(1)2223 1415 y Fr(\).)73 b(Using)42 b(the)h(pro)5 b(jection)42 b(of)386 1531 y(Gr)500 1546 y Fn(2)540 1531 y Fr(\(4\))27 b Fp(')h Fo(S)863 1495 y Fn(2)923 1531 y Fp(\002)21 b Fo(S)1087 1495 y Fn(2)1158 1531 y Fr(on)m(to)32 b(the)g(\014rst)g(factor,)g(w)m(e)h(obtain)e(a)g (homotop)m(y)h(b)s(et)m(w)m(een)386 1647 y Fp(C)438 1662 y Fn(0)510 1647 y Fr(and)h Fp(C)752 1662 y Fn(1)824 1647 y Fr(in)f Fo(T)14 b Fr(\()p Fo(S)1113 1611 y Fn(2)1174 1647 y Fp(\002)23 b Fo(S)1340 1611 y Fn(1)1379 1647 y Fr(\).)1853 b Fi(\003)486 1851 y Fr(After)28 b(w)m(e)g(ha)m(v)m(e)i (isotop)s(ed)d Fo( )32 b Fr(to)27 b(a)h(con)m(tact)g(di\013eomorphism)e (w)m(e)j(can)f(c)m(ho)s(ose)386 1967 y(a)40 b(mo)s(del)f(Engel)h (structure)h(on)f Fo(R)1695 1982 y Fn(3)1775 1967 y Fr(suc)m(h)i(that)e (the)h(orien)m(tation)d(of)i(the)h(con-)386 2083 y(tact)32 b(structure)i(and)f(the)g(homotop)m(y)f(class)h(of)f(the)h(in)m (tersection)g(line)e(\014eld)i(are)386 2200 y(preserv)m(ed)i(b)m(y)f (the)f(isotop)s(ed)f(attac)m(hing)f(map,)h(cf.)h(Theorem)g(4.15.)386 2351 y Fy(Prop)s(osition)44 b(5.8.)i Fm(The)41 b(Engel)g(structur)-5 b(e)43 b(extends)f(fr)-5 b(om)41 b Fo(M)53 b Fm(to)42 b Fo(M)c Fp([)3181 2366 y Fl( )3261 2351 y Fo(R)3335 2366 y Fn(3)386 2468 y Fm(if)g(and)f(only)g(if)h(the)g(c)-5 b(ontact)38 b(structur)-5 b(e)39 b(on)e Fo(@)2102 2483 y Fn(+)2162 2468 y Fo(M)48 b Fm(extends)37 b(to)h Fo(M)d Fp([)2968 2483 y Fl( )3046 2468 y Fo(R)3120 2483 y Fn(3)3197 2468 y Fm(as)j(a)386 2584 y(plane)c(\014eld.)1312 2869 y Fr(6.)48 b Fs(Existence)37 b(theorems)486 3043 y Fr(No)m(w)j(w)m(e)g (com)m(bine)f(the)h(to)s(ols)f(from)f(the)i(previous)g(sections)g(to)g (pro)m(v)m(e)g(ex-)386 3159 y(istence)46 b(theorems)f(for)g(Engel)g (structures.)82 b(In)46 b(Section)f(6.1)f(w)m(e)j(sho)m(w)f(that)386 3276 y(ev)m(ery)35 b(parallelizable)30 b(manifold)h(admits)i(an)g (orien)m(table)g(Engel)g(structure.)48 b(In)386 3392 y(Section)29 b(6.2)f(w)m(e)i(explain)e(the)h(construction)g(of)g(Engel) f(structures)j(on)e(the)g(con-)386 3508 y(nected)38 b(sum)g(of)e(t)m(w) m(o)i(Engel)f(manifolds)e(with)i Fo(S)2232 3472 y Fn(2)2296 3508 y Fp(\002)26 b Fo(S)2465 3472 y Fn(2)2541 3508 y Fr(if)36 b(the)i(c)m(haracteristic)386 3624 y(foliations)29 b(of)k(the)g(original)c(Engel)j(structures)i(satisfy)f(certain)f (conditions.)386 3860 y(6.1.)48 b Fy(Manifolds)e(with)f(trivial)f (tangen)m(t)i(bundle.)j Fr(In)41 b(the)f(pro)s(of)f(of)h(our)386 3976 y(main)31 b(result)i(Theorem)g(6.1)f(w)m(e)i(use)g(the)f(mo)s(del) e(Engel)h(structures)j(from)c(Sec-)386 4092 y(tion)36 b(4)h(and)g(the)h(results)g(from)e(Section)h(5)g(sho)m(wing)g(that)g(w) m(e)i(can)e(extend)i(an)386 4209 y(Engel)24 b(structure)i(from)d Fo(M)1386 4172 y Fk(0)1438 4209 y Fp(\032)28 b Fo(M)36 b Fr(to)24 b Fo(M)1888 4172 y Fk(0)1918 4209 y Fp([)6 b Fo(R)2064 4224 y Fl(l)2115 4209 y Fr(if)23 b Fo(M)2300 4172 y Fk(0)2349 4209 y Fr(has)i(transv)m(erse)h(b)s(oundary)386 4325 y(and)i(the)h(Engel)f(framing)e(extends)k(to)e(a)g(framing)e(of)i Fo(M)2469 4289 y Fk(0)2507 4325 y Fp([)13 b Fo(R)2660 4340 y Fl(l)2687 4325 y Fr(.)42 b(The)29 b(remaining)386 4441 y(problem)f(is)g(to)g(ensure)j(that)d(after)h(w)m(e)h(ha)m(v)m(e)g (constructed)g(an)f(Engel)f(structure)386 4557 y(on)j Fo(M)624 4521 y Fk(0)666 4557 y Fp([)19 b Fo(R)825 4572 y Fl(l)851 4557 y Fr(,)31 b(the)g(new)h(Engel)e(framing)f(extends)k (from)c Fo(M)2589 4521 y Fk(0)2632 4557 y Fp([)18 b Fo(R)2790 4572 y Fl(l)2847 4557 y Fr(to)31 b(the)g(whole)386 4674 y(of)h Fo(M)10 b Fr(.)44 b(This)33 b(allo)m(ws)e(us)j(to)e(p)s(erform)f (the)i(handle)g(attac)m(hmen)m(ts)g(successiv)m(ely)-8 b(.)386 4825 y Fy(Theorem)50 b(6.1.)d Fm(Every)f(p)-5 b(ar)g(al)5 b(lelizable)43 b Fr(4)p Fm({manifold)g(admits)i(an)g (orientable)386 4942 y(Engel)34 b(structur)-5 b(e.)386 5145 y(Pr)g(o)g(of.)41 b Fr(Let)f Fo(M)50 b Fr(b)s(e)40 b(a)g(closed)g(parallelizable)c(manifold)h(of)i(dimension)f(4)i(and)386 5261 y(\014x)30 b(a)g(trivialization)25 b Fo(T)14 b(M)39 b Fp(')28 b Fo(M)f Fp(\002)17 b Fh(R)1764 5225 y Fn(4)1839 5261 y Fr(of)30 b Fo(T)14 b(M)c Fr(.)43 b(W)-8 b(e)30 b(consider)g(a)g(round)g(handle)386 5378 y(decomp)s(osition)386 5591 y Fo(M)38 b Fr(=)622 5481 y Fj(\020)681 5591 y Fo(:)17 b(:)g(:)813 5481 y Fj(\020\020)932 5591 y Fo(:)g(:)g(:)1063 5481 y Fj(\020)1122 5591 y Fo(R)1196 5606 y Fn(0)1258 5591 y Fp([)1324 5612 y Fl( )1372 5589 y Fg(1)1370 5633 y(1)1434 5591 y Fo(R)1509 5550 y Fn(1)1508 5616 y(1)1548 5481 y Fj(\021)1624 5591 y Fo(:)g(:)g(:)22 b Fp([)1827 5616 y Fl( )1875 5578 y Ff(r)1904 5593 y Fg(1)1873 5637 y(1)1970 5591 y Fo(R)2045 5548 y Fl(r)2077 5557 y Fg(1)2044 5616 y Fn(1)2116 5481 y Fj(\021)2197 5591 y Fp([)2263 5612 y Fl( )2311 5589 y Fg(1)2309 5633 y(2)2373 5591 y Fo(R)2448 5550 y Fn(1)2447 5616 y(2)2487 5481 y Fj(\021)2564 5591 y Fo(:)17 b(:)g(:)k Fp([)2766 5616 y Fl( )2814 5578 y Ff(r)2843 5593 y Fg(2)2812 5637 y(2)2909 5591 y Fo(R)2984 5548 y Fl(r)3016 5557 y Fg(2)2983 5616 y Fn(2)3055 5481 y Fj(\021)3114 5591 y Fp([)3180 5606 y Fl( )3226 5615 y Fg(3)3265 5591 y Fo(R)3339 5606 y Fn(3)p eop %%Page: 47 47 47 46 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(47)386 459 y Fr(of)43 b Fo(M)55 b Fr(suc)m(h)45 b(that)f(there)g(is)g (exactly)g(one)g(round)g(3{handle)f(and)h(one)g(round)386 575 y(0{handle.)58 b(Suc)m(h)39 b(a)e(decomp)s(osition)f(exists)j(b)m (y)f(Theorem)g(4.3.)59 b(W)-8 b(e)38 b(will)d(fre-)386 691 y(quen)m(tly)29 b(isotop)s(e)f(the)h(attac)m(hing)f(maps)h(but)f (this)h(will)d(not)i(b)s(e)h(re\015ected)h(in)e(the)386 807 y(notation.)486 924 y(W)-8 b(e)37 b(start)g(with)g(the)g(round)g (handle)g(of)f(index)i(0.)56 b(By)38 b(Prop)s(osition)d(4.7)i(w)m(e)386 1040 y(can)29 b(c)m(ho)s(ose)h(a)f(mo)s(del)e(Engel)i(structure)i(on)d Fo(R)2120 1055 y Fn(0)2189 1040 y Fr(suc)m(h)j(that)e(the)g(Engel)g (framing)386 1156 y(on)41 b Fo(R)604 1171 y Fn(0)685 1156 y Fr(is)g(homotopic)f(to)h(the)h(original)c(framing.)68 b(In)41 b(particular)f(the)i(Engel)386 1272 y(framing)i(extends)49 b(from)c Fo(R)1448 1287 y Fn(0)1534 1272 y Fr(to)h Fo(M)10 b Fr(.)87 b(The)47 b(con)m(tact)g(structure)h(on)e Fo(@)3089 1287 y Fn(+)3149 1272 y Fo(R)3223 1287 y Fn(0)3309 1272 y Fr(is)386 1389 y(o)m(v)m(ert)m(wisted)34 b(b)m(y)g(construction.)486 1505 y(Let)29 b Fo(M)761 1464 y Fl(i)p Fk(\000)p Fn(1)751 1529 y(1)910 1505 y Fr(b)s(e)h(the)g(round)g(handle)f(b)s(o)s(dy)g (obtained)g(from)g Fo(R)2738 1520 y Fn(0)2777 1505 y Fo(;)17 b(R)2896 1469 y Fn(1)2895 1529 y(1)2936 1505 y Fo(;)g(:)g(:)g(:)e(;)i(R)3229 1464 y Fl(i)p Fk(\000)p Fn(1)3228 1529 y(1)3348 1505 y Fr(.)386 1623 y(Assume)45 b(that)f(w)m(e)h(ha)m(v)m(e)h(constructed)g(an)e(Engel)g(structure)i (on)e Fo(M)3024 1582 y Fl(i)p Fk(\000)p Fn(1)3014 1647 y(1)3187 1623 y Fr(suc)m(h)386 1741 y(that)26 b(the)g(con)m(tact)h (structure)g(on)f Fo(@)1681 1756 y Fn(+)1741 1741 y Fo(M)1845 1700 y Fl(i)p Fk(\000)p Fn(1)1835 1765 y(1)1990 1741 y Fr(is)f(o)m(v)m(ert)m(wisted.)43 b(Assume)27 b(moreo)m(v)m(er)386 1857 y(that)37 b(throughout)f(this)h(pro)s(cess)h(w)m(e)g(ha)m(v)m(e)g (homotop)s(ed)e(the)i(trivialization)32 b(of)386 1973 y Fo(M)55 b Fr(w)m(e)46 b(c)m(hose)g(at)e(the)h(b)s(eginning)e(suc)m(h) j(that)f(it)e(coincides)i(with)f(the)h(Engel)386 2089 y(trivialization)28 b(on)33 b Fo(M)1203 2048 y Fl(i)p Fk(\000)p Fn(1)1193 2114 y(1)1322 2089 y Fr(.)486 2207 y(Then)f(the)g(Engel)f(trivialization)26 b(on)32 b Fo(M)1991 2166 y Fl(i)p Fk(\000)p Fn(1)1981 2232 y(1)2141 2207 y Fr(can)f(b)s(e)h(extended)h(to)e Fo(M)3085 2166 y Fl(i)p Fk(\000)p Fn(1)3075 2232 y(1)3223 2207 y Fp([)3289 2229 y Fl( )3337 2206 y Ff(i)3335 2250 y Fg(1)386 2338 y Fo(R)461 2302 y Fl(i)460 2363 y Fn(1)500 2338 y Fr(.)42 b(By)30 b(Theorem)g(5.3,)g(w)m(e)g(can)g(isotop)s(e)f Fo( )2024 2302 y Fl(i)2020 2363 y Fn(1)2089 2338 y Fr(to)g(an)g(attac)m(hing)g (map)g(suc)m(h)i(that)386 2456 y(the)38 b(Engel)g(structure)h(on)e Fo(M)1504 2415 y Fl(i)p Fk(\000)p Fn(1)1494 2481 y(1)1661 2456 y Fr(extends)j(to)d(an)h(Engel)g(structure)h(on)e Fo(M)3232 2420 y Fl(i)3222 2481 y Fn(1)3299 2456 y Fr(=)386 2573 y Fo(M)490 2537 y Fl(i)480 2597 y Fn(1)545 2573 y Fp([)611 2594 y Fl( )659 2571 y Ff(i)657 2615 y Fg(1)721 2573 y Fo(R)796 2537 y Fl(i)795 2597 y Fn(1)871 2573 y Fr(using)f(a)h(mo)s(del)e(Engel)h(structure)i(on)f Fo(R)2422 2588 y Fn(1)2498 2573 y Fr(from)e(Section)i(4.4.)55 b(In)386 2704 y(order)23 b(to)f(ensure)i(that)e(the)h(con)m(tact)g (structure)h(on)e Fo(@)2311 2719 y Fn(+)2370 2704 y Fo(M)2474 2668 y Fl(i)2464 2728 y Fn(1)2527 2704 y Fr(is)g(again)f(o)m(v)m(ert)m (wisted,)386 2820 y(w)m(e)26 b(isotop)s(e)f Fo( )916 2784 y Fl(i)912 2845 y Fn(1)977 2820 y Fr(b)s(efore)h(the)g (application)d(of)i(Theorem)g(5.3)g(suc)m(h)i(that)f(its)e(image)386 2938 y(is)32 b(disjoin)m(t)f(from)h(an)g(o)m(v)m(ert)m(wisted)j(disc)d (in)g Fo(@)2073 2953 y Fn(+)2133 2938 y Fo(M)2237 2897 y Fl(i)p Fk(\000)p Fn(1)2227 2963 y(1)2356 2938 y Fr(.)486 3054 y(Let)23 b Fo(\015)702 3069 y Fk(\006)789 3054 y Fr(=)k Fp(f\006)p Fr(1)p Fp(g)t(\002)t(f)p Fr(0)p Fp(g)t(\002)t Fo(S)1503 3018 y Fn(1)1566 3054 y Fr(b)s(e)c(the)i(attac)m(hing)d(curv) m(es)k(of)d Fo(R)2737 3018 y Fl(i)2736 3079 y Fn(1)2776 3054 y Fr(.)40 b(Assume)25 b(that)386 3171 y Fo( )453 3134 y Fl(i)449 3195 y Fn(1)489 3171 y Fr(\()p Fo(\015)578 3186 y Fk(\006)636 3171 y Fr(\))31 b(is)g(transv)m(erse)j(to)d(an)g(o)m (v)m(ert)m(wisted)i(disc)e Fo(D)2304 3186 y Fl(ot)2399 3171 y Fr(and)g(let)g Fo(p)g Fr(b)s(e)g(a)g(p)s(oin)m(t)g(on)386 3287 y Fo(D)467 3302 y Fl(ot)563 3287 y Fr(whic)m(h)h(do)s(es)g(not)g (lie)e(on)i Fo( )1564 3251 y Fl(i)1560 3311 y Fn(1)1600 3287 y Fr(\()p Fo(\015)1689 3302 y Fk(\006)1748 3287 y Fr(\).)43 b(Then)33 b(use)g(the)f(\015o)m(w)h(of)e(a)h(radial)e(v)m (ector)386 3403 y(\014eld)41 b(cen)m(tered)i(at)e Fo(p)g Fr(to)f(isotop)s(e)h Fo( )1757 3367 y Fl(i)1753 3428 y Fn(1)1834 3403 y Fr(suc)m(h)h(that)f(the)h(image)d(of)i Fo(\015)2920 3418 y Fk(\006)3019 3403 y Fr(b)s(ecomes)386 3519 y(disjoin)m(t)36 b(from)h Fo(D)1055 3534 y Fl(ot)1119 3519 y Fr(.)59 b(The)38 b(remaining)e(steps)j(in)e(the)h(mo)s (di\014cation)d(of)i Fo( )t Fr(,)i(lik)m(e)386 3636 y(making)26 b(the)j(attac)m(hing)e(curv)m(es)j(Legendrian)d(and)h(p)s(erforming)e (stabilizations,)386 3752 y(can)d(b)s(e)g(carried)f(out)h(in)e(a)i (small)d(tubular)i(neigh)m(b)s(orho)s(o)s(d)g(whic)m(h)h(is)f(also)g (disjoin)m(t)386 3868 y(from)31 b Fo(D)697 3883 y Fl(ot)761 3868 y Fr(.)486 3984 y(Next)g(w)m(e)g(compare)f(the)h(Engel)f (trivialization)c(and)31 b(the)f(original)e(trivializa-)386 4100 y(tion)j(of)i Fo(M)43 b Fr(on)32 b Fo(M)1074 4064 y Fl(i)1064 4125 y Fn(1)1137 4100 y Fr(relativ)m(e)f(to)i Fo(M)1707 4059 y Fl(i)p Fk(\000)p Fn(1)1697 4125 y(1)1826 4100 y Fr(.)43 b(The)34 b(cylinder)1084 4397 y Fo(D)1168 4355 y Fn(1)1229 4397 y Fp(\002)23 b(f)p Fr(0)p Fp(g)e(\002)i Fo(S)1665 4355 y Fn(1)1732 4397 y Fp(\032)28 b Fo(R)1912 4355 y Fl(i)1911 4421 y Fn(1)1978 4397 y Fr(=)g Fo(D)2166 4355 y Fn(1)2227 4397 y Fp(\002)23 b Fo(D)2411 4355 y Fn(2)2472 4397 y Fp(\002)g Fo(S)2638 4355 y Fn(1)386 4691 y Fr(can)34 b(b)s(e)g(decomp)s(osed)g(in)m(to)f(a)g(1{cell)f Fo(e)1844 4706 y Fn(1)1913 4691 y Fr(=)d Fo(D)2102 4655 y Fn(1)2164 4691 y Fp(\002)23 b(f)p Fr(0)p Fp(g)g(\002)g(f)p Fr(1)p Fp(g)33 b Fr(and)h(a)f(2{cell)f Fo(e)3308 4706 y Fn(2)3348 4691 y Fr(.)386 4807 y(The)d(1{cell)e(is)h(attac)m(hed)h (to)f Fo(M)1556 4771 y Fl(i)p Fk(\000)p Fn(1)1704 4807 y Fr(using)g(the)h(restriction)e(of)h Fo( )2755 4771 y Fl(i)2751 4832 y Fn(1)2819 4807 y Fr(to)h Fo(@)5 b(e)3036 4822 y Fn(1)3105 4807 y Fr(and)28 b Fo(e)3335 4822 y Fn(2)386 4925 y Fr(is)d(attac)m(hed)i(to)e Fo(M)1085 4884 y Fl(i)p Fk(\000)p Fn(1)1075 4950 y(1)1213 4925 y Fp([)8 b Fo(e)1332 4940 y Fn(1)1372 4925 y Fr(.)41 b(Since)26 b Fo(\031)1743 4940 y Fn(1)1783 4925 y Fr(\(SO\(4\)\))h(=)h Fh(Z)2314 4940 y Fn(2)2376 4925 y Fr(there)f(are)f(t)m(w)m(o)g(homotop) m(y)386 5042 y(classes)33 b(of)g(orien)m(ted)f(framings)f(of)h Fo(T)14 b(M)43 b Fr(along)31 b Fo(e)2207 5057 y Fn(1)2247 5042 y Fr(.)486 5158 y(If)36 b(necessary)-8 b(,)39 b(w)m(e)e(mo)s(dify) e(the)h(mo)s(del)f(Engel)h(structure)h(on)f Fo(R)2897 5122 y Fl(i)2896 5182 y Fn(1)2972 5158 y Fr(suc)m(h)i(that)386 5274 y(the)i(new)g(Engel)f(framing)d(is)j(homotopic)f(to)g(the)i(giv)m (en)f(trivialization)c(along)386 5390 y Fo(e)431 5405 y Fn(1)507 5390 y Fr(relativ)m(e)h(to)g(the)h(endp)s(oin)m(ts)f(of)g Fo(e)1761 5405 y Fn(1)1801 5390 y Fr(.)55 b(F)-8 b(or)36 b(this)g(let)g Fo(\032)e Fr(:)g Fo(D)2630 5354 y Fn(1)2704 5390 y Fp(\000)-16 b(!)34 b Fr([0)p Fo(;)17 b Fr(2)p Fo(\031)t Fr(])35 b(b)s(e)i(a)386 5506 y(smo)s(oth)d(function)g(whic)m (h)h(is)f(constan)m(t)i(near)f(the)g(b)s(oundary)-8 b(,)35 b Fo(\032)p Fr(\()p Fp(\000)p Fr(1\))d(=)f(0)k(and)386 5623 y Fo(\032)p Fr(\(1\))30 b(=)f(2)p Fo(\031)t Fr(.)47 b(If)34 b(w)m(e)g(push)h(forw)m(ard)f(the)g(mo)s(del)e(Engel)i (structure)h(on)f Fo(R)3079 5638 y Fn(1)3152 5623 y Fr(using)p eop %%Page: 48 48 48 47 bop 386 259 a Fq(48)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fr(the)33 b(di\013eomorphism)553 659 y Fo(F)616 674 y Fn(1)684 659 y Fr(:)27 b Fo(R)813 618 y Fl(i)812 683 y Fn(1)880 659 y Fr(=)g Fo(D)1067 618 y Fn(1)1129 659 y Fp(\002)22 b Fo(D)1312 618 y Fn(2)1373 659 y Fp(\002)h Fo(S)1539 618 y Fn(1)1606 659 y Fp(\000)-16 b(!)27 b Fo(D)1878 618 y Fn(1)1939 659 y Fp(\002)c Fo(D)2123 618 y Fn(2)2184 659 y Fp(\002)g Fo(S)2350 618 y Fn(1)2417 659 y Fr(=)k Fo(R)2595 618 y Fl(i)2594 683 y Fn(1)1107 876 y Fr(\()p Fo(x;)17 b(y)1292 891 y Fn(1)1331 876 y Fo(;)g(y)1423 891 y Fn(2)1461 876 y Fo(;)g(t)p Fr(\))28 b Fp(7\000)-16 b(!)1859 816 y Fr(\()p Fo(x;)17 b Fr(cos)q(\()p Fo(\032)p Fr(\()p Fo(x)p Fr(\)\))p Fo(y)2432 831 y Fn(1)2493 816 y Fr(+)22 b(sin\()p Fo(\032)p Fr(\()p Fo(x)p Fr(\)\))p Fo(y)3016 831 y Fn(2)3056 816 y Fo(;)1836 933 y Fp(\000)17 b Fr(sin\()p Fo(\032)p Fr(\()p Fo(x)p Fr(\)\))p Fo(y)2355 948 y Fn(1)2416 933 y Fr(+)22 b(cos)q(\()p Fo(\032)p Fr(\()p Fo(x)p Fr(\)\))p Fo(y)2950 948 y Fn(2)2989 933 y Fo(;)17 b(t)p Fr(\))3180 876 y Fo(;)386 1140 y Fr(then)28 b(w)m(e)h(still)c(obtain)h(a)i(smo)s(oth)e(Engel)h(structure)i(on)f Fo(M)2556 1104 y Fl(i)2546 1164 y Fn(1)2613 1140 y Fr(b)m(y)g(the)g(c)m (hoice)g(of)f Fo(\032)p Fr(.)386 1256 y(Because)i Fo(F)817 1271 y Fn(1)884 1256 y Fr(in)m(terc)m(hanges)g(the)f(t)m(w)m(o)g (homotop)m(y)f(classes)h(of)f(framings)f(along)g Fo(e)3335 1271 y Fn(1)386 1372 y Fr(relativ)m(e)35 b Fo(@)5 b(e)837 1387 y Fn(1)878 1372 y Fr(,)37 b(the)f(trivialization)c(induced)k(b)m (y)h(the)g(new)g(Engel)e(structure)j(on)386 1488 y Fo(R)460 1503 y Fn(1)527 1488 y Fr(is)26 b(no)m(w)i(homotopic)e(to)g(the)i(giv)m (en)f(trivialization)c(of)j Fo(T)14 b(M)38 b Fr(along)26 b Fo(e)2994 1503 y Fn(1)3060 1488 y Fr(relativ)m(e)386 1605 y Fo(@)5 b(e)487 1620 y Fn(1)527 1605 y Fr(.)43 b(W)-8 b(e)29 b(homotop)f(the)h(original)d(framing)g(suc)m(h)31 b(it)c(coincides)i(with)g(the)g(Engel)386 1721 y(framing)35 b(on)j Fo(M)996 1680 y Fl(i)p Fk(\000)p Fn(1)986 1745 y(1)1140 1721 y Fp([)26 b Fo(e)1277 1736 y Fn(1)1317 1721 y Fr(.)58 b(Since)38 b Fo(\031)1717 1736 y Fn(2)1756 1721 y Fr(\(SO\(4\)\))f(is)g(trivial)e(the)j(same)f(is)g(true)h(for)386 1839 y Fo(M)490 1798 y Fl(i)p Fk(\000)p Fn(1)480 1864 y(1)631 1839 y Fp([)23 b Fo(e)765 1854 y Fn(1)827 1839 y Fp([)f Fo(e)960 1854 y Fn(2)1000 1839 y Fr(.)486 1955 y(Th)m(us)48 b(the)f(Engel)f(framing)e(on)j Fo(M)1843 1919 y Fl(i)1833 1980 y Fn(1)1919 1955 y Fr(is)f(no)m(w)i(homotopic)d (to)h(the)h(original)386 2074 y(framing)e(relativ)m(e)i(to)h Fo(M)1362 2032 y Fl(i)p Fk(\000)p Fn(1)1352 2098 y(1)1481 2074 y Fr(.)89 b(In)48 b(particular)d(it)i(extends)j(from)c Fo(M)3031 2037 y Fl(i)3021 2098 y Fn(1)3109 2074 y Fr(to)h Fo(M)10 b Fr(.)386 2190 y(Hence)38 b(w)m(e)h(can)e(iterate)g(the)g(pro) s(cedure)i(to)d(attac)m(h)i(round)f(handles)h(of)e(index)386 2306 y(1.)53 b(During)34 b(this)i(pro)s(cess)h(w)m(e)f(ensure)i(that)d (the)h(con)m(tact)h(structure)g(on)e Fo(@)3181 2321 y Fn(+)3241 2306 y Fo(M)3345 2270 y Fl(i)3335 2331 y Fn(1)386 2422 y Fr(is)f(o)m(v)m(ert)m(wisted)i(for)d(all)g Fo(i)h Fr(b)m(y)h(c)m(ho)s(osing)f(the)h(attac)m(hing)e(regions)h(disjoin)m(t) f(from)386 2538 y Fo(D)467 2553 y Fl(ot)531 2538 y Fr(.)486 2655 y(In)j(the)h(next)g(step)h(w)m(e)f(attac)m(h)g(round)g(2{handles.) 54 b(Assume)38 b(that)e(w)m(e)h(ha)m(v)m(e)386 2771 y(already)26 b(attac)m(hed)g(the)h(\014rst)g Fo(i)9 b Fp(\000)g Fr(1)26 b(round)g(2{handles)g(suc)m(h)i(that)e(on)g(the)g(result-)386 2887 y(ing)k(handle)h(b)s(o)s(dy)f Fo(M)1208 2846 y Fl(i)p Fk(\000)p Fn(1)1198 2912 y(2)1358 2887 y Fr(there)i(is)e(an)h(Engel)g (structure)h(extending)g(the)f(Engel)386 3003 y(structure)j(on)e Fo(M)1033 3018 y Fn(1)1073 3003 y Fr(.)43 b(In)33 b(addition)e(w)m(e)j (assume)f(that)f(the)h(con)m(tact)g(structure)h(on)386 3120 y Fo(@)437 3135 y Fn(+)496 3120 y Fo(M)600 3078 y Fl(i)p Fk(\000)p Fn(1)590 3144 y(2)752 3120 y Fr(is)e(o)m(v)m(ert)m (wisted.)46 b(Consider)33 b(the)g(attac)m(hing)e(map)1371 3321 y Fo( )1438 3280 y Fl(i)1434 3346 y Fn(2)1502 3321 y Fr(:)c Fo(@)1607 3336 y Fk(\000)1667 3321 y Fo(R)1742 3280 y Fl(i)1741 3346 y Fn(2)1808 3321 y Fp(\000)-16 b(!)27 b Fo(@)2047 3336 y Fn(+)2107 3321 y Fo(M)2211 3280 y Fl(i)p Fk(\000)p Fn(1)2201 3346 y(2)2363 3321 y Fo(:)386 3523 y Fr(of)33 b Fo(R)573 3487 y Fl(i)572 3548 y Fn(2)611 3523 y Fr(.)46 b(The)35 b(con)m(tact)e(structure)i(on)e Fo(@)1836 3538 y Fn(+)1896 3523 y Fo(M)2000 3482 y Fl(i)p Fk(\000)p Fn(1)1990 3547 y(2)2152 3523 y Fr(is)g(orien)m(ted)h(and)f (it)f(has)i(an)f(ori-)386 3639 y(en)m(table)28 b(section,)i(namely)d (the)i(in)m(tersection)f(line)f(\014eld.)42 b(Th)m(us)30 b(the)e(Euler)h(class)386 3756 y(of)f(the)i(con)m(tact)g(structure,)h (view)m(ed)f(as)f(a)g(bundle,)h(v)-5 b(anishes.)43 b(By)29 b(assumption,)386 3872 y(the)42 b(con)m(tact)g(structure)h(is)e(o)m(v)m (ert)m(wisted.)72 b(According)41 b(to)h(Theorem)f(5.4)h(and)386 3988 y(Theorem)31 b(3.13)e(w)m(e)j(can)e(isotop)s(e)g Fo( )1716 3952 y Fl(i)1712 4013 y Fn(2)1782 3988 y Fr(suc)m(h)i(that)e (the)h(resulting)e(map)h(preserv)m(es)386 4104 y(singular)39 b(foliations.)65 b(Recall)39 b(that)h(in)g(Theorem)h(5.4)f(w)m(e)i (also)e(ensured)i(that)386 4220 y(the)f(attac)m(hing)g(region)f(of)g Fo(R)1497 4184 y Fl(i)1496 4245 y Fn(2)1577 4220 y Fr(is)g(con)m (tained)i(in)e(a)g(neigh)m(b)s(orho)s(o)s(d)g(of)h Fo( )3153 4184 y Fl(i)3149 4245 y Fn(2)3189 4220 y Fr(\()p Fo(T)3298 4184 y Fn(2)3284 4245 y(0)3337 4220 y Fr(\))386 4337 y(whic)m(h)33 b(is)f(disjoin)m(t)f(from)h(some)g(o)m(v)m(ert)m(wisted)j (disc.)486 4453 y(In)40 b(order)f(to)h(\014nd)g(a)f(mo)s(del)f(Engel)i (structure)h(on)e Fo(R)2512 4417 y Fl(i)2511 4478 y Fn(2)2590 4453 y Fr(whic)m(h)i(extends)g(the)386 4571 y(Engel)34 b(structure)i(on)f Fo(M)1322 4530 y Fl(i)p Fk(\000)p Fn(1)1312 4595 y(2)1476 4571 y Fr(to)f(an)h(Engel)f(structure)i(on)f Fo(M)2671 4535 y Fl(i)2661 4596 y Fn(2)2732 4571 y Fr(=)c Fo(M)2943 4530 y Fl(i)p Fk(\000)p Fn(1)2933 4595 y(2)3086 4571 y Fp([)3152 4593 y Fl( )3200 4570 y Ff(i)3198 4614 y Fg(2)3261 4571 y Fo(R)3336 4535 y Fl(i)3335 4596 y Fn(2)386 4691 y Fr(w)m(e)j(ha)m(v)m(e)f(to)g(pro)m(v)m(e)h(the)f(follo) m(wing)c(claim.)486 4807 y Fm(Claim)c(1)h(:)41 b(The)26 b(Engel)g(trivialization)g(on)g Fo(M)2167 4766 y Fl(i)p Fk(\000)p Fn(1)2157 4832 y(2)2313 4807 y Fm(extends)f(to)i(a)f (trivialization)386 4924 y(of)33 b Fo(T)14 b(M)45 b Fm(over)33 b Fo(e)968 4939 y Fn(2)1035 4924 y Fp(\032)c Fo(R)1216 4888 y Fl(i)1215 4948 y Fn(2)1254 4924 y Fm(.)45 b(The)33 b(extension)g(is)g(unique)h(up)g(to)g(homotopy)f(r)-5 b(elative)386 5040 y Fo(@)5 b(e)487 5055 y Fn(2)527 5040 y Fm(.)386 5259 y(Pr)-5 b(o)g(of)35 b(of)f(Claim)g(1.)41 b Fr(First)c(w)m(e)i(decomp)s(ose)f(eac)m(h)g(round)g(2{handle)f Fo(R)3063 5211 y Fl(j)3062 5283 y Fn(2)3102 5259 y Fo(;)17 b(j)42 b Fp(\024)36 b Fo(i)386 5383 y Fr(in)m(to)f(one)h(ordinary)f (handle)g Fo(h)1538 5335 y Fl(j)1538 5407 y Fn(2)1613 5383 y Fr(of)g(index)h(2)f(and)h(one)g(ordinary)f(handle)g Fo(h)3221 5335 y Fl(j)3221 5407 y Fn(3)3296 5383 y Fr(of)386 5506 y(index)k(3)g(as)g(in)f(\(16\))o(.)63 b(Then)40 b(w)m(e)g(rearrange)f(the)g(handles)g Fo(h)2692 5459 y Fl(j)2692 5531 y Fn(2)2732 5506 y Fo(;)17 b(h)2832 5459 y Fl(j)2832 5531 y Fn(3)2871 5506 y Fo(;)g(j)44 b Fp(\024)39 b Fo(i)g Fr(suc)m(h)386 5623 y(that)29 b(the)h(ordinary)e (handles)i(of)e(index)i(2)f(are)g(attac)m(hed)h(to)f Fo(M)2707 5638 y Fn(1)2776 5623 y Fr(indep)s(enden)m(tly)p eop %%Page: 49 49 49 48 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(49)386 459 y Fr(and)45 b(the)h(remaining)d(ordinary)i(3{handles)g (are)h(attac)m(hed)g(to)f(the)h(resulting)386 575 y(handle)32 b(b)s(o)s(dy)-8 b(.)486 691 y(The)25 b(Engel)f(trivialization)c(on)k Fo(M)1742 650 y Fl(i)p Fk(\000)p Fn(1)1732 716 y(2)1886 691 y Fr(extends)i(o)m(v)m(er)g Fo(h)2490 655 y Fl(i)2490 716 y Fn(2)2554 691 y Fd(b)s(efore)e Fr(w)m(e)i(rearrange)386 807 y(the)h(handle)f(decomp)s(osition)f(if)h(and)g(only)h(if)e(the)i (Engel)f(trivialization)d(extends)386 924 y(from)36 b Fo(M)715 939 y Fn(1)791 924 y Fr(o)m(v)m(er)i Fo(h)1060 887 y Fl(i)1060 948 y Fn(2)1137 924 y Fd(after)e Fr(the)i(rearrangemen) m(t.)56 b(By)38 b(construction)f(the)g(Engel)386 1040 y(framing)46 b(on)h Fo(M)1006 1055 y Fn(1)1094 1040 y Fr(is)h(homotopic)e(to)h(the)i(framing)c(of)j Fo(M)58 b Fr(w)m(e)49 b(c)m(hose)h(at)d(the)386 1156 y(b)s(eginning)35 b(of)g(the)i(pro)s(of.)54 b(In)36 b(particular)f(the)i(Engel)f(framing) e(extends)k(from)386 1275 y Fo(M)480 1290 y Fn(1)552 1275 y Fr(to)32 b Fo(h)727 1239 y Fl(i)727 1300 y Fn(2)800 1275 y Fr(after)g(the)h(rearrangemen)m(t)g(of)f(the)h(handles)f Fo(h)2533 1228 y Fl(j)2533 1299 y Fn(2)2573 1275 y Fo(;)17 b(h)2673 1228 y Fl(j)2673 1299 y Fn(2)2712 1275 y Fo(;)g(j)34 b Fp(\024)28 b Fo(i)p Fr(.)486 1393 y(This)37 b(pro)m(v)m(es)i(that)f (the)g(Engel)f(framing)e(can)j(b)s(e)g(extended)h(from)d Fo(M)3131 1352 y Fl(i)p Fk(\000)p Fn(1)3121 1417 y(2)3288 1393 y Fr(to)386 1511 y Fo(M)490 1470 y Fl(i)p Fk(\000)p Fn(1)480 1535 y(2)637 1511 y Fp([)28 b Fo(e)776 1475 y Fl(i)776 1535 y Fn(2)815 1511 y Fr(.)68 b(This)41 b(extension)g(is)f (unique)h(up)g(to)g(homotop)m(y)f(relativ)m(e)g(to)g Fo(@)5 b(e)3334 1475 y Fl(i)3334 1535 y Fn(2)386 1627 y Fr(since)33 b Fo(\031)680 1642 y Fn(2)720 1627 y Fr(\(SO\(4\)\))27 b(=)g Fp(f)p Fr(0)p Fp(g)p Fr(.)1940 b Fi(\003)486 1801 y Fr(By)31 b(Prop)s(osition)f(5.6)g(w)m(e)i(can)g(extend)g(the)g(Engel) f(structure)h(from)e Fo(M)3138 1759 y Fl(i)p Fk(\000)p Fn(1)3128 1825 y(2)3288 1801 y Fr(to)386 1917 y Fo(M)490 1881 y Fl(i)480 1941 y Fn(2)557 1917 y Fr(using)37 b(a)g(mo)s(del)f (Engel)h(structure)i(from)d(Section)h(4.5)g(on)g Fo(R)2856 1881 y Fl(i)2855 1941 y Fn(2)2895 1917 y Fr(.)58 b(Since)37 b(the)386 2033 y(con)m(tact)30 b(structure)h(on)f Fo(@)1326 2048 y Fn(+)1385 2033 y Fo(M)1489 1997 y Fl(i)1479 2058 y Fn(2)1549 2033 y Fr(is)f(again)f(o)m(v)m(ert)m(wisted)k(w)m(e)e(can)g (iterate)f(the)h(pro-)386 2149 y(cedure.)43 b(When)28 b(the)g(last)f(round)h(2{handle)e(is)h(attac)m(hed)i(w)m(e)f(ha)m(v)m (e)h(constructed)386 2265 y(an)k(Engel)f(structure)i(on)f Fo(M)1442 2280 y Fn(2)1482 2265 y Fr(.)44 b(In)34 b(order)f(to)f (\014nish)h(the)g(construction)g(w)m(e)h(ha)m(v)m(e)386 2382 y(to)e(extend)i(the)f(Engel)f(structure)i(o)m(v)m(er)g(the)f (round)g(3{handle)f Fo(R)2825 2397 y Fn(3)2864 2382 y Fr(.)486 2498 y Fm(Claim)h(2)i(:)45 b(The)34 b(Engel)g(trivialization)h (extends)f(fr)-5 b(om)34 b Fo(M)2648 2513 y Fn(2)2723 2498 y Fm(to)h Fo(M)10 b Fm(.)386 2669 y(Pr)-5 b(o)g(of)35 b(of)f(Claim)g(2.)41 b Fr(W)-8 b(e)53 b(decomp)s(ose)f(all)f(round)h (2{handles)g(in)m(to)f(ordinary)386 2788 y(handles)e Fo(h)811 2741 y Fl(j)811 2812 y Fn(2)850 2788 y Fo(;)17 b(h)950 2741 y Fl(j)950 2812 y Fn(3)1038 2788 y Fr(for)48 b(1)54 b Fp(\024)h Fo(j)61 b Fp(\024)55 b Fo(r)1715 2803 y Fn(2)1803 2788 y Fr(of)48 b(index)h(2)f(and)g(3)h(and)f(w)m(e)i (rearrange)386 2904 y(the)36 b(handles)f(suc)m(h)i(that)f(the)f (2{handles)h(are)f(attac)m(hed)h(to)f Fo(M)2756 2919 y Fn(1)2796 2904 y Fr(.)52 b(W)-8 b(e)36 b(also)f(de-)386 3026 y(comp)s(ose)i Fo(R)858 3041 y Fn(3)934 3026 y Fr(in)m(to)f(an)h (ordinary)f(3{handle)2087 2999 y Fj(b)2089 3026 y Fo(h)2145 3041 y Fn(3)2221 3026 y Fr(and)h(one)g(ordinary)f(4{handle)384 3125 y Fj(b)386 3152 y Fo(h)442 3167 y Fn(4)482 3152 y Fr(.)45 b(In)34 b(Claim)c(1)j(w)m(e)i(ha)m(v)m(e)f(sho)m(wn)h(that)e (the)g(Engel)g(trivialization)c(on)k Fo(M)3174 3167 y Fn(1)3247 3152 y Fr(ex-)386 3268 y(tends)d(to)e Fo(M)852 3283 y Fn(1)905 3268 y Fp([)14 b Fo(h)1041 3232 y Fn(1)1041 3293 y(2)1095 3268 y Fp([)g Fo(:)j(:)g(:)c Fp([)h Fo(h)1439 3224 y Fl(r)1471 3233 y Fg(2)1439 3292 y Fn(2)1539 3268 y Fr(and)28 b(that)h(all)d(suc)m(h)k(extensions)g(are)e(homotopic.)386 3384 y(Therefore,)33 b(the)f(Engel)g(trivialization)27 b(on)32 b Fo(M)2098 3399 y Fn(1)2158 3384 y Fp([)22 b Fo(h)2302 3348 y Fn(1)2302 3409 y(2)2362 3384 y Fp([)f Fo(:)c(:)g(:)f(h)2636 3341 y Fl(r)2668 3350 y Fg(2)2636 3409 y Fn(2)2738 3384 y Fr(also)31 b(extends)j(to)386 3500 y Fo(M)10 b Fr(.)486 3617 y(Next)36 b(w)m(e)h(reduce)g(the)f (problem)f(to)g(trivializations)d(of)j(bundles)i(of)e(rank)h(3.)386 3733 y(The)29 b(\014rst)g(comp)s(onen)m(t)f Fo(W)42 b Fr(of)28 b(the)g(Engel)g(trivialization)c(is)k(transv)m(erse)i(to)e Fo(@)5 b(M)3334 3748 y Fn(2)386 3849 y Fr(b)m(y)38 b(construction.)56 b(Th)m(us)38 b Fo(W)51 b Fr(extends)38 b(to)f(a)f(v)m(ector)i(\014eld)f (without)f(zero)s(es)i(on)386 3965 y Fo(M)10 b Fr(.)50 b(W)-8 b(e)36 b(equip)e Fo(M)46 b Fr(with)34 b(an)h(almost)e (quaternionic)h(structure)i(suc)m(h)g(that)e(the)386 4082 y(Engel)26 b(framing)f(and)i Fo(W)m(;)17 b(I)8 b(W)m(;)17 b(J)9 b(W)m(;)17 b(K)7 b(W)40 b Fr(coincide)26 b(on)h Fo(M)2518 4097 y Fn(1)2568 4082 y Fp([)10 b Fo(h)2700 4045 y Fn(1)2700 4106 y(2)2756 4082 y Fo(:)17 b(:)g(:)10 b Fp([)g Fo(h)3013 4038 y Fl(r)3045 4047 y Fg(2)3013 4106 y Fn(2)3084 4082 y Fr(.)42 b(Then)386 4198 y(w)m(e)31 b(can)f(c)m(ho)s(ose)h(a)f(trivialization)c(of)j(the)i(orthogonal)d (complemen)m(t)h Fp(W)3071 4162 y Fk(?)3161 4198 y Fr(of)g Fo(W)386 4314 y Fr(in)35 b Fo(M)10 b Fr(.)54 b(\(This)36 b(tric)m(k)f(can)h(b)s(e)g(found)g(in)f([Gei)o(].\))53 b(F)-8 b(or)35 b(the)i(pro)s(of)d(of)i(Claim)d(2)j(it)386 4430 y(su\016ces)k(to)e(sho)m(w)h(that)f(w)m(e)h(can)f(extend)i(the)e (trivialization)c(of)j Fp(W)2946 4394 y Fk(?)3044 4430 y Fr(induced)386 4546 y(b)m(y)c(the)g(Engel)g(structure.)486 4663 y(On)g(the)h(2{sk)m(eleton)h(of)e Fo(M)44 b Fr(the)34 b(SO\(3\){bundle)g Fp(W)2442 4627 y Fk(?)2535 4663 y Fr(is)f(trivial.)45 b(Therefore)386 4779 y(w)m(e)29 b(can)f(lift)d(it)i (to)g(an)h(SU\(2\){bundle.)41 b(\(Recall)26 b(that)i(Spin)o(\(3\))g(=)f (SU)q(\(2\))g(=)g Fo(S)3270 4743 y Fn(3)3310 4779 y Fr(.\))386 4895 y(Since)d Fo(\031)687 4910 y Fn(2)726 4895 y Fr(\(SU)q(\(2\)\))f (is)g(trivial,)g(the)h(trivialization)19 b(of)k Fp(W)2412 4859 y Fk(?)2495 4895 y Fr(induced)h(b)m(y)h(the)f(Engel)386 5027 y(structure)34 b(extends)g(from)e Fo(M)1484 5042 y Fn(2)1556 5027 y Fr(to)1673 5000 y Fj(b)1675 5027 y Fo(h)1731 5042 y Fn(3)1771 5027 y Fr(.)43 b(W)-8 b(e)33 b(\014x)g(suc)m(h)h(an)f(extension.)486 5148 y(The)g(obstruction)f(for) g(the)h(extension)g(of)f(the)h(trivialization)28 b(from)j Fo(M)3129 5163 y Fn(2)3191 5148 y Fp([)3277 5122 y Fj(b)3279 5148 y Fo(h)3335 5163 y Fn(3)386 5264 y Fr(to)24 b Fo(M)34 b Fr(of)24 b Fp(W)834 5228 y Fk(?)918 5264 y Fr(induced)g(b)m(y)h(the)g (Engel)e(structure)j(is)d(a)h(co)s(cycle)h Fo(x)f Fr(in)f(the)i (cellular)386 5390 y(co)s(c)m(hain)37 b(group)g Fo(C)1101 5354 y Fn(4)1141 5390 y Fr(\()p Fo(M)5 b(;)17 b(M)1416 5405 y Fn(2)1481 5390 y Fp([)1571 5364 y Fj(b)1573 5390 y Fo(h)1629 5405 y Fn(3)1669 5390 y Fr(;)g Fo(\031)1768 5405 y Fn(3)1807 5390 y Fr(\(SU\(2\)\))36 b(=)g Fh(Z)p Fr(\))e(and)k Fo(x)g Fr(ma)m(y)f(dep)s(end)i(on)386 5506 y(the)f(c)m(hoice)g(of)f(extensions)i(of)e(the)h(trivialization)33 b(o)m(v)m(er)39 b(the)f(3{handles.)59 b(The)386 5623 y(co)s(cycle)40 b Fo(x)h Fr(represen)m(ts)h(a)e(class)g([)p Fo(x)p Fr(])h Fp(2)g Fo(H)1962 5587 y Fn(4)2001 5623 y Fr(\()p Fo(M)5 b(;)17 b(M)2276 5638 y Fn(2)2343 5623 y Fp([)2434 5596 y Fj(b)2436 5623 y Fo(h)2492 5638 y Fn(3)2532 5623 y Fr(;)g Fh(Z)p Fr(\))37 b(whic)m(h)j(do)s(es)h(not)p eop %%Page: 50 50 50 49 bop 386 259 a Fq(50)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fr(dep)s(end)31 b(on)f(the)h(c)m(hoice)f(of)g(trivializations)c (on)k(the)h(3{handles)f(b)s(ecause)h(it)e(is)h(a)386 575 y(primary)h(obstruction,)i(cf.)43 b(Theorem)33 b(34.2)f(in)g ([Ste].)486 691 y(W)-8 b(e)41 b(ha)m(v)m(e)h(already)e(sho)m(wn)i(that) f(the)g(Engel)g(trivialization)36 b(extends)43 b(from)386 807 y(the)32 b(2{sk)m(eleton)f(to)g(the)h(whole)f(of)g Fo(M)10 b Fr(.)44 b(Th)m(us)32 b([)p Fo(x)p Fr(])d(=)e(0.)43 b(Since)31 b(there)h(is)f(exactly)386 939 y(one)44 b(4{handle)f(w)m(e)i (ha)m(v)m(e)g Fo(C)1467 903 y Fn(4)1507 939 y Fr(\()p Fo(M)5 b(;)17 b(M)1782 954 y Fn(2)1851 939 y Fp([)1945 913 y Fj(b)1948 939 y Fo(h)2004 954 y Fn(3)2043 939 y Fr(;)g Fh(Z)p Fr(\))44 b(=)j Fo(H)2450 903 y Fn(4)2489 939 y Fr(\()p Fo(M)5 b(;)17 b(M)2764 954 y Fn(2)2834 939 y Fp([)2928 913 y Fj(b)2930 939 y Fo(h)2986 954 y Fn(3)3026 939 y Fr(;)g Fh(Z)p Fr(\))41 b(and)386 1055 y(hence)34 b Fo(x)29 b Fr(=)f(0.)44 b(This)33 b(implies)d(that)j(the)g (Engel)g(trivialization)28 b(on)33 b Fo(M)2979 1070 y Fn(2)3052 1055 y Fr(extends)386 1171 y(to)f Fo(M)10 b Fr(.)2661 b Fi(\003)486 1347 y Fr(Since)32 b(the)g(Engel)g(framing)e (extends)j(from)e Fo(M)2217 1362 y Fn(2)2289 1347 y Fr(to)g Fo(M)43 b Fr(the)33 b(same)e(is)h(true)g(for)386 1464 y(the)41 b(con)m(tact)g(structure)h(on)e Fo(@)1534 1479 y Fn(+)1594 1464 y Fo(M)1688 1479 y Fn(2)1768 1464 y Fr(view)m(ed)i(as)f(a)f(plane)g(\014eld.)68 b(The)41 b(con)m(tact)386 1580 y(structure)35 b(on)e Fo(@)992 1595 y Fn(+)1052 1580 y Fo(M)1146 1595 y Fn(2)1219 1580 y Fr(is)g(o)m(v)m(ert)m(wisted)i(b)m(y)g(construction.)46 b(By)34 b(Prop)s(osition)e(5.8)386 1696 y(w)m(e)41 b(can)f(extend)h (the)f(Engel)f(structure)i(from)e Fo(M)2258 1711 y Fn(2)2337 1696 y Fr(to)g Fo(M)10 b Fr(.)66 b(This)40 b(\014nishes)g(the)386 1812 y(pro)s(of)32 b(of)g(the)h(theorem.)2003 b Fi(\003)486 1988 y Fr(Let)31 b(us)h(compare)f(the)h(pro)s(of)f(of)g(Theorem)h(6.1)f (with)g(the)h(follo)m(wing)d(c)m(harac-)386 2104 y(terization)i(of)h (parallelizable)d(4{manifolds.)386 2245 y Fy(Theorem)f(6.2)g Fr(\(Hirzebruc)m(h,)f(Hopf,)f([HH]\))p Fy(.)36 b Fm(A)n(n)27 b(orientable)g Fr(4)p Fm({manifold)e(has)386 2361 y(trivial)35 b(tangent)f(bund)5 b(le)35 b(if)f(and)h(only)f(if)556 2502 y Fr(\(i\))40 b Fm(the)35 b(Euler)g(char)-5 b(acteristic)35 b(vanishes,)529 2618 y Fr(\(ii\))k Fm(the)c(se)-5 b(c)g(ond)34 b(Stiefel{Whitney)h(class)f Fo(w)2178 2633 y Fn(2)2217 2618 y Fr(\()p Fo(M)10 b Fr(\))36 b Fm(is)e(zer)-5 b(o)35 b(and)502 2734 y Fr(\(iii\))j Fm(the)d(signatur)-5 b(e)35 b(of)g Fo(M)45 b Fm(is)35 b(zer)-5 b(o.)486 2875 y Fr(Our)36 b(construction)g(relies)f(on)h(round)h(handle)e(decomp)s(ositions)g (and)h(there-)386 2991 y(fore)28 b(condition)f(\(i\))h(in)g(Theorem)g (4.3)h(is)f(used)h(at)g(all)d(stages)j(of)f(the)h(pro)s(of.)42 b(The)386 3107 y(second)35 b(Stiefel{Whitney)d(class)h Fo(w)1714 3122 y Fn(2)1754 3107 y Fr(\()p Fo(M)10 b Fr(\))34 b(of)e(an)i(orien)m(table)e(4{manifold)e Fo(M)44 b Fr(is)386 3224 y(zero)25 b(if)f(and)h(only)g(if)f Fo(T)14 b(M)35 b Fr(is)25 b(trivial)d(on)j(the)h(2{sk)m(eleton)f(of)g Fo(M)10 b Fr(.)41 b(Th)m(us)27 b(condition)386 3340 y(\(ii\))36 b(of)h(Theorem)h(6.2)g(is)f(used)i(when)g(w)m(e)g(attac)m(h)f(round)g (handles)g(of)g(index)g(1)386 3456 y(and)f(2)f(in)g(order)h(to)g (establish)f(certain)g(prop)s(erties)h(of)f(rotation)g(n)m(um)m(b)s (ers.)57 b(It)386 3572 y(is)32 b(also)g(used)h(when)h(w)m(e)g(lift)c Fp(W)1561 3536 y Fk(?)1653 3572 y Fr(to)j(an)f(SU\(2\){bundle)h(in)e (Claim)g(2.)486 3689 y(Finally)h(the)j(obstruction)g(for)g(the)g (extension)h(of)f(a)g(section)g(of)f(the)i(SU\(2\){)386 3805 y(bundle)27 b(app)s(earing)g(at)g(the)h(end)g(of)e(the)i(pro)s(of) f(of)g(Theorem)g(6.1)g(can)h(b)s(e)f(view)m(ed)386 3921 y(as)22 b(the)g(primary)f(obstruction)g(to)h(the)g(construction)g(of)f (a)h(section)f(of)h(the)g(SU\(2\){)386 4037 y(bundle)j(o)m(v)m(er)g Fo(M)36 b Fr(since)25 b(all)d(sections)j(of)f(a)h(SU\(2\){bundle)f(o)m (v)m(er)i Fo(M)2826 4052 y Fn(1)2871 4037 y Fp([)6 b Fo(h)2999 4001 y Fn(1)2999 4062 y(2)3056 4037 y Fo(:)17 b(:)g(:)5 b Fp([)h Fo(h)3304 3994 y Fl(r)3336 4003 y Fg(2)3304 4062 y Fn(2)386 4154 y Fr(are)39 b(homotopic.)62 b(According)38 b(to)h([GoS])g(p.)g(31,)i(an)e(SU\(2\){bundle)g(is)f (trivial)386 4270 y(if)47 b(and)h(only)f(if)g(its)h(second)h(Chern)g (class)f(v)-5 b(anishes.)91 b(On)48 b(the)g(other)g(hand)386 4386 y(the)i(second)h(Chern)g(class)f(of)f(the)h(SU)q(\(2\){bundle)f (is)g Fp(\000)p Fo(p)2651 4401 y Fn(1)2691 4386 y Fr(\()p Fp(W)2835 4350 y Fk(?)2895 4386 y Fr(\))p Fo(=)p Fr(4.)95 b(Since)386 4502 y Fo(p)435 4517 y Fn(1)474 4502 y Fr(\()p Fo(T)14 b(M)c Fr(\))33 b(=)f Fo(p)915 4517 y Fn(1)954 4502 y Fr(\()p Fp(W)1098 4466 y Fk(?)1158 4502 y Fr(\),)k(the)f(v)-5 b(anishing)34 b(of)h Fo(x)g Fr(corresp)s(onds)i(to)e(\(iii\))d(b)m(y)k (the)g(sig-)386 4618 y(nature)d(theorem)f(of)g(Hirzebruc)m(h.)386 4809 y(6.2.)48 b Fy(Connected)g(sums)g(of)g(Engel)e(manifolds.)i Fr(Let)42 b Fo(M)2731 4824 y Fn(1)2771 4809 y Fo(;)17 b(M)2909 4824 y Fn(2)2989 4809 y Fr(b)s(e)42 b(mani-)386 4925 y(folds)34 b(with)h(Engel)g(structures)i Fp(D)1655 4940 y Fn(1)1695 4925 y Fo(;)17 b Fp(D)1816 4940 y Fn(2)1855 4925 y Fr(.)51 b(The)36 b(connected)h(sum)e Fo(M)2897 4940 y Fn(1)2937 4925 y Fr(#)p Fo(M)3112 4940 y Fn(2)3187 4925 y Fr(do)s(es)386 5042 y(not)42 b(admit)f(an)h(Engel)g(structure)i (b)s(ecause)g(its)e(Euler)g(c)m(haracteristic)g(is)g Fp(\000)p Fr(2.)386 5158 y(On)f(the)g(other)f(hand)h(the)g(Euler)g(c)m (haracteristic)f(of)g Fo(M)2513 5173 y Fn(1)2553 5158 y Fr(#)p Fo(M)2728 5173 y Fn(2)2768 5158 y Fr(#\()p Fo(S)2953 5122 y Fn(2)3020 5158 y Fp(\002)28 b Fo(S)3191 5122 y Fn(2)3231 5158 y Fr(\))40 b(is)386 5274 y(zero.)53 b(Moreo)m(v)m(er,)38 b(if)c Fo(M)1284 5289 y Fn(1)1324 5274 y Fo(;)17 b(M)1462 5289 y Fn(2)1537 5274 y Fr(ha)m(v)m(e)37 b(trivial)c(tangen)m(t)j (bundle,)h(then)f(the)g(same)386 5390 y(is)c(true)h(for)f Fo(M)933 5405 y Fn(1)973 5390 y Fr(#)p Fo(M)1148 5405 y Fn(2)1188 5390 y Fr(#\()p Fo(S)1373 5354 y Fn(2)1434 5390 y Fp(\002)23 b Fo(S)1600 5354 y Fn(2)1639 5390 y Fr(\))33 b(b)m(y)g(Theorem)g(6.2.)486 5506 y(Therefore)47 b(it)f(is)g(natural)g(to)g(try)h(to)f(construct)i(an)e(Engel)h (structure)h(on)386 5623 y Fo(M)480 5638 y Fn(1)520 5623 y Fr(#)p Fo(M)695 5638 y Fn(2)735 5623 y Fr(#\()p Fo(S)920 5587 y Fn(2)992 5623 y Fp(\002)33 b Fo(S)1168 5587 y Fn(2)1207 5623 y Fr(\))48 b(from)f Fp(D)1616 5638 y Fn(1)1703 5623 y Fr(and)h Fp(D)1985 5638 y Fn(2)2072 5623 y Fr(suc)m(h)i(that)e (the)g(resulting)f(Engel)p eop %%Page: 51 51 51 50 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(51)386 459 y Fr(structure)40 b(coincides)e(with)g(the)h(original)c (Engel)j(structures)i(one)f(large)e(parts)386 575 y(of)44 b Fo(M)603 590 y Fn(1)688 575 y Fr(and)h Fo(M)984 590 y Fn(2)1024 575 y Fr(.)80 b(In)45 b(this)f(section)h(w)m(e)h(sho)m(w)g (that)f(this)f(is)h(p)s(ossible)f(under)386 691 y(certain)32 b(assumptions)g(on)h Fp(D)1478 706 y Fn(1)1550 691 y Fr(and)f Fp(D)1816 706 y Fn(2)1856 691 y Fr(.)486 807 y(F)-8 b(or)31 b(this)i(w)m(e)g(use)h(mo)s(del)d(Engel)h(structures)j (on)d(round)h(handles)g(of)f(index)h(1)386 924 y(and)38 b(2.)60 b(As)39 b(mo)s(del)e(Engel)h(structure)h Fp(D)1944 887 y Fn(\(1\))2076 924 y Fr(on)g Fo(R)2292 939 y Fn(1)2369 924 y Fr(w)m(e)h(use)f Fp(D)2770 939 y Fn(1)2847 924 y Fr(from)e(Prop)s(o-)386 1040 y(sition)d(4.9.)51 b(On)35 b Fo(R)1097 1055 y Fn(2)1172 1040 y Fr(w)m(e)h(will)d(use)j(a)f(mo)s (del)f(Engel)h(structure)h Fp(D)2828 1004 y Fn(\(2\))2957 1040 y Fr(whic)m(h)g(did)386 1156 y(not)c(app)s(ear)h(y)m(et.)44 b(W)-8 b(e)33 b(de\014ne)h Fp(D)1612 1120 y Fn(\(2\))1739 1156 y Fr(as)e(the)h(span)g(of)679 1397 y Fo(W)42 b Fr(=)944 1329 y Fo(@)p 926 1374 93 4 v 926 1465 a(@)5 b(t)1050 1397 y Fp(\000)23 b Fr(1)p Fo(=)p Fr(2)17 b Fo(y)1362 1412 y Fn(1)1453 1329 y Fo(@)p 1410 1374 144 4 v 1410 1465 a(@)5 b(y)1514 1480 y Fn(1)1586 1397 y Fp(\000)23 b Fo(y)1734 1412 y Fn(2)1826 1329 y Fo(@)p 1783 1374 V 1783 1465 a(@)5 b(y)1887 1480 y Fn(2)1959 1397 y Fr(+)22 b(1)p Fo(=)p Fr(2)17 b Fo(x)2313 1329 y(@)p 2286 1374 113 4 v 2286 1465 a(@)5 b(x)696 1666 y(X)36 b Fr(=)27 b(cos)q(\()p Fo(t)p Fr(\))1175 1526 y Fj(\022)1248 1666 y Fo(y)1296 1681 y Fn(2)1388 1599 y Fo(@)p 1345 1644 144 4 v 1345 1735 a(@)5 b(y)1449 1750 y Fn(1)1521 1666 y Fr(+)1657 1599 y Fo(@)p 1629 1644 113 4 v 1629 1735 a(@)g(x)1751 1526 y Fj(\023)1847 1666 y Fr(+)22 b(sin)o(\()p Fo(t)p Fr(\))2192 1526 y Fj(\022)2265 1666 y Fr(1)p Fo(=)p Fr(2)17 b Fo(x)2537 1599 y(@)p 2494 1644 144 4 v 2494 1735 a(@)5 b(y)2598 1750 y Fn(1)2670 1666 y Fr(+)2821 1599 y Fo(@)p 2778 1644 V 2778 1735 a(@)g(y)2882 1750 y Fn(2)2932 1526 y Fj(\023)3054 1666 y Fo(:)386 1915 y Fr(This)31 b(mo)s(del)f(Engel)h(structure)h(can)f(b)s(e)h(obtained)e (using)h(Prop)s(osition)e(4.5.)43 b(Its)386 2031 y(c)m(haracteristic)32 b(foliation)d(is)j(spanned)i(and)f(orien)m(ted)g(b)m(y)g Fo(W)14 b Fr(.)486 2148 y(The)48 b(mo)s(del)e(Engel)h(structures)i Fp(D)1844 2111 y Fn(\(1\))1985 2148 y Fr(and)f Fp(D)2270 2111 y Fn(\(2\))2411 2148 y Fr(are)f(v)m(ery)i(similar)c(when)386 2264 y(one)37 b(iden)m(ti\014es)f Fo(R)1056 2279 y Fn(1)1133 2264 y Fr(and)g Fo(R)1400 2279 y Fn(2)1476 2264 y Fr(in)g(the)h(ob)m (vious)g(w)m(a)m(y)-8 b(.)57 b(The)37 b(curv)m(es)i Fo(\015)2921 2279 y Fk(\006)3014 2264 y Fr(=)34 b Fp(f)p Fr(0)p Fp(g)24 b(\002)386 2380 y(f\006)p Fr(1)p Fp(g)h(\002)g Fo(S)805 2344 y Fn(1)881 2380 y Fr(in)36 b Fo(@)1050 2395 y Fn(+)1110 2380 y Fo(R)1184 2395 y Fn(2)1258 2380 y Fp(')g Fo(@)1422 2395 y Fk(\000)1481 2380 y Fo(R)1555 2395 y Fn(1)1632 2380 y Fr(are)g(Legendrian)h(and)g(the)g(rotation)e(n)m(um)m(b)s(er)386 2499 y(of)g(the)h(in)m(tersection)f(line)f(\014eld)h(along)f(these)j (curv)m(es)g(is)e Fp(\000)p Fr(1)h(for)e(b)s(oth)h Fp(D)3142 2463 y Fn(\(1\))3272 2499 y Fr(on)386 2615 y Fo(R)460 2630 y Fn(1)537 2615 y Fr(and)i Fp(D)811 2579 y Fn(\(2\))943 2615 y Fr(on)g Fo(R)1157 2630 y Fn(2)1196 2615 y Fr(.)58 b(On)38 b(a)f(neigh)m(b)s(orho)s(o)s(d)f(of)h(the)g(b)s(oundary)h(of)f Fo(R)3074 2630 y Fn(1)3149 2615 y Fr(=)f Fo(R)3335 2630 y Fn(2)386 2732 y Fr(the)42 b(ev)m(en)h(con)m(tact)f(structures)i (induced)e(b)m(y)g Fp(D)2209 2695 y Fn(\(1\))2345 2732 y Fr(and)f Fp(D)2623 2695 y Fn(\(2\))2759 2732 y Fr(are)g(homotopic)386 2848 y(through)32 b(the)h(family)e(of)h(ev)m(en)i(con)m(tact)f (structures)h(de\014ned)g(b)m(y)386 3041 y(\(30\))281 b Fo(\014)896 3056 y Fl(s)961 3041 y Fr(=)27 b Fp(\000)p Fr(\(1)c Fp(\000)f Fr(2)p Fo(s)p Fr(\))p Fo(dy)1582 3056 y Fn(1)1643 3041 y Fr(+)g(1)p Fo(=)p Fr(2)17 b Fo(y)1953 3056 y Fn(1)2007 3041 y Fo(dt)22 b Fp(\000)h Fo(y)2263 3056 y Fn(2)2319 3041 y Fo(dx)f Fp(\000)h Fr(1)p Fo(=)p Fr(2)17 b Fo(x)g(dy)2882 3056 y Fn(2)386 3233 y Fr(with)40 b Fo(s)i Fp(2)h Fr([0)p Fo(;)17 b Fr(1].)68 b(By)42 b(\(20\))f(the)g (ev)m(en)i(con)m(tact)e(structure)h(on)f Fo(R)2884 3248 y Fn(1)2965 3233 y Fr(is)f(de\014ned)386 3350 y(b)m(y)i Fo(\014)585 3365 y Fn(0)666 3350 y Fr(while)f(the)h(ev)m(en)h(con)m (tact)g(structure)f(on)g Fo(R)2337 3365 y Fn(2)2418 3350 y Fr(is)f(de\014ned)i(b)m(y)g Fo(\014)3070 3365 y Fn(1)3109 3350 y Fr(.)71 b(The)386 3466 y(c)m(haracteristic)44 b(foliations)d(of)j(the)g(ev)m(en)i(con)m(tact)e(structures)i (de\014ned)g(b)m(y)f Fo(\014)3338 3481 y Fl(s)386 3582 y Fr(are)35 b(transv)m(erse)i(to)e Fo(@)1188 3597 y Fk(\000)1247 3582 y Fo(R)1321 3597 y Fn(1)1393 3582 y Fp(')d Fo(@)1553 3597 y Fn(+)1612 3582 y Fo(R)1686 3597 y Fn(2)1761 3582 y Fr(and)j Fo(@)2004 3597 y Fn(+)2064 3582 y Fo(R)2138 3597 y Fn(1)2209 3582 y Fp(')d Fo(@)2369 3597 y Fk(\000)2429 3582 y Fo(R)2503 3597 y Fn(2)2578 3582 y Fr(for)i(all)f Fo(s)p Fr(.)51 b(Hence)36 b Fo(\014)3338 3597 y Fl(s)386 3698 y Fr(induces)d(a)g(family)d(of)i(con)m(tact)h(forms)f(on)g Fo(@)2023 3713 y Fk(\000)2083 3698 y Fo(R)2157 3713 y Fn(1)2229 3698 y Fr(and)h Fo(@)2470 3713 y Fn(+)2529 3698 y Fo(R)2603 3713 y Fn(1)2643 3698 y Fr(.)486 3815 y(W)-8 b(e)24 b(can)g(also)g(obtain)f(con)m(tact)h(em)m(b)s(eddings)h (of)e Fo(@)2310 3830 y Fn(+)2370 3815 y Fo(R)2444 3830 y Fn(2)2508 3815 y Fr(from)g(attac)m(hing)g(maps)386 3931 y(of)h Fo(R)563 3946 y Fn(1)603 3931 y Fr(.)40 b(Consider)25 b(an)g(Engel)f(manifold)e Fo(M)35 b Fr(with)24 b(transv)m(erse)j(b)s (oundary)e(and)f(let)386 4047 y Fo( )449 4062 y Fn(1)516 4047 y Fr(:)k Fo(@)622 4062 y Fk(\000)682 4047 y Fo(R)756 4062 y Fn(1)823 4047 y Fp(\000)-16 b(!)27 b Fo(@)1062 4062 y Fn(+)1122 4047 y Fo(M)39 b Fr(b)s(e)28 b(an)g(em)m(b)s(edding)g (whic)m(h)g(preserv)m(es)j(orien)m(ted)d(con)m(tact)386 4163 y(structures.)49 b(By)34 b(Remark)f(4.10)h(w)m(e)g(can)g(isotop)s (e)g Fo( )2337 4178 y Fn(1)2410 4163 y Fr(suc)m(h)h(that)f(the)g(image) e(of)386 4279 y(the)h(resulting)f(em)m(b)s(edding)g(lies)g(in)g(a)h (small)d(neigh)m(b)s(ourho)s(o)s(d)i(of)g Fo( )2910 4294 y Fn(1)2950 4279 y Fr(\()p Fo(\015)3039 4294 y Fk(\006)3098 4279 y Fr(\).)44 b(The)386 4396 y(new)33 b(em)m(b)s(edding)g(is)f (again)f(called)g Fo( )1777 4411 y Fn(1)1817 4396 y Fr(.)44 b(F)-8 b(or)31 b(a)i(suitable)e(cut)i(o\013)f(function)g Fo(\032)386 4598 y Fr(\(31\))827 4572 y Fj(e)816 4598 y Fo(\014)871 4613 y Fl(s)936 4598 y Fr(=)27 b Fp(\000)p Fr(\(1)c Fp(\000)f Fr(2)p Fo(\032s)p Fr(\))p Fo(dy)1607 4613 y Fn(1)1668 4598 y Fr(+)g(1)p Fo(=)p Fr(2)17 b Fo(y)1978 4613 y Fn(1)2033 4598 y Fo(dt)22 b Fp(\000)g Fo(y)2288 4613 y Fn(2)2344 4598 y Fo(dx)g Fp(\000)h Fr(1)p Fo(=)p Fr(2)17 b Fo(x)g(dy)2907 4613 y Fn(2)386 4791 y Fr(de\014nes)45 b(a)f(family)d(of)i(con)m(tact)i(structures)g(on)f Fo(M)54 b Fr(whic)m(h)44 b(is)g(constan)m(t)g(a)m(w)m(a)m(y)386 4907 y(from)26 b(a)h(neigh)m(b)s(orho)s(o)s(d)f(of)h Fo( )1463 4922 y Fn(1)1503 4907 y Fr(\()p Fo(\015)1592 4922 y Fk(\006)1651 4907 y Fr(\))g(and)h(whic)m(h)f(is)g(de\014ned)i(b) m(y)f Fo(\014)2783 4922 y Fl(s)2848 4907 y Fr(near)f Fo( )3122 4922 y Fn(1)3162 4907 y Fr(\()p Fo(\015)3251 4922 y Fk(\006)3310 4907 y Fr(\).)386 5023 y(This)d(homotop)m(y)g (through)g(con)m(tact)g(structures)i(has)e(compact)g(supp)s(ort.)41 b(Using)386 5140 y(Gra)m(y's)31 b(theorem)f(\(Theorem)g(2.2\))g(w)m(e)h (obtain)e(an)h(isotop)m(y)g Fo(f)2668 5155 y Fl(s)2735 5140 y Fr(of)f Fo(M)41 b Fr(suc)m(h)32 b(that)386 5271 y Fo(f)445 5235 y Fk(\003)434 5296 y Fn(1)495 5245 y Fj(e)484 5271 y Fo(\014)539 5286 y Fn(1)618 5271 y Fr(is)40 b(a)f(p)s(ositiv)m(e)h(m)m(ultiple)d(of)1699 5245 y Fj(e)1688 5271 y Fo(\014)1743 5286 y Fn(0)1783 5271 y Fr(.)65 b(Moreo)m(v)m(er,) 43 b Fo(f)2387 5286 y Fl(s)2464 5271 y Fr(maps)d(the)g(Legendrian)386 5387 y(curv)m(es)35 b Fo( )748 5402 y Fn(1)787 5387 y Fr(\()p Fo(\015)876 5402 y Fk(\006)935 5387 y Fr(\))d(to)h(themselv)m (es)g(for)g(all)d Fo(s)p Fr(.)486 5506 y(The)39 b(orien)m(tation)e(of)h (the)h(con)m(tact)g(structure)h(induced)f(b)m(y)g Fp(D)2849 5470 y Fn(\(1\))2982 5506 y Fr(on)f Fo(@)3174 5521 y Fk(\000)3234 5506 y Fo(R)3308 5521 y Fn(1)3348 5506 y Fr(,)386 5623 y(resp)s(ectiv)m(ely)24 b Fp(D)988 5587 y Fn(\(2\))1104 5623 y Fr(on)e Fo(@)1280 5638 y Fn(+)1340 5623 y Fo(R)1414 5638 y Fn(2)1454 5623 y Fr(,)i(is)e(giv)m(en)h(b)m(y)g (the)g(restriction)f(of)g Fo(d\014)2785 5638 y Fn(0)2824 5623 y Fr(,)j(resp)s(ectiv)m(ely)p eop %%Page: 52 52 52 51 bop 386 259 a Fq(52)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fo(d\014)492 474 y Fn(1)531 459 y Fr(,)33 b(to)f(the)h(con)m (tact)g(structure.)45 b(This)33 b(implies)d(that)1252 626 y Fo( )1315 641 y Fn(2)1383 626 y Fr(=)d Fo(f)1534 641 y Fn(1)1596 626 y Fp(\016)22 b Fo( )1731 641 y Fn(1)1798 626 y Fr(:)28 b Fo(@)1904 641 y Fn(+)1964 626 y Fo(R)2038 641 y Fn(2)2105 626 y Fp(\000)-16 b(!)27 b Fo(@)2344 641 y Fn(+)2404 626 y Fo(M)386 794 y Fr(preserv)m(es)33 b Fm(oriente)-5 b(d)30 b Fr(con)m(tact)i(structures)g(\(but)f(the)g (orien)m(tations)e(of)i(the)g(c)m(har-)386 910 y(acteristic)k (foliations)e(do)j(not)g(matc)m(h\).)54 b(Since)36 b Fo(f)2235 925 y Fl(s)2308 910 y Fr(preserv)m(es)j(the)e(Legendrian)386 1026 y(curv)m(es)j Fo( )753 1041 y Fn(1)793 1026 y Fr(\()p Fo(\015)882 1041 y Fk(\006)941 1026 y Fr(\),)f(the)g(comp)s(osition)d Fo( )1833 1041 y Fn(2)1910 1026 y Fr(=)i Fo(f)2072 1041 y Fn(1)2137 1026 y Fp(\016)26 b Fo( )2276 1041 y Fn(1)2354 1026 y Fr(preserv)m(es)41 b(the)e(homotop)m(y)386 1145 y(class)33 b(of)f(the)h(in)m(tersection)f(line)f(\014eld)i(of)f Fp(D)2004 1109 y Fn(\(2\))2098 1145 y Fr(.)486 1261 y(W)-8 b(e)30 b(will)e(use)j(the)f(em)m(b)s(edding)g Fo( )1718 1276 y Fn(2)1788 1261 y Fr(in)f(the)i(pro)s(of)e(of)h(the)g(follo)m (wing)e(theorem.)386 1403 y Fy(Theorem)39 b(6.3.)j Fm(L)-5 b(et)37 b Fo(M)1343 1418 y Fn(1)1383 1403 y Fo(;)17 b(M)1521 1418 y Fn(2)1596 1403 y Fm(b)-5 b(e)36 b(manifolds)f(with)h(Engel)f (structur)-5 b(es)37 b Fp(D)3175 1418 y Fn(1)3215 1403 y Fo(;)17 b Fp(D)3336 1418 y Fn(2)386 1520 y Fm(such)32 b(that)g(b)-5 b(oth)31 b(char)-5 b(acteristic)32 b(foliations)e(admit)i (close)-5 b(d)31 b(tr)-5 b(ansversals.)43 b(Then)386 1636 y Fo(M)480 1651 y Fn(1)520 1636 y Fr(#)p Fo(M)695 1651 y Fn(2)735 1636 y Fr(#\()p Fo(S)920 1600 y Fn(2)968 1636 y Fp(\002)8 b Fo(S)1119 1600 y Fn(2)1159 1636 y Fr(\))29 b Fm(c)-5 b(arries)28 b(an)g(Engel)g(structur)-5 b(e)30 b(which)e(c)-5 b(oincides)27 b(with)i Fp(D)3306 1651 y Fn(1)3345 1636 y Fm(,)386 1752 y(r)-5 b(esp)g(e)g(ctively)31 b Fp(D)969 1767 y Fn(2)1008 1752 y Fm(,)h(away)g(fr)-5 b(om)31 b(a)g(neighb)-5 b(orho)g(o)g(d)30 b(of)i(the)f(tr)-5 b(ansversals)31 b(wher)-5 b(e)31 b(al)5 b(l)386 1868 y(c)-5 b(onne)g(cte)g(d)35 b(sums)g(ar)-5 b(e)35 b(p)-5 b(erforme)g(d.)46 b(The)35 b(char)-5 b(acteristic)35 b(foliation)g(of)h(the)f(new)386 1984 y(Engel)f(structur)-5 b(e)36 b(again)e(admits)h(a)f(close)-5 b(d)34 b(tr)-5 b(ansversal.)386 2163 y(Pr)g(o)g(of.)41 b Fr(Let)33 b(us)h(assume)f (for)g(the)g(momen)m(t)g(that)f(the)i(manifolds)c(are)j(orien)m(ted.) 386 2280 y(Then)e(the)g(c)m(haracteristic)f(foliations)d(are)j (canonically)e(orien)m(ted.)43 b(F)-8 b(or)29 b Fo(i)f Fr(=)f(1)p Fo(;)17 b Fr(2)386 2396 y(w)m(e)34 b(cut)f Fo(M)792 2411 y Fl(i)854 2396 y Fr(along)e(a)i(closed)g(transv)m(ersal) h Fo(N)2057 2411 y Fl(i)2118 2396 y Fr(of)f(the)g(c)m(haracteristic)g (foliation.)386 2512 y(The)26 b(resulting)e(manifolds)e(are)j(still)e (denoted)i(b)m(y)h Fo(M)2327 2527 y Fn(1)2392 2512 y Fr(and)f Fo(M)2668 2527 y Fn(2)2708 2512 y Fr(.)40 b(The)26 b(b)s(oundary)386 2628 y(of)31 b Fo(M)590 2643 y Fl(i)619 2628 y Fo(;)17 b(i)27 b Fr(=)h(1)p Fo(;)17 b Fr(2)31 b(has)h(t)m(w)m(o)h(connected)g(comp)s(onen)m(ts)g Fo(@)2399 2643 y Fn(+)2458 2628 y Fo(M)2552 2643 y Fl(i)2608 2628 y Fp(')c Fo(N)2792 2643 y Fl(i)2848 2628 y Fp(')f Fo(@)3004 2643 y Fk(\000)3063 2628 y Fo(M)3157 2643 y Fl(i)3218 2628 y Fr(and)386 2745 y(there)33 b(is)f(a)h(natural)e(iden)m (ti\014cation)1100 2912 y Fo(')c Fr(:)h Fo(@)1297 2927 y Fn(+)1357 2912 y Fo(M)1451 2927 y Fn(1)1513 2912 y Fp([)22 b Fo(@)1652 2927 y Fn(+)1712 2912 y Fo(M)1806 2927 y Fn(2)1873 2912 y Fp(\000)-16 b(!)27 b Fo(@)2112 2927 y Fk(\000)2172 2912 y Fo(M)2266 2927 y Fn(1)2328 2912 y Fp([)22 b Fo(@)2467 2927 y Fk(\000)2527 2912 y Fo(M)2621 2927 y Fn(2)386 3079 y Fr(preserving)41 b(orien)m(ted)f(con)m (tact)g(structures)i(and)e(the)h(in)m(tersection)f(line)e(\014elds)386 3196 y(together)33 b(with)f(their)g(orien)m(tations)f(if)h Fp(D)1927 3211 y Fn(1)1966 3196 y Fo(;)17 b Fp(D)2087 3211 y Fn(2)2159 3196 y Fr(are)32 b(b)s(oth)g(orien)m(ted.)486 3312 y(Let)i Fo(U)728 3327 y Fn(1)798 3312 y Fp(\032)d Fo(@)957 3327 y Fn(+)1016 3312 y Fo(M)1110 3327 y Fn(1)1184 3312 y Fr(and)j Fo(U)1441 3327 y Fn(2)1511 3312 y Fp(\032)d Fo(@)1670 3327 y Fn(+)1729 3312 y Fo(M)1823 3327 y Fn(2)1897 3312 y Fr(b)s(e)k(t)m(w)m(o)f(balls.)47 b(W)-8 b(e)34 b(will)e(only)i(need)h(to)386 3428 y(mo)s(dify)j(in)m(tersection)j (line)d(\014elds)j(on)f Fo(U)1911 3443 y Fn(1)1951 3428 y Fo(;)17 b(U)2061 3443 y Fn(2)2140 3428 y Fr(and)41 b Fo(')p Fr(\()p Fo(U)2506 3443 y Fn(1)2545 3428 y Fr(\))p Fo(;)17 b(')p Fr(\()p Fo(U)2795 3443 y Fn(2)2834 3428 y Fr(\).)67 b(Therefore)386 3544 y(it)33 b(is)g(su\016cien)m(t)i(to)e (orien)m(t)g Fp(D)1472 3559 y Fn(1)1512 3544 y Fo(;)17 b Fp(D)1633 3559 y Fn(2)1705 3544 y Fr(only)33 b(along)g Fo(U)2248 3559 y Fn(1)2287 3544 y Fo(;)17 b(U)2397 3559 y Fn(2)2437 3544 y Fr(.)46 b(F)-8 b(rom)33 b(this)g(w)m(e)i(obtain)386 3661 y(coheren)m(t)45 b(orien)m(tations)e(of)h(the)h(in)m(tersection)f (line)f(\014elds)h(on)h Fo(U)2845 3676 y Fn(1)2884 3661 y Fo(;)17 b(')p Fr(\()p Fo(U)3096 3676 y Fn(1)3135 3661 y Fr(\))45 b(and)386 3777 y Fo(U)452 3792 y Fn(2)492 3777 y Fo(;)17 b(')p Fr(\()p Fo(U)704 3792 y Fn(2)743 3777 y Fr(\).)486 3893 y(In)53 b Fo(U)694 3908 y Fn(1)734 3893 y Fo(;)17 b(U)844 3908 y Fn(2)936 3893 y Fr(w)m(e)54 b(c)m(ho)s(ose)g(Legendrian)f(unknots)h Fo(K)2431 3908 y Fn(1)2471 3893 y Fo(;)17 b(K)2598 3908 y Fn(2)2690 3893 y Fr(with)52 b(Th)m(urston{)386 4009 y(Bennequin)d(in)m(v)-5 b(arian)m(t)46 b Fp(\000)p Fr(2)i(and)g(rotation)e(n)m(um)m(b)s(er)i Fp(\000)p Fr(1.)89 b(One)48 b(can)g(obtain)386 4126 y Fo(K)469 4141 y Fn(1)508 4126 y Fo(;)17 b(K)635 4141 y Fn(2)720 4126 y Fr(b)m(y)46 b(negativ)m(e)f(stabilization)e(of)h(the) i(standard)g(Legendrian)f(unknot)386 4242 y(with)32 b(Th)m (urston{Bennequin)j(in)m(v)-5 b(arian)m(t)31 b Fp(\000)p Fr(1,)i(cf.)g(\(10\))o(.)486 4358 y(Let)c Fo( )720 4373 y Fn(1)789 4358 y Fr(b)s(e)h(an)f(em)m(b)s(edding)g(of)g Fo(@)1700 4373 y Fk(\000)1760 4358 y Fo(R)1834 4373 y Fn(1)1903 4358 y Fr(in)m(to)f Fo(@)2148 4373 y Fn(+)2208 4358 y Fo(M)2302 4373 y Fn(1)2357 4358 y Fp([)16 b Fo(@)2490 4373 y Fn(+)2550 4358 y Fo(M)2644 4373 y Fn(2)2713 4358 y Fr(whic)m(h)30 b(preserv)m(es)386 4474 y(orien)m(ted)g(con)m(tact)i (structures)g(suc)m(h)g(that)e Fo( )2047 4489 y Fn(1)2117 4474 y Fr(maps)g Fo(\015)2421 4489 y Fn(+)2510 4474 y Fr(to)h Fo(K)2711 4489 y Fn(1)2780 4474 y Fr(and)g Fo(\015)3019 4489 y Fk(\000)3108 4474 y Fr(to)f Fo(K)3308 4489 y Fn(2)3348 4474 y Fr(.)386 4590 y(Because)37 b(the)f(rotation)e(n)m(um)m(b)s(ers)j (along)d Fo(\015)2021 4605 y Fk(\006)2115 4590 y Fr(and)i Fo(K)2391 4605 y Fn(1)2430 4590 y Fo(;)17 b(K)2557 4605 y Fn(2)2632 4590 y Fr(equal)35 b Fp(\000)p Fr(1,)i(w)m(e)f(can)386 4707 y(apply)44 b(v)m(ertical)f(mo)s(di\014cation)f(to)i Fo(@)1779 4722 y Fn(+)1838 4707 y Fo(M)1932 4722 y Fn(1)2002 4707 y Fp([)31 b Fo(@)2150 4722 y Fn(+)2209 4707 y Fo(M)2303 4722 y Fn(2)2387 4707 y Fr(suc)m(h)46 b(that)e Fo( )2905 4722 y Fn(1)2989 4707 y Fr(preserv)m(es)386 4823 y(orien)m(ted)33 b(in)m(tersection)f(line)f(\014elds.)486 4939 y(Using)h(the)h (discussion)g(ab)s(o)m(v)m(e)g(w)m(e)h(obtain)d(an)i(attac)m(hing)e (map)1268 5107 y Fo( )1331 5122 y Fn(2)1398 5107 y Fr(:)d Fo(@)1504 5122 y Fn(+)1564 5107 y Fo(R)1638 5122 y Fn(2)1705 5107 y Fp(\000)-16 b(!)27 b Fo(@)1944 5122 y Fk(\000)2004 5107 y Fo(M)2098 5122 y Fn(1)2160 5107 y Fp([)22 b Fo(@)2299 5122 y Fk(\000)2359 5107 y Fo(M)2453 5122 y Fn(2)386 5274 y Fr(for)39 b Fo(R)616 5289 y Fn(2)696 5274 y Fr(with)h(the)g(mo)s (del)f(Engel)h(structure)h Fp(D)2184 5238 y Fn(\(2\))2318 5274 y Fr(whic)m(h)g(preserv)m(es)i(orien)m(ted)386 5390 y(con)m(tact)29 b(structures)i(and)e(the)g(homotop)m(y)g(t)m(yp)s(e)h (of)e(the)h(in)m(tersection)g(line)f(\014eld.)386 5506 y(Here)40 b(w)m(e)h(do)e(not)h(stic)m(k)g(to)f(the)h(con)m(v)m(en)m (tion)h(that)e Fo(R)2420 5521 y Fn(2)2499 5506 y Fr(is)g(attac)m(hed)i (using)e(an)386 5623 y(em)m(b)s(edding)26 b(of)g Fo(@)1029 5638 y Fk(\000)1089 5623 y Fo(R)1163 5638 y Fn(2)1229 5623 y Fr(but)h(notice)f(that)h(the)g(orien)m(tation)e(of)h(the)h(c)m (haracteristic)p eop %%Page: 53 53 53 52 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(53)386 459 y Fr(foliations)20 b(matc)m(h.)40 b(After)24 b(a)f(suitable)f(v)m(ertical)h(mo)s(di\014cation)e(of)i(the)g(b)s (oundary)-8 b(,)386 575 y(w)m(e)34 b(obtain)d(an)i(Engel)f(structure)i (on)1158 763 y Fo(M)39 b Fr(=)27 b(\(\()p Fo(M)1564 778 y Fn(1)1626 763 y Fp([)c Fo(M)1809 778 y Fn(2)1848 763 y Fr(\))f Fp([)1974 778 y Fl( )2020 787 y Fg(1)2082 763 y Fo(R)2156 778 y Fn(1)2196 763 y Fr(\))g Fp([)2322 778 y Fl( )2368 787 y Fg(2)2429 763 y Fo(R)2503 778 y Fn(2)2575 763 y Fo(:)386 951 y Fr(Next)38 b(w)m(e)h(w)m(an)m(t)g(to)e(iden)m (tify)g Fo(@)1557 966 y Fn(+)1617 951 y Fo(M)48 b Fr(with)38 b Fo(@)2038 966 y Fk(\000)2097 951 y Fo(M)49 b Fr(suc)m(h)39 b(that)e(orien)m(ted)h(con)m(tact)386 1067 y(structures)47 b(and)e(the)g(homotop)m(y)g(class)g(of)f(the)i(in)m(tersection)f(line)e (\014elds)j(are)386 1183 y(preserv)m(ed.)f(Aw)m(a)m(y)32 b(from)d Fo(U)1420 1198 y Fn(1)1477 1183 y Fp([)18 b Fo(U)1627 1198 y Fn(2)1697 1183 y Fr(w)m(e)31 b(can)g(use)g Fo(')p Fr(.)43 b(This)30 b(map)g(has)g(an)g(ob)m(vious)386 1300 y(extension)39 b(to)f(a)g(map)f Fo(@)1308 1315 y Fk(\000)1368 1300 y Fo(M)48 b Fp(\000)-16 b(!)37 b Fo(@)1759 1315 y Fn(+)1818 1300 y Fo(M)49 b Fr(whic)m(h)39 b(coincides)f(with)g (the)h(natural)386 1416 y(iden)m(ti\014cation)31 b(of)h Fo(@)1136 1431 y Fn(+)1196 1416 y Fo(R)1270 1431 y Fn(1)1337 1416 y Fp(\032)c Fo(@)1493 1431 y Fn(+)1553 1416 y Fo(M)43 b Fr(with)32 b Fo(@)1963 1431 y Fk(\000)2023 1416 y Fo(R)2097 1431 y Fn(2)2164 1416 y Fp(\032)c Fo(@)2320 1431 y Fk(\000)2380 1416 y Fo(M)10 b Fr(.)486 1532 y(Unfortunately)27 b(the)g(con)m(tact)h (structures)h(on)e Fo(@)2245 1547 y Fn(+)2305 1532 y Fo(R)2379 1547 y Fn(1)2446 1532 y Fr(and)g Fo(@)2681 1547 y Fk(\000)2740 1532 y Fo(R)2814 1547 y Fn(2)2881 1532 y Fr(are)h(not)f(pre-)386 1648 y(serv)m(ed)38 b(b)m(y)f(this)f (iden)m(ti\014cation.)53 b(Ho)m(w)m(ev)m(er,)40 b(the)c(family)e(of)i (distributions)f(de-)386 1764 y(\014ned)g(b)m(y)i(\(31\))d(a)m(w)m(a)m (y)i(from)d Fo(@)1502 1779 y Fn(+)1562 1764 y Fo(R)1636 1779 y Fn(1)1710 1764 y Fr(together)i(with)f(the)h(restriction)f(of)41 b(\(30\))34 b(to)386 1881 y Fo(@)437 1896 y Fn(+)496 1881 y Fo(R)570 1896 y Fn(1)638 1881 y Fp(')28 b Fo(@)794 1896 y Fk(\000)854 1881 y Fo(R)928 1896 y Fn(2)967 1881 y Fr(,)j(is)f(a)h(family)d(of)i(con)m(tact)h(structures.)45 b(By)31 b(Theorem)g(2.2)f(there)386 1997 y(is)i(an)g(iden)m (ti\014cation)1496 2144 y Fj(e)-67 b Fo(')27 b Fr(:)h Fo(@)1681 2159 y Fk(\000)1741 2144 y Fo(M)38 b Fp(\000)-16 b(!)27 b Fo(@)2112 2159 y Fn(+)2172 2144 y Fo(M)386 2312 y Fr(whic)m(h)38 b(preserv)m(es)j(orien)m(ted)d(con)m(tact)h (structures)h(and)e(whic)m(h)g(coincides)g(with)386 2428 y Fo(')32 b Fr(a)m(w)m(a)m(y)i(from)e(a)g(neigh)m(b)s(orho)s(o)s(d)f (of)h(the)h(attac)m(hing)f(region)g(of)g Fo(R)2838 2443 y Fn(1)2878 2428 y Fr(.)486 2544 y(Next)26 b(w)m(e)h(compare)e(the)h (in)m(tersection)g(line)f(\014elds)h(on)f Fo(@)2515 2559 y Fn(+)2575 2544 y Fo(M)36 b Fr(and)26 b Fo(@)2939 2559 y Fk(\000)2999 2544 y Fo(M)10 b Fr(.)42 b(Near)386 2660 y Fo(T)457 2624 y Fn(2)443 2685 y(0)536 2660 y Fr(=)e Fp(f)p Fo(x)g Fr(=)g(0)p Fp(g)g(\032)g Fo(@)1220 2675 y Fn(+)1280 2660 y Fo(R)1354 2675 y Fn(1)1433 2660 y Fr(the)h(Gra)m(y)e(isotop)m(y)h(constructed)i(in)d(the)h(pro)s(of)f(of) 386 2777 y(Theorem)33 b(2.2)f(is)g(induced)h(b)m(y)h(the)f(Legendrian)f (v)m(ector)i(\014eld)386 3021 y(\(32\))655 b Fo(Z)34 b Fr(=)1492 2953 y(8)p Fo(y)1589 2968 y Fn(2)p 1430 2998 260 4 v 1430 3089 a Fr(1)22 b(+)g Fo(y)1651 3055 y Fn(2)1647 3113 y(2)1716 2880 y Fj(\022)1789 3021 y Fo(y)1837 3036 y Fn(2)1904 2953 y Fo(@)p 1886 2998 93 4 v 1886 3089 a(@)5 b(t)2010 3021 y Fr(+)2118 2953 y(1)p 2118 2998 49 4 v 2118 3089 a(2)2177 3021 y Fo(y)2225 3036 y Fn(1)2302 2953 y Fo(@)p 2274 2998 113 4 v 2274 3089 a(@)g(x)2396 2880 y Fj(\023)2519 3021 y Fo(:)386 3270 y Fr(F)-8 b(rom)27 b(\(32\))h(one)g(sees)i(that)41 b Fj(e)-68 b Fo(')29 b Fr(maps)f(the)h(Legendrian)f(curv)m(es)i Fp(f)p Fo(x)e Fr(=)g Fo(y)3001 3285 y Fn(1)3067 3270 y Fr(=)g(0)p Fp(g)f(\032)386 3386 y Fo(@)437 3401 y Fn(+)496 3386 y Fo(R)570 3401 y Fn(1)640 3386 y Fr(to)j(themselv)m(es.)44 b(The)31 b(rotation)d(n)m(um)m(b)s(er)j(along)e(these)i(curv)m(es)h(is)e Fp(\000)p Fr(1)g(for)386 3502 y(b)s(oth)i(mo)s(del)f(Engel)h (structures)j Fp(D)1715 3466 y Fn(\(1\))1841 3502 y Fr(and)e Fp(D)2111 3466 y Fn(\(2\))2205 3502 y Fr(.)486 3619 y(The)45 b(rotation)e(n)m(um)m(b)s(er)i(of)f(the)g(in)m(tersection)h(line)e (\014eld)h(with)g(resp)s(ect)i(to)386 3735 y(the)32 b(Legendrian)f (line)g(\014eld)g Fp(L)h Fr(spanned)h(b)m(y)f Fo(y)2113 3750 y Fn(2)2152 3735 y Fo(@)2203 3750 y Fl(t)2254 3735 y Fr(+)20 b(1)p Fo(=)p Fr(2)d Fo(y)2562 3750 y Fn(1)2600 3735 y Fo(@)2651 3750 y Fl(x)2727 3735 y Fr(is)31 b(zero)h(for)f(b)s (oth)386 3854 y Fp(D)466 3818 y Fn(\(1\))596 3854 y Fr(and)37 b Fp(D)870 3818 y Fn(\(2\))1000 3854 y Fr(\(cf.)f(Prop)s(osition)f (4.9\).)54 b(Since)37 b Fo(Z)43 b Fr(is)36 b(tangen)m(t)g(to)g Fp(L)p Fr(,)h(the)g(\015o)m(w)386 3970 y(of)e(this)h(v)m(ector)g (\014eld)g(preserv)m(es)i Fp(L)p Fr(.)53 b(This)36 b(implies)d(that)j (the)g(iden)m(ti\014cation)46 b Fj(e)-67 b Fo(')386 4087 y Fr(preserv)m(es)35 b(the)e(homotop)m(y)g(class)f(of)g(the)h(in)m (tersection)g(line)e(\014eld)h(along)g Fo(T)3186 4050 y Fn(2)3172 4111 y(0)3225 4087 y Fr(.)486 4203 y(No)m(w)39 b(let)f Fo(\015)43 b Fr(b)s(e)c(a)g(curv)m(e)h(in)e Fo(@)1619 4218 y Fn(+)1678 4203 y Fo(M)50 b Fr(whic)m(h)39 b(in)m(tersects)h Fo(T)2616 4167 y Fn(2)2602 4227 y(0)2693 4203 y Fp(\032)e Fo(@)2859 4218 y Fn(+)2919 4203 y Fo(R)2993 4218 y Fn(1)3071 4203 y Fr(exactly)386 4319 y(once)30 b(suc)m(h)i(that)d(the)i(endp)s (oin)m(ts)f(of)f Fo(\015)35 b Fr(lie)29 b(outside)g(of)h Fo(U)2469 4334 y Fn(1)2538 4319 y Fr(and)g Fo(U)2791 4334 y Fn(2)2831 4319 y Fr(.)43 b(W)-8 b(e)30 b(ha)m(v)m(e)h(to)386 4435 y(ensure)c(that)f(the)g(in)m(tersection)g(line)e(\014elds)j(are)e (homotopic)g(along)f Fo(\015)31 b Fr(relativ)m(e)25 b(to)386 4551 y(the)f(endp)s(oin)m(ts.)41 b(F)-8 b(or)22 b(this)i(w)m(e)g(use)h (the)e(\015exibilit)m(y)f(men)m(tioned)h(in)g(Remark)g(2.21:)386 4668 y(Cho)s(osing)g(suitable)h(v)-5 b(alues)23 b(of)h Fo(k)j Fr(for)d(the)g(v)m(ertical)f(mo)s(di\014cation)f(of)h(the)i(b)s (ound-)386 4784 y(ary)33 b(when)h(w)m(e)g(attac)m(h)e Fo(R)1330 4799 y Fn(2)1370 4784 y Fr(,)h(w)m(e)h(can)f(ensure)h(that)45 b Fj(e)-67 b Fo(')32 b Fr(preserv)m(es)k(the)d(homotop)m(y)386 4900 y(class)45 b(of)g(the)h(in)m(tersection)f(line)f(\014eld)h(along)f Fo(\015)50 b Fr(relativ)m(e)45 b(to)g(the)g(endp)s(oin)m(ts.)386 5016 y(Then)h Fj(e)-67 b Fo(')32 b Fr(preserv)m(es)j(the)e(homotop)m(y) g(class)f(of)g(the)h(in)m(tersection)g(line)e(\014eld.)486 5133 y(After)36 b(a)f(v)m(ertical)h(mo)s(di\014cation)d(of)j(the)g(b)s (oundary)h(w)m(e)g(can)f(iden)m(tify)f Fo(@)3210 5148 y Fn(+)3270 5133 y Fo(M)386 5249 y Fr(with)48 b Fo(@)675 5264 y Fk(\000)735 5249 y Fo(M)59 b Fr(using)i Fj(e)-68 b Fo(')49 b Fr(suc)m(h)h(that)e(w)m(e)i(obtain)d(an)i(Engel)f (structure)i(on)e(the)386 5379 y(resulting)29 b(manifold)1197 5354 y Fj(f)1186 5379 y Fo(M)11 b Fr(.)43 b(The)31 b(new)g(Engel)e (structure)j(coincides)e(with)f Fp(D)3148 5394 y Fn(1)3218 5379 y Fr(and)386 5495 y Fp(D)463 5510 y Fn(2)549 5495 y Fr(a)m(w)m(a)m(y)49 b(from)d(su\016cien)m(tly)i(big)e(neigh)m(b)s (orho)s(o)s(ds)g(of)h Fo(N)2602 5510 y Fn(1)2689 5495 y Fr(and)g Fo(N)2971 5510 y Fn(2)3011 5495 y Fr(.)87 b(These)386 5623 y(transv)m(erse)35 b(h)m(yp)s(ersurfaces)g(are)d(also) g(con)m(tained)h(in)2380 5598 y Fj(f)2369 5623 y Fo(M)11 b Fr(.)p eop %%Page: 54 54 54 53 bop 386 259 a Fq(54)1096 b(THOMAS)25 b(V)n(OGEL)486 476 y Fr(It)34 b(remains)f(to)h(sho)m(w)i(that)1546 450 y Fj(f)1536 476 y Fo(M)45 b Fr(is)34 b(di\013eomorphic)e(to)i Fo(M)2601 491 y Fn(1)2641 476 y Fr(#)p Fo(M)2816 491 y Fn(2)2856 476 y Fr(#\()p Fo(S)3041 439 y Fn(2)3104 476 y Fp(\002)23 b Fo(S)3270 439 y Fn(2)3310 476 y Fr(\).)386 592 y(In)34 b(order)f(to)h(pro)m(v)m(e)g(this,)g(w)m(e)g(decomp)s(ose)g Fo(R)2077 607 y Fn(1)2146 592 y Fr(=)29 b Fo(h)2307 607 y Fn(1)2370 592 y Fp([)23 b Fo(h)2515 607 y Fn(2)2588 592 y Fr(and)33 b Fo(R)2852 607 y Fn(2)2921 592 y Fr(=)c Fo(h)3082 556 y Fk(0)3082 616 y Fn(2)3145 592 y Fp([)23 b Fo(R)3308 607 y Fn(3)3348 592 y Fr(,)386 708 y(where)39 b Fo(h)729 723 y Fn(1)769 708 y Fo(;)17 b(h)869 723 y Fn(2)908 708 y Fo(;)g(h)1008 672 y Fk(0)1008 733 y Fn(2)1048 708 y Fo(;)g(h)1148 723 y Fn(3)1225 708 y Fr(are)39 b(ordinary)e (handles)i(of)f(index)g(1)p Fo(;)17 b Fr(2)p Fo(;)g Fr(2)p Fo(;)g Fr(3,)38 b(as)h(in)e(\(16\).)386 824 y(The)f(left)f(part)g(of)g (Figure)f(14)h(sho)m(ws)i(the)e(attac)m(hing)g(curv)m(e)i(and)e(the)h (framing)386 941 y(of)e Fo(h)555 956 y Fn(2)630 941 y Fr(after)h Fo(h)919 956 y Fn(1)993 941 y Fr(is)g(attac)m(hed.)51 b(One)35 b(end)h(of)e Fo(h)2101 956 y Fn(1)2176 941 y Fr(lies)g(in)g Fo(U)2529 956 y Fn(1)2603 941 y Fr(and)i(the)f(other)g (end)386 1057 y(lies)c(in)h Fo(U)734 1072 y Fn(2)774 1057 y Fr(.)486 1173 y(Recall)45 b(that)h Fo(R)1093 1188 y Fn(1)1179 1173 y Fr(and)h Fo(R)1457 1188 y Fn(2)1543 1173 y Fr(are)g(attac)m(hed)g(to)f Fo(M)2359 1188 y Fn(1)2431 1173 y Fp([)32 b Fo(M)2623 1188 y Fn(2)2709 1173 y Fr(in)46 b(a)g(symmetric)386 1289 y(w)m(a)m(y)-8 b(.)44 b(If)30 b(w)m(e)h(discard)e Fo(h)1238 1304 y Fn(2)1308 1289 y Fr(and)h Fo(h)1551 1253 y Fk(0)1551 1314 y Fn(2)1621 1289 y Fr(and)g(iden)m(tify)f(the)h(b)s(oundary)h(comp)s(onen)m(ts)f (of)386 1405 y Fo(M)480 1420 y Fn(1)542 1405 y Fp([)22 b Fo(M)724 1420 y Fn(2)797 1405 y Fr(with)32 b Fo(h)1075 1420 y Fn(1)1115 1405 y Fo(;)17 b(h)1215 1420 y Fn(3)1286 1405 y Fr(attac)m(hed,)34 b(then)f(w)m(e)h(obtain)d Fo(M)2475 1420 y Fn(1)2515 1405 y Fr(#)p Fo(M)2690 1420 y Fn(2)2730 1405 y Fr(.)486 1522 y(Using)25 b(an)h(isotop)m(y)h(of)e(the)i(attac)m (hing)e(curv)m(es)k(of)c Fo(h)2356 1537 y Fn(2)2422 1522 y Fr(w)m(e)i(can)g(separate)g Fo(h)3167 1537 y Fn(2)3206 1522 y Fr(,)h(re-)386 1638 y(sp)s(ectiv)m(ely)d Fo(h)884 1602 y Fk(0)884 1663 y Fn(2)924 1638 y Fr(,)h(from)d Fo(h)1255 1653 y Fn(1)1294 1638 y Fr(,)j(resp)s(ectiv)m(ely)f Fo(h)1926 1653 y Fn(3)1966 1638 y Fr(,)h(suc)m(h)g(that)e Fo(h)2490 1653 y Fn(2)2553 1638 y Fr(and)h Fo(h)2791 1602 y Fk(0)2791 1663 y Fn(2)2855 1638 y Fr(are)f(attac)m(hed)386 1754 y(along)k(symmetric)g(unknots)i(in)e Fo(@)1643 1769 y Fn(+)1703 1754 y Fo(M)1797 1769 y Fn(1)1852 1754 y Fp([)15 b Fo(@)1984 1769 y Fn(+)2044 1754 y Fo(M)2138 1769 y Fn(2)2178 1754 y Fr(,)29 b(resp)s(ectiv)m(ely)h Fo(@)2813 1769 y Fk(\000)2873 1754 y Fo(M)2967 1769 y Fn(1)3022 1754 y Fp([)15 b Fo(@)3154 1769 y Fk(\000)3214 1754 y Fo(M)3308 1769 y Fn(2)3348 1754 y Fr(,)386 1870 y(whic)m(h)33 b(do)g(not)f(meet)g(the)h(attac)m(hing)f(region)g(of)g Fo(h)2271 1885 y Fn(1)2311 1870 y Fr(,)g(resp)s(ectiv)m(ely)i Fo(h)2958 1885 y Fn(3)2997 1870 y Fr(.)446 2908 y @beginspecial 0 @llx 0 @lly 395 @urx 112 @ury 3443 @rwi @setspecial %%BeginDocument: unknotkirby.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: unknotkirby.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta3 %%CreationDate: Wed Oct 13 11:12:51 2004 %%For: tvogel@math2h (T. Vogel) %%Orientation: Portrait %%BoundingBox: 0 0 395 112 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -4.0 117.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /endash 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 2947 m -1000 -1000 l 7650 -1000 l 7650 2947 l cp clip 0.06000 0.06000 sc /Times-Roman-iso ff 165.00 scf sf 3195 1830 m gs 1 -1 sc (2) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 3061 1750 m gs 1 -1 sc (U) col0 sh gr /Times-Roman-iso ff 165.00 scf sf 480 1800 m gs 1 -1 sc (1) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 358 1706 m gs 1 -1 sc (U) col0 sh gr % Arc 7.500 slw gs n 6342.3 444.0 349.1 125.8 51.2 arcn gs col0 s gr gr % Arc gs [60] 0 sd n 6353.0 1047.1 392.3 -123.2 -58.7 arc gs col0 s gr gr [] 0 sd % Ellipse n 6346 716 214 215 0 360 DrawEllipse gs col0 s gr % Arc gs n 4774.3 443.9 349.1 125.8 51.2 arcn gs col0 s gr gr % Arc gs [60] 0 sd n 4784.5 1045.7 390.8 -123.3 -58.6 arc gs col0 s gr gr [] 0 sd % Ellipse n 4777 716 214 215 0 360 DrawEllipse gs col0 s gr /Times-Roman-iso ff 165.00 scf sf 5138 394 m gs 1 -1 sc (1) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 5016 300 m gs 1 -1 sc (U) col0 sh gr /Times-Roman-iso ff 165.00 scf sf 6048 394 m gs 1 -1 sc (2) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 5914 314 m gs 1 -1 sc (U) col0 sh gr % Polyline [60] 0 sd n 5566 99 m 5566 1935 l gs col0 s gr [] 0 sd % Polyline n 6297 1585 m 6298 1584 l 6301 1580 l 6307 1571 l 6313 1562 l 6319 1554 l 6323 1547 l 6327 1540 l 6330 1534 l 6334 1528 l 6337 1521 l 6340 1514 l 6342 1505 l 6343 1497 l 6343 1489 l 6344 1479 l 6343 1470 l 6343 1460 l 6343 1450 l 6343 1441 l 6342 1433 l 6341 1423 l 6339 1414 l 6337 1405 l 6335 1397 l 6333 1388 l 6330 1380 l 6327 1372 l 6324 1364 l 6320 1356 l 6317 1349 l 6312 1341 l 6307 1332 l 6302 1325 l 6296 1317 l 6291 1309 l 6285 1301 l 6278 1292 l 6271 1284 l 6264 1277 l 6256 1270 l 6247 1264 l 6237 1259 l 6226 1253 l 6214 1249 l 6203 1244 l 6191 1239 l 6180 1235 l 6169 1232 l 6158 1229 l 6147 1227 l 6136 1226 l 6125 1225 l 6114 1223 l 6103 1223 l 6093 1222 l 6083 1223 l 6073 1225 l 6064 1227 l 6056 1230 l 6047 1233 l 6039 1237 l 6030 1241 l 6022 1245 l 6013 1249 l 6004 1254 l 5996 1259 l 5987 1265 l 5978 1270 l 5970 1276 l 5961 1282 l 5953 1289 l 5946 1297 l 5939 1306 l 5933 1316 l 5926 1326 l 5920 1337 l 5914 1348 l 5908 1359 l 5903 1370 l 5899 1381 l 5896 1392 l 5894 1404 l 5892 1415 l 5891 1427 l 5890 1439 l 5889 1451 l 5888 1462 l 5888 1473 l 5889 1483 l 5890 1494 l 5892 1504 l 5894 1514 l 5896 1524 l 5899 1533 l 5901 1542 l 5903 1551 l 5906 1561 l 5909 1571 l 5913 1580 l 5916 1588 l 5920 1597 l 5925 1605 l 5931 1613 l 5937 1621 l 5943 1629 l 5950 1637 l 5958 1644 l 5966 1651 l 5973 1655 l 5981 1660 l 5989 1663 l 5998 1667 l 6006 1671 l 6015 1674 l 6024 1677 l 6033 1680 l 6042 1682 l 6052 1684 l 6063 1685 l 6074 1686 l 6085 1687 l 6095 1688 l 6104 1688 l 6113 1689 l 6123 1689 l 6132 1689 l 6140 1689 l 6147 1688 l 6155 1687 l 6163 1686 l 6172 1684 l 6182 1680 l 6194 1675 l 6204 1671 l 6210 1668 l 6211 1668 l gs col0 s gr % Polyline n 6214 1304 m 6213 1305 l 6210 1312 l 6204 1324 l 6197 1337 l 6192 1348 l 6188 1357 l 6185 1365 l 6182 1372 l 6179 1378 l 6177 1385 l 6175 1392 l 6173 1401 l 6172 1409 l 6172 1418 l 6172 1427 l 6172 1438 l 6172 1448 l 6172 1458 l 6172 1467 l 6173 1475 l 6175 1485 l 6177 1494 l 6179 1502 l 6181 1510 l 6184 1518 l 6187 1526 l 6190 1534 l 6194 1542 l 6197 1549 l 6200 1557 l 6205 1565 l 6210 1574 l 6215 1581 l 6220 1589 l 6226 1597 l 6232 1605 l 6238 1614 l 6245 1622 l 6252 1629 l 6260 1636 l 6269 1642 l 6280 1648 l 6291 1653 l 6302 1658 l 6314 1663 l 6326 1668 l 6337 1672 l 6348 1675 l 6359 1678 l 6370 1679 l 6381 1681 l 6392 1682 l 6403 1683 l 6414 1684 l 6424 1684 l 6434 1683 l 6443 1681 l 6452 1679 l 6461 1676 l 6469 1673 l 6477 1669 l 6485 1665 l 6494 1661 l 6502 1657 l 6511 1652 l 6519 1647 l 6528 1641 l 6537 1636 l 6546 1630 l 6554 1624 l 6562 1617 l 6570 1609 l 6577 1600 l 6584 1591 l 6590 1580 l 6596 1570 l 6602 1559 l 6608 1548 l 6613 1537 l 6617 1526 l 6620 1515 l 6622 1503 l 6624 1492 l 6625 1480 l 6626 1468 l 6627 1456 l 6628 1445 l 6628 1434 l 6627 1424 l 6626 1413 l 6625 1403 l 6623 1393 l 6620 1383 l 6618 1374 l 6616 1365 l 6614 1356 l 6611 1346 l 6607 1336 l 6604 1327 l 6600 1318 l 6596 1310 l 6591 1301 l 6585 1293 l 6579 1285 l 6573 1277 l 6566 1269 l 6558 1262 l 6550 1255 l 6543 1251 l 6535 1247 l 6527 1243 l 6518 1240 l 6509 1236 l 6500 1233 l 6492 1230 l 6484 1227 l 6473 1224 l 6463 1221 l 6453 1218 l 6443 1216 l 6433 1213 l 6422 1212 l 6414 1211 l 6405 1211 l 6396 1212 l 6387 1212 l 6377 1212 l 6368 1213 l 6359 1214 l 6351 1215 l 6343 1217 l 6333 1220 l 6323 1223 l 6311 1227 l 6301 1231 l 6293 1234 l 6290 1236 l 6289 1236 l gs col0 s gr /Times-Roman-iso ff 210.00 scf sf 5817 1211 m gs 1 -1 sc (0) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 6420 1841 m gs 1 -1 sc (-4) col0 sh gr % Polyline n 939 620 m 2712 620 l gs col0 s gr % Polyline n 929 909 m 2756 909 l gs col0 s gr % Polyline [60] 0 sd n 957 1007 m 2698 1007 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 939 490 m 2680 490 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 3444 878 m 3444 879 l 3441 882 l 3436 890 l 3429 902 l 3419 917 l 3410 933 l 3403 949 l 3399 963 l 3397 975 l 3399 986 l 3403 995 l 3410 1005 l 3419 1014 l 3430 1023 l 3442 1033 l 3455 1042 l 3469 1052 l 3481 1061 l 3493 1071 l 3503 1080 l 3510 1090 l 3516 1099 l 3521 1110 l 3524 1122 l 3525 1133 l 3525 1145 l 3525 1156 l 3524 1168 l 3523 1180 l 3521 1191 l 3519 1203 l 3516 1214 l 3512 1225 l 3506 1237 l 3499 1249 l 3492 1260 l 3484 1272 l 3476 1284 l 3468 1295 l 3460 1307 l 3451 1317 l 3442 1328 l 3434 1336 l 3425 1345 l 3417 1353 l 3407 1362 l 3398 1370 l 3389 1378 l 3379 1387 l 3369 1394 l 3359 1402 l 3349 1409 l 3339 1415 l 3328 1420 l 3317 1424 l 3305 1428 l 3292 1431 l 3279 1433 l 3266 1435 l 3252 1436 l 3238 1438 l 3225 1439 l 3212 1440 l 3199 1441 l 3187 1442 l 3176 1442 l 3163 1442 l 3151 1441 l 3139 1439 l 3128 1437 l 3117 1435 l 3107 1433 l 3096 1430 l 3085 1427 l 3074 1424 l 3064 1420 l 3054 1415 l 3044 1409 l 3034 1403 l 3024 1396 l 3014 1389 l 3004 1382 l 2995 1375 l 2986 1367 l 2977 1360 l 2969 1352 l 2962 1344 l 2955 1335 l 2950 1326 l 2944 1317 l 2939 1308 l 2934 1299 l 2929 1290 l 2924 1280 l 2919 1270 l 2913 1260 l 2907 1249 l 2901 1238 l 2895 1227 l 2889 1215 l 2883 1203 l 2876 1191 l 2870 1179 l 2864 1168 l 2859 1156 l 2853 1145 l 2848 1134 l 2843 1123 l 2838 1112 l 2833 1100 l 2828 1089 l 2824 1078 l 2819 1067 l 2814 1057 l 2809 1048 l 2804 1040 l 2797 1032 l 2791 1025 l 2784 1020 l 2777 1015 l 2770 1011 l 2763 1007 l 2756 1003 l 2750 1001 l 2741 1000 l 2731 1002 l 2720 1007 l 2710 1011 l 2704 1014 l 2703 1014 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 950 1001 m 949 1001 l 946 1003 l 939 1007 l 927 1014 l 913 1022 l 898 1032 l 884 1041 l 872 1051 l 862 1061 l 853 1072 l 846 1082 l 840 1094 l 834 1107 l 829 1121 l 824 1135 l 820 1150 l 815 1165 l 810 1180 l 805 1194 l 798 1209 l 791 1222 l 783 1236 l 775 1248 l 767 1259 l 759 1271 l 750 1284 l 740 1296 l 731 1309 l 722 1322 l 712 1335 l 702 1347 l 692 1359 l 681 1370 l 670 1380 l 659 1389 l 647 1397 l 634 1404 l 621 1410 l 606 1415 l 591 1419 l 576 1423 l 560 1427 l 544 1430 l 527 1433 l 511 1436 l 495 1438 l 480 1440 l 465 1442 l 451 1442 l 437 1442 l 421 1440 l 406 1437 l 391 1433 l 377 1427 l 362 1420 l 347 1414 l 333 1406 l 319 1399 l 305 1393 l 292 1386 l 280 1380 l 268 1374 l 255 1367 l 243 1360 l 233 1353 l 222 1346 l 213 1340 l 203 1333 l 194 1326 l 185 1318 l 176 1310 l 168 1300 l 161 1289 l 154 1277 l 148 1264 l 141 1249 l 136 1235 l 130 1220 l 125 1206 l 120 1192 l 117 1180 l 115 1170 l 115 1159 l 117 1148 l 122 1137 l 128 1126 l 134 1116 l 139 1109 l 142 1106 l 142 1105 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 249 1040 m 252 1037 l 257 1030 l 266 1019 l 276 1006 l 286 993 l 295 980 l 301 968 l 304 957 l 304 947 l 301 937 l 296 926 l 289 915 l 280 903 l 271 892 l 261 880 l 250 869 l 241 858 l 231 850 l 222 842 l 212 836 l 202 832 l 190 829 l 178 827 l 166 826 l 154 825 l 143 824 l 132 821 l 123 817 l 115 812 l 107 804 l 101 794 l 96 783 l 92 771 l 88 759 l 85 748 l 83 737 l 82 726 l 83 716 l 87 704 l 92 690 l 100 676 l 107 662 l 112 652 l 115 648 l 115 647 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 228 620 m 231 618 l 238 613 l 248 607 l 260 598 l 272 589 l 283 580 l 292 571 l 299 563 l 304 555 l 307 546 l 309 536 l 310 526 l 311 515 l 311 504 l 311 493 l 311 482 l 309 471 l 307 460 l 304 450 l 299 440 l 292 430 l 283 420 l 273 411 l 263 401 l 252 391 l 242 382 l 233 372 l 227 362 l 222 353 l 219 341 l 217 330 l 217 318 l 219 307 l 220 296 l 222 284 l 225 273 l 228 262 l 231 253 l 236 244 l 240 235 l 245 226 l 250 217 l 256 208 l 261 199 l 268 191 l 274 183 l 282 175 l 291 168 l 300 161 l 311 154 l 323 147 l 335 140 l 347 134 l 359 128 l 370 122 l 380 117 l 390 113 l 401 109 l 411 107 l 420 105 l 427 104 l 436 104 l 445 103 l 456 103 l 470 103 l 481 103 l 493 104 l 506 105 l 520 105 l 535 106 l 550 107 l 566 108 l 581 109 l 595 110 l 608 112 l 621 115 l 632 118 l 644 123 l 656 129 l 666 135 l 676 142 l 686 150 l 695 158 l 704 165 l 713 173 l 722 181 l 730 190 l 737 199 l 744 209 l 750 219 l 756 229 l 762 240 l 767 250 l 772 261 l 778 272 l 783 283 l 789 294 l 795 306 l 800 318 l 806 332 l 811 346 l 817 360 l 822 374 l 828 387 l 833 400 l 838 410 l 842 419 l 848 429 l 852 435 l 855 438 l 858 440 l 863 444 l 870 450 l 877 456 l 887 463 l 900 473 l 915 483 l 929 493 l 939 500 l 944 503 l 945 504 l gs col0 s gr [] 0 sd % Polyline n 924 902 m 923 902 l 918 901 l 910 900 l 899 900 l 890 902 l 880 909 l 873 916 l 866 924 l 859 933 l 851 943 l 842 953 l 834 964 l 825 976 l 817 988 l 809 1000 l 800 1014 l 794 1025 l 787 1036 l 781 1049 l 775 1062 l 768 1076 l 762 1090 l 755 1104 l 749 1119 l 742 1133 l 735 1147 l 729 1160 l 722 1173 l 714 1185 l 707 1196 l 698 1208 l 689 1220 l 680 1231 l 671 1242 l 662 1253 l 654 1263 l 645 1273 l 635 1283 l 625 1292 l 614 1301 l 602 1309 l 588 1315 l 575 1319 l 561 1323 l 546 1326 l 529 1329 l 512 1332 l 494 1334 l 476 1336 l 457 1338 l 439 1339 l 422 1340 l 406 1340 l 390 1339 l 376 1337 l 363 1334 l 349 1329 l 337 1322 l 325 1313 l 314 1303 l 304 1292 l 294 1281 l 284 1270 l 274 1258 l 265 1246 l 256 1234 l 248 1222 l 240 1210 l 233 1197 l 227 1183 l 221 1169 l 216 1153 l 211 1137 l 206 1121 l 202 1105 l 198 1089 l 194 1073 l 190 1059 l 187 1045 l 184 1033 l 182 1019 l 181 1005 l 180 991 l 180 976 l 181 960 l 182 946 l 183 934 l 184 927 l 184 924 l 184 923 l gs col0 s gr % Polyline n 180 726 m 180 725 l 180 722 l 180 714 l 180 701 l 179 685 l 179 669 l 179 653 l 179 639 l 180 626 l 180 615 l 181 604 l 181 593 l 182 581 l 183 570 l 184 558 l 185 546 l 186 535 l 187 524 l 189 514 l 190 504 l 192 492 l 195 480 l 199 467 l 203 452 l 207 439 l 209 430 l 211 426 l 211 425 l gs col0 s gr % Polyline n 276 307 m 279 303 l 285 297 l 294 286 l 305 273 l 317 260 l 329 247 l 340 237 l 352 229 l 363 222 l 373 217 l 385 213 l 397 210 l 409 206 l 423 202 l 436 199 l 450 196 l 463 193 l 476 191 l 489 190 l 502 189 l 514 190 l 526 192 l 538 195 l 550 199 l 562 204 l 575 209 l 587 214 l 599 220 l 611 226 l 623 233 l 634 239 l 644 247 l 654 255 l 663 264 l 671 274 l 679 285 l 687 297 l 693 309 l 700 321 l 707 334 l 713 346 l 720 359 l 727 372 l 734 385 l 741 399 l 747 411 l 753 424 l 760 438 l 766 452 l 773 467 l 779 483 l 786 499 l 792 515 l 799 530 l 805 544 l 811 557 l 817 569 l 822 579 l 827 588 l 835 599 l 841 606 l 847 609 l 853 610 l 858 611 l 863 612 l 869 613 l 875 615 l 884 617 l 894 617 l 906 616 l 917 616 l 923 615 l 924 615 l gs col0 s gr % Polyline 30.000 slw n 1047 360 m 1045 363 l 1041 368 l 1033 376 l 1024 388 l 1013 401 l 1002 416 l 991 430 l 981 443 l 973 454 l 967 465 l 961 476 l 955 489 l 950 502 l 946 516 l 943 530 l 939 545 l 936 559 l 933 574 l 930 588 l 927 601 l 924 615 l 921 628 l 919 641 l 917 654 l 915 667 l 913 679 l 911 692 l 910 705 l 908 718 l 907 731 l 907 745 l 907 757 l 908 769 l 909 781 l 910 794 l 911 807 l 913 820 l 914 833 l 916 846 l 917 858 l 919 871 l 922 884 l 924 896 l 927 908 l 930 920 l 933 933 l 937 945 l 941 957 l 944 969 l 948 981 l 952 994 l 957 1005 l 961 1017 l 966 1029 l 971 1040 l 977 1051 l 983 1062 l 990 1073 l 998 1085 l 1006 1097 l 1014 1108 l 1022 1120 l 1030 1131 l 1038 1141 l 1044 1150 l 1049 1157 l 1053 1163 l 1054 1164 l 1055 1166 l 1056 1167 l 1056 1168 l 1057 1169 l 1058 1169 l 1058 1170 l 1059 1170 l 1058 1169 l 1058 1168 l 1057 1167 l 1057 1166 l 1057 1165 l 1056 1165 l 1056 1164 l 1056 1163 l 1055 1162 l 1055 1161 l 1054 1161 l 1054 1160 l 1054 1159 l 1053 1159 l 1053 1158 l gs col0 s gr % Polyline n 2607 372 m 2609 375 l 2614 380 l 2621 389 l 2631 401 l 2642 414 l 2653 429 l 2664 443 l 2674 456 l 2683 468 l 2689 479 l 2695 490 l 2701 503 l 2706 516 l 2710 530 l 2713 544 l 2716 558 l 2719 573 l 2722 587 l 2725 601 l 2728 614 l 2731 628 l 2733 641 l 2736 655 l 2738 667 l 2740 680 l 2741 693 l 2743 706 l 2745 719 l 2746 732 l 2747 745 l 2747 759 l 2747 771 l 2746 783 l 2745 795 l 2744 808 l 2743 820 l 2742 833 l 2741 846 l 2739 858 l 2738 871 l 2736 883 l 2733 896 l 2731 908 l 2728 920 l 2725 932 l 2721 945 l 2717 957 l 2714 969 l 2709 982 l 2705 994 l 2701 1006 l 2697 1018 l 2692 1030 l 2687 1042 l 2682 1053 l 2676 1064 l 2670 1075 l 2663 1087 l 2656 1098 l 2648 1110 l 2640 1122 l 2632 1134 l 2625 1145 l 2618 1155 l 2612 1164 l 2607 1171 l 2603 1177 l 2602 1179 l 2601 1180 l 2600 1181 l 2600 1182 l 2599 1183 l 2598 1184 l 2598 1183 l 2598 1182 l 2598 1181 l 2599 1180 l 2599 1179 l 2599 1178 l 2600 1177 l 2600 1176 l 2601 1176 l 2601 1175 l 2601 1174 l 2602 1173 l 2602 1172 l 2602 1171 l 2603 1171 l 2603 1170 l gs col0 s gr % Polyline 7.500 slw [60] 0 sd n 3527 812 m 3530 808 l 3534 802 l 3542 791 l 3551 778 l 3560 764 l 3567 750 l 3573 739 l 3576 729 l 3576 720 l 3574 712 l 3570 703 l 3565 694 l 3558 686 l 3551 677 l 3543 668 l 3536 660 l 3528 653 l 3520 646 l 3511 641 l 3502 637 l 3491 635 l 3479 634 l 3467 634 l 3454 635 l 3441 636 l 3429 636 l 3417 636 l 3407 633 l 3397 629 l 3388 623 l 3380 616 l 3371 609 l 3363 600 l 3354 591 l 3346 581 l 3339 572 l 3334 562 l 3330 553 l 3328 543 l 3329 533 l 3333 521 l 3339 507 l 3348 490 l 3359 473 l 3369 457 l 3378 444 l 3384 435 l 3387 432 l 3387 431 l gs col0 s gr [] 0 sd % Polyline n 2746 902 m 2747 902 l 2750 902 l 2758 904 l 2770 905 l 2785 908 l 2800 911 l 2815 915 l 2828 920 l 2838 926 l 2847 934 l 2854 942 l 2860 952 l 2866 964 l 2871 977 l 2876 990 l 2881 1004 l 2886 1018 l 2891 1033 l 2896 1046 l 2902 1060 l 2907 1073 l 2913 1085 l 2919 1097 l 2925 1110 l 2931 1122 l 2937 1135 l 2943 1147 l 2949 1160 l 2955 1173 l 2961 1185 l 2967 1197 l 2974 1209 l 2981 1220 l 2989 1230 l 2997 1240 l 3006 1249 l 3015 1258 l 3023 1268 l 3032 1277 l 3041 1286 l 3050 1294 l 3060 1302 l 3070 1310 l 3081 1317 l 3093 1323 l 3107 1327 l 3120 1330 l 3134 1332 l 3150 1333 l 3166 1333 l 3184 1333 l 3203 1333 l 3222 1332 l 3241 1331 l 3259 1330 l 3277 1329 l 3293 1327 l 3308 1325 l 3321 1323 l 3333 1321 l 3347 1316 l 3358 1311 l 3368 1304 l 3376 1296 l 3383 1288 l 3389 1280 l 3395 1272 l 3402 1264 l 3408 1256 l 3414 1249 l 3420 1240 l 3425 1231 l 3430 1222 l 3433 1212 l 3437 1203 l 3440 1194 l 3443 1185 l 3446 1177 l 3449 1166 l 3452 1153 l 3454 1139 l 3456 1126 l 3457 1119 l 3457 1118 l gs col0 s gr % Polyline n 3484 1007 m 3484 1006 l 3485 1002 l 3487 992 l 3489 979 l 3492 964 l 3493 950 l 3495 936 l 3495 923 l 3495 911 l 3494 898 l 3493 884 l 3492 868 l 3491 853 l 3489 837 l 3488 822 l 3487 807 l 3485 795 l 3484 783 l 3482 770 l 3480 758 l 3477 744 l 3474 731 l 3471 719 l 3469 711 l 3468 707 l 3468 706 l gs col0 s gr % Polyline n 3464 576 m 3464 575 l 3464 571 l 3463 561 l 3462 546 l 3460 527 l 3458 507 l 3456 488 l 3453 472 l 3450 458 l 3446 445 l 3441 433 l 3435 421 l 3429 410 l 3422 398 l 3414 386 l 3407 375 l 3399 364 l 3392 353 l 3384 343 l 3376 334 l 3368 325 l 3359 318 l 3351 310 l 3342 302 l 3333 295 l 3324 288 l 3315 282 l 3305 276 l 3295 271 l 3285 267 l 3274 264 l 3262 263 l 3250 263 l 3237 263 l 3224 264 l 3211 265 l 3198 266 l 3186 267 l 3175 267 l 3165 267 l 3154 267 l 3144 267 l 3136 266 l 3128 265 l 3120 264 l 3112 264 l 3104 265 l 3096 267 l 3088 270 l 3080 274 l 3071 278 l 3063 283 l 3055 288 l 3047 294 l 3039 300 l 3032 307 l 3025 315 l 3019 325 l 3013 336 l 3007 347 l 3001 358 l 2995 369 l 2989 381 l 2983 392 l 2978 401 l 2974 411 l 2969 421 l 2965 431 l 2960 442 l 2956 452 l 2952 462 l 2947 472 l 2943 481 l 2939 490 l 2934 500 l 2929 510 l 2925 520 l 2921 529 l 2916 538 l 2912 546 l 2907 555 l 2902 563 l 2896 571 l 2891 578 l 2885 585 l 2879 592 l 2873 599 l 2866 606 l 2860 611 l 2853 615 l 2846 617 l 2838 618 l 2829 618 l 2820 617 l 2812 616 l 2803 615 l 2795 615 l 2787 615 l 2776 616 l 2764 617 l 2750 618 l 2737 619 l 2730 620 l 2729 620 l gs col0 s gr % Polyline [60] 0 sd n 3435 340 m 3435 339 l 3436 333 l 3436 320 l 3437 302 l 3438 283 l 3437 266 l 3435 253 l 3430 242 l 3424 235 l 3417 228 l 3409 221 l 3400 215 l 3391 210 l 3381 204 l 3371 199 l 3361 193 l 3351 188 l 3340 183 l 3329 178 l 3316 173 l 3304 168 l 3291 163 l 3277 158 l 3264 154 l 3250 149 l 3236 146 l 3223 143 l 3209 142 l 3197 142 l 3185 143 l 3173 144 l 3160 146 l 3147 148 l 3135 151 l 3122 153 l 3109 156 l 3097 159 l 3085 163 l 3073 167 l 3062 171 l 3049 177 l 3037 184 l 3025 193 l 3014 201 l 3002 210 l 2991 220 l 2980 229 l 2969 238 l 2960 247 l 2951 255 l 2942 266 l 2935 276 l 2929 286 l 2924 297 l 2919 307 l 2914 318 l 2908 329 l 2902 340 l 2897 349 l 2892 359 l 2888 370 l 2883 380 l 2878 391 l 2874 402 l 2869 412 l 2864 422 l 2859 431 l 2853 439 l 2845 448 l 2837 455 l 2829 463 l 2821 469 l 2812 475 l 2803 481 l 2793 486 l 2783 490 l 2774 492 l 2763 493 l 2750 494 l 2735 493 l 2719 493 l 2704 492 l 2692 491 l 2684 490 l 2681 490 l 2680 490 l gs col0 s gr [] 0 sd /Times-Roman-iso ff 210.00 scf sf 1423 1230 m gs 1 -1 sc (1-handle) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 1595 3124 a Fs(Figure)k(14.)486 3425 y Fr(Since)k(the)h (Th)m(urston{Bennequin)i(in)m(v)-5 b(arian)m(t)41 b(of)h Fo(K)2487 3440 y Fn(1)2569 3425 y Fr(and)g Fo(K)2851 3440 y Fn(2)2933 3425 y Fr(is)g Fp(\000)p Fr(2,)j(the)386 3542 y(framing)30 b(of)i(the)h(attac)m(hing)f(curv)m(e)i(of)e Fo(h)1883 3557 y Fn(2)1955 3542 y Fr(is)g Fp(\000)p Fr(4)h(with)f(resp) s(ect)h(to)g(the)f(ob)m(vious)386 3658 y(Seifert)21 b(surface)i(of)e (the)i(attac)m(hing)e(curv)m(e.)41 b(The)23 b(2{handles)e Fo(h)2656 3673 y Fn(2)2696 3658 y Fo(;)c(h)2796 3622 y Fk(0)2796 3683 y Fn(2)2857 3658 y Fr(are)22 b(attac)m(hed)386 3774 y(b)m(y)40 b(a)f(doubling)e(construction.)63 b(The)40 b(Kirb)m(y)f(diagram)e(therefore)i(consists)h(of)386 3890 y(spheres)31 b(represen)m(ting)e(the)g(attac)m(hing)f(regions)h (of)f Fo(h)2353 3905 y Fn(1)2392 3890 y Fr(,)i(the)f(attac)m(hing)f (curv)m(e)i(of)386 4007 y Fo(h)442 4022 y Fn(2)514 4007 y Fr(and)j(a)g(zero{framed)f(meridian)f(of)h(the)h(attac)m(hing)g(curv) m(e)h(of)e Fo(h)2881 4022 y Fn(2)2921 4007 y Fr(,)h(cf.)g([GoS],)386 4123 y(p.)48 b(133.)90 b(The)49 b(ordinary)e(handles)i(of)f(index)g(2)g (therefore)h(accoun)m(t)g(for)e(the)386 4239 y(summand)23 b Fo(S)888 4203 y Fn(2)932 4239 y Fp(\002)t Fo(S)1079 4203 y Fn(2)1118 4239 y Fr(.)41 b(The)25 b(righ)m(t)e(part)g(of)h (Figure)e(14)i(sho)m(ws)h(the)f(Kirb)m(y)g(diagram)386 4355 y(of)32 b Fo(M)591 4370 y Fn(1)653 4355 y Fp([)23 b Fo(M)836 4370 y Fn(2)908 4355 y Fr(with)32 b Fo(h)1186 4370 y Fn(1)1226 4355 y Fo(;)17 b(h)1326 4370 y Fn(2)1365 4355 y Fo(;)g(h)1465 4319 y Fk(0)1465 4380 y Fn(2)1537 4355 y Fr(attac)m(hed.)486 4472 y(This)43 b(pro)m(v)m(es)h(the)f(claim) e(under)i(the)h(assumption)e(that)h Fo(M)2786 4487 y Fn(1)2868 4472 y Fr(and)g Fo(M)3162 4487 y Fn(2)3245 4472 y Fr(are)386 4588 y(orien)m(ted.)f(If)28 b(this)f(is)g(not)h(the)g (case)h(w)m(e)g(can)f(nev)m(ertheless)i(orien)m(t)e Fp(W)2928 4603 y Fl(i)2984 4588 y Fr(along)e(the)386 4704 y(comp)s(onen)m(ts)31 b(of)f Fo(@)5 b(M)1180 4719 y Fl(i)1240 4704 y Fr(after)30 b(w)m(e)h(ha)m(v)m(e)h(cut)f(along)e(the)h(h)m(yp)s(ersurfaces.)46 b(F)-8 b(or)29 b(the)386 4820 y(pro)s(of)g(it)g(su\016ces)i(to)f(orien) m(t)f Fp(W)1552 4835 y Fl(i)1610 4820 y Fr(along)g Fo(U)1934 4835 y Fl(i)1992 4820 y Fr(suc)m(h)i(that)e(it)g(p)s(oin)m(ts)h(out)m (w)m(ards)h(and)386 4936 y(along)c Fo(')p Fr(\()p Fo(U)10 b Fr(\))29 b(suc)m(h)h(that)e(it)f(p)s(oin)m(ts)h(in)m(w)m(ards.)43 b(All)27 b(constructions)i(carry)g(o)m(v)m(er)g(to)386 5053 y(this)j(situation.)2318 b Fi(\003)486 5274 y Fr(In)22 b(order)g(to)g(apply)g(Theorem)h(6.3,)h(one)e(has)h(to)f(\014nd)h (Engel)e(structures)j(whose)386 5390 y(c)m(haracteristic)30 b(foliation)c(admits)i(a)i(closed)g(transv)m(ersal.)43 b(This)30 b(is)g(true)g(for)f(the)386 5506 y(Engel)41 b(structures)i(from)c(the)j(pro)s(of)e(of)h(Theorem)g(6.1.)69 b(W)-8 b(e)41 b(can)h(also)e(apply)386 5623 y(Theorem)47 b(6.3)f(to)g(Engel)g(structures)i(obtained)e(b)m(y)h(prolongation)d (after)j(w)m(e)p eop %%Page: 55 55 55 54 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(55)386 459 y Fr(p)s(erturb)26 b(them)g(sligh)m(tly)f(in)g(the)h (neigh)m(b)s(orho)s(o)s(d)f(of)h(a)g(leaf)f(of)g(the)i(c)m (haracteristic)386 575 y(foliation)i(\(as)k(in)f(Prop)s(osition)e (4.5\).)386 715 y Fy(Corollary)50 b(6.4.)e Fm(If)d Fr(\()p Fo(N)1349 730 y Fn(1)1388 715 y Fo(;)17 b Fp(C)1484 730 y Fn(1)1524 715 y Fr(\))45 b Fm(and)g Fr(\()p Fo(N)1923 730 y Fn(2)1962 715 y Fo(;)17 b Fp(C)2058 730 y Fn(2)2098 715 y Fr(\))45 b Fm(ar)-5 b(e)45 b(c)-5 b(ontact)46 b(manifolds,)g (then)386 831 y Fh(P)p Fp(C)497 846 y Fn(1)539 831 y Fr(#)p Fh(P)p Fp(C)731 846 y Fn(2)773 831 y Fr(#\()p Fo(S)958 795 y Fn(2)1019 831 y Fp(\002)23 b Fo(S)1185 795 y Fn(2)1224 831 y Fr(\))35 b Fm(admits)f(an)h(Engel)f(structur)-5 b(e.)486 971 y Fr(Using)32 b(this)g(corollary)f(one)i(obtains)f(Engel)g (structures)i(on)1084 1134 y Fo(M)1178 1149 y Fl(n)1253 1134 y Fr(=)28 b Fo(n)p Fr(\()p Fo(S)1519 1093 y Fn(3)1580 1134 y Fp(\002)23 b Fo(S)1746 1093 y Fn(1)1785 1134 y Fr(\)#\()p Fo(n)g Fp(\000)f Fr(1\)\()p Fo(S)2313 1093 y Fn(2)2374 1134 y Fp(\002)h Fo(S)2540 1093 y Fn(2)2579 1134 y Fr(\))32 b Fo(:)386 1298 y Fr(One)45 b(can)h(sho)m(w)g(that)f (it)f(is)h(imp)s(ossible)e(to)i(construct)h(an)f(Engel)g(structure)386 1414 y(on)e Fo(M)626 1429 y Fl(n)716 1414 y Fr(using)g(the)h(metho)s(d) e(of)h(Geiges)g(or)f(prolongation,)i(although)e Fo(M)3219 1429 y Fl(n)3309 1414 y Fr(is)386 1530 y(actually)22 b(the)h(total)f(space)i(of)f(an)g(orien)m(table)f(circle)g(bundle)i(o)m (v)m(er)g(a)f(3{manifold.)386 1646 y(Ho)m(w)m(ev)m(er,)45 b(the)40 b(Euler)h(class)f(of)g(all)f(circle)g(bundles)i(with)f(total)f (space)i Fo(M)3221 1661 y Fl(n)3309 1646 y Fr(is)386 1763 y(o)s(dd)36 b(\(notice)h(that)f(since)h Fo(M)1464 1778 y Fl(n)1548 1763 y Fr(is)f(orien)m(table,)h(the)g(c)m (haracteristic)f(foliation)d(of)386 1879 y(all)i(Engel)j(structures)h (on)e Fo(M)1499 1894 y Fl(n)1584 1879 y Fr(are)h(orien)m(table\).)57 b(Since)38 b(the)g(tangen)m(t)f(bundle)386 1995 y(of)28 b(an)h(orien)m(table)f(3{manifold)e(is)i(trivial,)f Fo(M)2074 2010 y Fl(n)2151 1995 y Fr(can)i(not)g(arise)f(as)h(circle)f(bundle)386 2111 y(of)k(a)g(subbundle)i(of)e(rank)h(2)f(of)g(the)h(3{manifold.)486 2227 y(Let)26 b(us)h(return)f(to)g(the)g(pro)s(of)g(of)f(Theorem)i(6.3) e(and)i(discuss)g(the)f(meaning)f(of)386 2344 y(the)e(assumption)f (that)h(b)s(oth)g(Engel)f(structures)j(ha)m(v)m(e)f(c)m(haracteristic)f (foliations)386 2460 y(whic)m(h)36 b(admit)d(a)i(closed)g(transv)m (ersal.)52 b(W)-8 b(e)35 b(do)h(not)f(mak)m(e)g(explicit)e(use)k(of)d (the)386 2576 y(fact)26 b(that)g Fo(N)855 2591 y Fn(1)921 2576 y Fr(and)g Fo(N)1182 2591 y Fn(2)1247 2576 y Fr(are)h Fm(close)-5 b(d)25 b Fr(transv)m(ersals)i(in)f(the)g(pro)s(of.)41 b(But)26 b(implicitly)-8 b(,)386 2692 y(this)46 b(assumption)g(is)g (used)i(when)g(w)m(e)g(apply)e(v)m(ertical)g(mo)s(di\014cation)e(of)i (the)386 2809 y(b)s(oundary)-8 b(.)71 b(If)41 b Fo(N)1066 2824 y Fn(1)1147 2809 y Fr(or)g Fo(N)1353 2824 y Fn(2)1434 2809 y Fr(ha)m(v)m(e)i(b)s(oundary)-8 b(,)44 b(then)e(w)m(e)h(ha)m(v)m (e)g(to)e(ensure)i(that)386 2925 y(ev)m(erything)c(is)e(w)m(ell)f (de\014ned)k(at)d(b)s(oundary)h(p)s(oin)m(ts)f(of)g Fo(N)2578 2940 y Fl(i)2644 2925 y Fr(when)i(w)m(e)f(iden)m(tify)386 3041 y(the)48 b(b)s(oundary)g(comp)s(onen)m(ts)g(of)g Fo(M)10 b Fr(.)89 b(One)48 b(can)g(replace)g(the)g(existence)h(of)386 3157 y(closed)36 b(transv)m(ersals)g(in)f(Theorem)h(6.3)f(b)m(y)h(an)g (assumption)f(on)g(the)h(b)s(eha)m(vior)386 3274 y(of)c(the)h(Engel)f (structure)i(along)d(lea)m(v)m(es)j(of)e(the)h(c)m(haracteristic)f (foliation.)486 3390 y(Let)23 b Fp(W)8 b Fr(\()p Fo(p)p Fr(\))24 b(b)s(e)g(a)f(leaf)f(of)h(the)h(c)m(haracteristic)f(foliation) d(of)j(an)g(Engel)g(structure)386 3506 y Fp(D)s Fr(.)69 b(W)-8 b(e)42 b(assume)g(that)f Fp(W)8 b Fr(\()p Fo(p)p Fr(\))42 b(is)f(not)g(closed.)70 b(Using)41 b(nearb)m(y)i(lea)m(v)m(es) f(of)f(the)386 3622 y(c)m(haracteristic)34 b(foliation)d(w)m(e)k (obtain)e(iden)m(ti\014cations)g(of)h Fp(E)2615 3637 y Fl(p)2654 3622 y Fo(=)p Fp(W)2801 3637 y Fl(p)2875 3622 y Fr(with)g Fp(E)3152 3637 y Fl(q)3189 3622 y Fo(=)p Fp(W)3336 3637 y Fl(q)386 3738 y Fr(for)e(all)e Fo(q)37 b Fr(on)32 b Fp(W)8 b Fr(\()p Fo(p)p Fr(\))33 b(b)m(y)h(Lemma)d(2.7.)43 b(This)33 b(allo)m(ws)e(us)i(to)g(de\014ne)1265 3903 y Fo(\016)f Fr(:)27 b Fp(W)8 b Fr(\()p Fo(p)p Fr(\))29 b Fp(\000)-17 b(!)28 b Fh(P)p Fr(\()p Fp(E)1992 3918 y Fl(p)2033 3903 y Fo(=)p Fp(W)2180 3918 y Fl(p)2220 3903 y Fr(\))f Fp(')h Fo(S)2456 3862 y Fn(1)1579 4054 y Fo(q)k Fp(7\000)-17 b(!)28 b Fr([)p Fp(D)1946 4069 y Fl(q)1984 4054 y Fr(])386 4223 y(where)33 b(w)m(e)g(iden)m(tify)e Fp(E)1217 4238 y Fl(q)1255 4223 y Fo(=)p Fp(W)1402 4238 y Fl(q)1472 4223 y Fr(with)h Fp(E)1747 4238 y Fl(p)1786 4223 y Fo(=)p Fp(W)1933 4238 y Fl(p)1973 4223 y Fr(.)43 b(This)32 b(map)f(is)g(called)g Fm(development)386 4339 y(map)36 b Fr(of)h Fp(W)8 b Fr(\()p Fo(p)p Fr(\))38 b(and)f(it)f(is)h (an)g(immersion)e(since)i([)p Fp(D)s Fo(;)17 b Fp(D)s Fr(])35 b(=)h Fp(E)9 b Fr(,)37 b(cf.)h([Ad,)g(Mo2].)386 4455 y(F)-8 b(or)26 b(a)h(giv)m(en)h(orien)m(tation)d(of)i Fp(W)8 b Fr(\()p Fo(p)p Fr(\))27 b(let)g Fp(W)1977 4419 y Fk(\006)2036 4455 y Fr(\()p Fo(p)p Fr(\))g(b)s(e)h(the)f(segmen)m(ts) i(of)d Fp(W)36 b Fr(whic)m(h)386 4572 y(lie)31 b(on)h(the)h(t)m(w)m(o)h (sides)f(of)f Fo(p)p Fr(.)43 b(No)m(w)34 b(w)m(e)f(de\014ne)h(the)f Fm(twisting)i(numb)-5 b(er)1112 4735 y Fr(t)m(w)1219 4694 y Fn(+)1278 4735 y Fr(\()p Fp(W)8 b Fr(\()p Fo(p)p Fr(\)\))28 b(=)1717 4650 y Fj(\014)1717 4710 y(\014)1750 4735 y Fo(\016)1797 4694 y Fk(\000)p Fn(1)1891 4735 y Fr(\([)p Fp(D)2033 4750 y Fl(p)2073 4735 y Fr(]\))22 b Fp(\\)h(W)2355 4694 y Fn(+)2414 4735 y Fr(\()p Fo(p)p Fr(\))2539 4650 y Fj(\014)2539 4710 y(\014)2621 4735 y Fo(;)386 4902 y Fr(i.e.)43 b(t)m(w)661 4866 y Fn(+)720 4902 y Fr(\()p Fo(p)p Fr(\))31 b(is)h(the)g(n)m(um)m(b)s(er)h(of)e (half)g(t)m(wists)h(of)g Fp(D)i Fr(around)e Fp(W)41 b Fr(in)31 b Fp(E)40 b Fr(when)33 b(one)386 5018 y(mo)m(v)m(es)h(from)f Fo(p)g Fr(in)g(the)h(sense)h(of)e(the)h(orien)m(tation)e(along)g(the)i (leaf.)45 b(W)-8 b(e)34 b(de\014ne)386 5134 y(t)m(w)493 5098 y Fk(\000)552 5134 y Fr(\()p Fp(W)8 b Fr(\()p Fo(p)p Fr(\)\))33 b(similarly)-8 b(.)41 b(Finally)-8 b(,)30 b(if)i Fp(W)8 b Fr(\()p Fo(p)p Fr(\))33 b(is)g(closed)g(w)m(e)h (de\014ne)g(t)m(w)r(\()p Fp(W)8 b Fr(\()p Fo(p)p Fr(\)\))33 b(as)386 5250 y(the)42 b(degree)g(of)f Fo(\016)47 b Fr(:)42 b Fp(W)8 b Fr(\()p Fo(p)p Fr(\))43 b(=)g Fo(S)1613 5214 y Fn(1)1695 5250 y Fp(\000)-16 b(!)42 b Fo(S)1964 5214 y Fn(1)2003 5250 y Fr(.)70 b(No)m(w)42 b(w)m(e)g(can)g(state)g(a)f(mo)s (di\014ed)386 5367 y(v)m(ersion)33 b(of)f(Theorem)h(6.3.)386 5506 y Fy(Theorem)e(6.5.)37 b Fm(L)-5 b(et)31 b Fo(M)1324 5521 y Fn(1)1363 5506 y Fo(;)17 b(M)1501 5521 y Fn(2)1571 5506 y Fm(c)-5 b(arry)30 b(Engel)f(structur)-5 b(es)31 b Fp(D)2608 5521 y Fn(1)2647 5506 y Fo(;)17 b Fp(D)2768 5521 y Fn(2)2837 5506 y Fm(such)29 b(that)i(the)386 5623 y(char)-5 b(acteristic)36 b(foliation)g(has)g(non{close)-5 b(d)35 b(le)-5 b(aves)36 b Fp(W)2431 5638 y Fn(1)2471 5623 y Fr(\()p Fo(p)2558 5638 y Fn(1)2597 5623 y Fr(\))h Fm(thr)-5 b(ough)37 b Fo(p)3073 5638 y Fn(1)3143 5623 y Fp(2)32 b Fo(M)3335 5638 y Fn(1)p eop %%Page: 56 56 56 55 bop 386 259 a Fq(56)1096 b(THOMAS)25 b(V)n(OGEL)386 459 y Fm(and)34 b Fp(W)673 474 y Fn(2)713 459 y Fr(\()p Fo(p)800 474 y Fn(2)839 459 y Fr(\))h Fm(thr)-5 b(ough)35 b Fo(p)1311 474 y Fn(2)1378 459 y Fp(2)28 b Fo(M)1566 474 y Fn(2)1641 459 y Fm(with)35 b(the)g(pr)-5 b(op)g(erty)386 620 y Fr(\(33\))446 b(t)m(w)1113 579 y Fk(\006)1172 620 y Fr(\()p Fp(W)1308 635 y Fn(1)1348 620 y Fr(\()p Fo(p)1435 635 y Fn(1)1474 620 y Fr(\)\))28 b Fp(\025)g Fo(C)42 b Fm(and)51 b Fr(t)m(w)2108 579 y Fk(\006)2167 620 y Fr(\()p Fp(W)2303 635 y Fn(2)2343 620 y Fr(\()p Fo(p)2430 635 y Fn(2)2469 620 y Fr(\)\))27 b Fp(\025)i Fo(C)386 781 y Fm(for)37 b(some)f(p)-5 b(ositive)36 b(c)-5 b(onstant)36 b Fo(C)44 b Fm(which)36 b(is)g(indep)-5 b(endent)36 b(of)g(the)h(Engel) f(struc-)386 897 y(tur)-5 b(es.)486 1013 y(Then)43 b(ther)-5 b(e)45 b(is)f(an)g(Engel)g(structur)-5 b(e)45 b Fp(D)j Fm(on)c Fo(M)2333 1028 y Fn(1)2372 1013 y Fr(#)p Fo(M)2547 1028 y Fn(2)2588 1013 y Fr(#\()p Fo(S)2773 977 y Fn(2)2841 1013 y Fp(\002)30 b Fo(S)3014 977 y Fn(2)3053 1013 y Fr(\))45 b Fm(which)386 1129 y(c)-5 b(oincides)24 b(with)h Fp(D)1074 1144 y Fn(1)1114 1129 y Fo(;)17 b Fp(D)1235 1144 y Fn(2)1299 1129 y Fm(outside)25 b(of)h(neighb)-5 b(orho)g(o)g(ds)23 b(of)i Fo(p)2492 1144 y Fn(1)2559 1129 y Fp(2)k(W)2752 1144 y Fn(1)2817 1129 y Fm(and)c Fo(p)3046 1144 y Fn(2)3113 1129 y Fp(2)j(W)3305 1144 y Fn(2)3345 1129 y Fm(.)386 1246 y(Mor)-5 b(e)g(over)26 b(one)h(c)-5 b(an)26 b(cho)-5 b(ose)25 b Fp(D)30 b Fm(such)c(that)h (ther)-5 b(e)27 b(is)f(a)h(le)-5 b(af)26 b(of)h(the)f(char)-5 b(acteristic)386 1362 y(foliation)36 b(which)h(satis\014es)45 b Fr(\(33\))o Fm(.)52 b(If)37 b Fp(W)1886 1377 y Fl(i)1914 1362 y Fr(\()p Fo(p)2001 1377 y Fl(i)2029 1362 y Fr(\))h Fm(is)f(close)-5 b(d)36 b(one)h(c)-5 b(an)36 b(r)-5 b(eplac)g(e)44 b Fr(\(33\))386 1478 y Fm(by)35 b Fr(t)m(w)r(\()p Fp(W)756 1493 y Fl(i)785 1478 y Fr(\()p Fo(p)872 1493 y Fl(i)900 1478 y Fr(\)\))27 b Fp(\025)h Fr(2\()p Fo(C)h Fr(+)22 b(1\))35 b Fm(for)f Fo(i)28 b Fr(=)g(1)p Fo(;)17 b Fr(2)p Fm(.)486 1617 y Fr(W)-8 b(e)38 b(just)h(giv)m(e)g(a)f(sk)m(etc)m(h)j (of)d(the)h(pro)s(of)e(and)i(omit)d(the)j(details.)61 b(Let)38 b Fo(p)3203 1632 y Fn(1)3243 1617 y Fo(;)17 b(p)3336 1632 y Fn(2)386 1733 y Fr(satisfy)29 b(the)h(assumption)e(of)h (the)g(theorem.)43 b(F)-8 b(or)28 b Fo(i)g Fr(=)f(1)p Fo(;)17 b Fr(2)29 b(w)m(e)h(cut)f Fo(M)2957 1748 y Fl(i)3015 1733 y Fr(along)f(an)386 1849 y(em)m(b)s(edded)h(closed)g(3{ball)c Fo(N)1486 1864 y Fl(i)1543 1849 y Fr(transv)m(erse)30 b(to)e Fp(W)2213 1864 y Fl(i)2242 1849 y Fr(,)h(whic)m(h)f(is)g(so)g (small)e(that)i(the)386 1965 y(t)m(wisting)f(n)m(um)m(b)s(ers)j(of)d (all)g(lea)m(v)m(es)i(of)f(the)h(c)m(haracteristic)f(foliation)d (satisfy)k(\(33\))386 2082 y(after)f(w)m(e)i(ha)m(v)m(e)f(cut)g(out)f Fo(N)1383 2097 y Fl(i)1412 2082 y Fr(.)42 b(Aw)m(a)m(y)29 b(from)f(the)g(b)s(oundary)h(of)f Fo(N)2757 2097 y Fl(i)2786 2082 y Fr(,)h(the)g(resulting)386 2198 y(space)37 b(is)e(a)h(manifold)d (with)i(b)s(oundary)-8 b(.)53 b(W)-8 b(e)37 b(can)f(p)s(erform)e(all)g (constructions)386 2314 y(from)i(the)h(pro)s(of)f(of)g(Theorem)h(6.3)g (as)g(long)f(as)h(w)m(e)h(do)e(not)h(c)m(hange)h(an)m(ything)386 2430 y(near)33 b Fo(@)5 b(N)737 2445 y Fl(i)766 2430 y Fr(.)486 2546 y(V)-8 b(ertical)25 b(mo)s(di\014cations)f(of)i(the)h (b)s(oundary)g(are)g(the)g(only)f(op)s(erations)f(in)h(the)386 2663 y(pro)s(of)39 b(of)h(Theorem)h(6.3)f(whic)m(h)h(a\013ect)f Fo(@)5 b(N)2043 2678 y Fl(i)2073 2663 y Fr(.)66 b(The)42 b(problematic)c(step)j(is)f(the)386 2779 y(v)m(ertical)31 b(mo)s(di\014cation)d(of)j(the)h(b)s(oundary)g(used)h(for)e(the)g(iden) m(ti\014cation)f(of)h(the)386 2895 y(b)s(oundary)25 b(comp)s(onen)m(ts) g(of)f Fo(M)10 b Fr(.)41 b(The)26 b(v)m(ertical)e(mo)s(di\014cation)d (of)j(the)h(b)s(oundary)386 3011 y(needed)33 b(at)f(this)f(p)s(oin)m(t) h(is)f(not)h(allo)m(w)m(ed)f(to)g(c)m(hange)i(an)m(ything)f(near)g Fo(@)5 b(N)3085 3026 y Fl(i)3114 3011 y Fr(.)43 b(This)386 3137 y(means)e(that)h(the)g(function)1500 3111 y Fj(b)1482 3137 y Fo(f)52 b Fr(used)43 b(in)e(Section)g(2.4)g(for)g(the)h (description)f(of)386 3254 y(a)35 b(v)m(ertical)f(mo)s(di\014cation)e (of)j(the)g(b)s(oundary)h(has)f(to)g(b)s(e)g(zero)g(near)g(b)s(oundary) 386 3379 y(p)s(oin)m(ts)d(of)h Fo(N)869 3394 y Fl(i)897 3379 y Fr(.)45 b(The)33 b(condition)f(\(33\))g(allo)m(ws)g(us)i(to)e(c) m(ho)s(ose)2668 3353 y Fj(b)2650 3379 y Fo(f)39 b Fp(\021)29 b Fr(0)j(on)h(a)g(collar)386 3496 y(of)f(the)h(b)s(oundary)g(of)f Fo(N)1293 3511 y Fl(i)1321 3496 y Fr(.)486 3612 y(W)-8 b(e)25 b(did)g(not)g(determine)h(b)s(ounds)g(for)f Fo(C)7 b Fr(.)41 b(Since)25 b(there)h(are)g(Engel)f(structures)386 3728 y(\(e.g.)52 b(the)35 b(standard)h(Engel)f(structure)i(on)e Fh(R)2102 3692 y Fn(4)2147 3728 y Fr(\))g(where)i(the)f(rotation)d(n)m (um)m(b)s(er)386 3844 y(along)g(all)f(lea)m(v)m(es)k(of)e(the)h(c)m (haracteristic)g(foliation)c(is)j(zero,)h(condition)e(\(33\))h(is)386 3961 y(not)e(ful\014lled)f(in)h(general.)1590 4170 y Fs(References)386 4328 y Fx([Ad])141 b(J.)29 b(Adac)n(hi,)g Fc(Engel)j(structur)l(es)e(with)i(trivial)g(char)l(acteristic)h (foliations)p Fx(,)f(Algebraic)681 4427 y(&)27 b(Geometric)g(T)-7 b(op)r(ology)27 b(V)-7 b(ol.)27 b(2)g(\(2002\),)g(239{255.)386 4527 y([Aeb])104 b(B.)23 b(Aebisc)n(her,)g(M.)g(Borer,)f(M.)h(K\177)-42 b(alin,)23 b(Ch.)g(Leuen)n(b)r(erger,)g(H.)g(Reimann,)h Fb(Symplec-)681 4626 y(tic)k(Geometry)p Fx(,)f(Progress)e(in)j(Math.)f (V)-7 b(ol.)28 b(124,)e(Birkh\177)-42 b(auser)26 b(V)-7 b(erlag)27 b(\(1994\).)386 4726 y([As])154 b(D.)31 b(Asimo)n(v,)f Fc(R)l(ound)h(hand)t(les)j(and)e(non{singular)h(Morse{Smale)h(\015ows)p Fx(,)d(Ann.)g(of)681 4826 y(Math.)d(\(2\),)g(102)e(\(1975\),)g(no.)i (1,)f(41{54.)386 4925 y([Col])124 b(V.)42 b(Colin,)i Fc(R)l(e)l(c)l(ol)t(lement)e(de)g(vari)n(\023)-40 b(et)n(\023)g(es)44 b(de)e(c)l(ontact)g(tendues)p Fx(,)i(Bull.)e(So)r(c.)f(Math.)681 5025 y(F)-7 b(rance)27 b(127)f(\(1999\),)h(no.)g(1,)g(43{69.)386 5125 y([El1])127 b(Y.)58 b(Eliash)n(b)r(erg,)64 b Fc(Classi\014c)l (ation)59 b(of)g(overtwiste)l(d)g(c)l(ontact)e(structur)l(es)f(on)i Fx(3)p Fc({)681 5224 y(manifolds)p Fx(,)30 b(In)n(v)n(en)n(t.)d(Math.)h (98)e(\(1989\),)h(no.)g(3,)g(623{637.)386 5324 y([El2])127 b(Y.)22 b(Eliash)n(b)r(erg,)g Fc(T)-6 b(op)l(olo)l(gic)l(al)27 b(char)l(acterization)f(of)g(Stein)e(manifolds)i(of)g(dimension)681 5423 y Fv(>)d Fx(2,)k(In)n(ternat.)g(J.)h(Math.)f(1)g(\(1990\),)g(no.)g (1,)g(29{46.)386 5523 y([ElG])104 b(Y.)36 b(Eliash)n(b)r(erg,)f(M.)g (Gromo)n(v,)h Fc(Convex)h(symple)l(ctic)h(manifolds)p Fx(,)h(Pro)r(c.)34 b(Symp)r(os.)681 5623 y(Pure)27 b(Math.)h(52)e(P)n (art)g(2)i(\(1991\),)e(135{162.)p eop %%Page: 57 57 57 56 bop 1149 259 a Fq(EXISTENCE)33 b(OF)g(ENGEL)h(STR)n(UCTURES)685 b(57)386 459 y Fx([ElM])93 b(Y.)19 b(Eliash)n(b)r(erg,)g(N.)g(Mishac)n (hev,)h Fb(In)n(tro)r(duction)e(to)g(the)h Fv(h)p Fb({principle)p Fx(,)h(Grad.)e(Studies)681 558 y(in)28 b(Math.)g(48,)e(AMS)j(\(2002\).) 386 658 y([Eng])104 b(F.)45 b(Engel,)j Fc(Zur)d(Invariantenthe)l(orie)h (der)g(Systeme)f(Pfa\013)9 b('scher)48 b(Gleichungen)p Fx(,)681 758 y(Leipz.)28 b(Ber.)f(Band)g(41)g(\(1889\),)f(157{176.)386 857 y([EH1])88 b(J.)28 b(Etn)n(yre,)g(K.)g(Honda,)g Fc(Knots)i(and)h(c) l(ontact)f(ge)l(ometry)h(I)f(:)h(T)-6 b(orus)30 b(knots)h(and)g(the)681 957 y(\014gur)l(e)e(eight)i(knot)p Fx(,)c(J.)h(Symplectic)g(Geom.)f(1)g (\(2001\),)g(no.)g(1,)g(63{120.)386 1056 y([EH2])88 b(J.)37 b(Etn)n(yre,)h(K.)e(Honda,)j Fc(Knots)e(and)i(c)l(ontact)f(ge)l(ometry) g(II)g(:)h(Conne)l(cte)l(d)f(sums)p Fx(,)681 1156 y(Adv.)28 b(Math.)g(179)e(\(2003\),)g(no.)i(1,)f(59{74.)386 1256 y([Gei])124 b(H.)85 b(J.)e(Geiges,)98 b Fc(R)l(eview)82 b(of)h Fx([Mo2)o(],)98 b(h)n(ttp://www.ams.org/mathscinet,)681 1355 y(MR)28 b(2001h:53127.)386 1455 y([Gi1])119 b(E.)46 b(Giroux,)j Fc(Convexit)n(\023)-40 b(e)47 b(en)g(top)l(olo)l(gie)h(de)f (c)l(ontact)p Fx(,)k(Comm.)46 b(Math.)g(Helv.)g(66)681 1555 y(\(1991\),)26 b(no.)i(4,)f(637{677.)386 1654 y([Gi2])119 b(E.)32 b(Giroux,)h Fc(G)n(\023)-40 b(eom)n(\023)g(etrie)36 b(de)f(Contact:)48 b(de)35 b(la)g(Dimension)g(tr)l(ois)f(vers)h(les)f (dimen-)681 1754 y(sions)c(Sup)n(\023)-40 b(erieur)l(es)p Fx(,)28 b(Pro)r(ceeding)e(of)i(the)g(ICM)f(2002,)f(V)-7 b(ol.)28 b(I)r(I)r(I)g(\(1-3\),)f(405{414.)386 1853 y([GoS])96 b(R.)24 b(E.)g(Gompf,)h(A.)f(I.)g(Stipsicz,)h(4)p Fb({manifolds)e(and)h (Kirb)n(y)f(calculus)p Fx(,)h(Grad.)f(Studies)681 1953 y(in)28 b(Math.)g(V)-7 b(ol.)27 b(20,)g(Amer.)h(Math.)g(So)r(c.)f (1999.)386 2053 y([Gr])151 b(J.)24 b(Gra)n(y)-7 b(,)24 b Fc(Some)j(glob)l(al)h(pr)l(op)l(erties)f(of)h(c)l(ontact)e(structur)l (es)p Fx(,)d(Ann.)i(of)f(Math.)h(\(2\),)g(69)681 2152 y(\(1959\),)h(421{450.)386 2252 y([HH])125 b(F.)30 b(Hirzebruc)n(h,)e (H.)i(Hopf,)g Fc(F)-6 b(elder)31 b(von)h(Fl\177)-42 b(achenelementen)32 b(in)f(4{dimensionalen)681 2352 y(Mannigfaltigkeiten)p Fx(,)f(Math.)e(Ann.)g(136)f(\(1958\),)f(156{172.)386 2451 y([Ho])145 b(K.)30 b(Honda,)h Fc(On)g(the)h(classi\014c)l(ation)i (of)f(tight)f(c)l(ontact)g(structur)l(es)e(I)p Fx(,)h(Geometry)e(&)681 2551 y(T)-7 b(op)r(ology)26 b(4)h(\(2000\),)g(309{368.)386 2651 y([Ka1])100 b(Y.)41 b(Kanda,)h Fc(The)h(classi\014c)l(ation)f(of)h (tight)e(c)l(ontact)g(structur)l(es)f(on)i(the)f Fx(3)p Fc({torus)p Fx(,)681 2750 y(Comm.)28 b(in)g(Anal.)f(and)h(Geom.)f(5)g (\(1997\),)g(no.)g(3,)g(413{438.)386 2850 y([Ka2])100 b(Y.)28 b(Kanda,)e Fc(On)i(the)i(Thurston{Benne)l(quin)f(invariant)h (of)g(L)l(e)l(gendrian)g(knots)f(and)681 2949 y(non{exactness)44 b(of)h(the)g(Benne)l(quin)-8 b('s)44 b(ine)l(quality)p Fx(,)k(In)n(v)n(en)n(t.)43 b(Math.)g(133)f(\(1998\),)681 3049 y(no.)27 b(2,)h(227{242.)386 3149 y([KMS])62 b(M.)37 b(Kazarian,)g(R.)g(Mon)n(tgomery)-7 b(,)37 b(B.)g(Shapiro,)h Fc(Char)l(acteristic)i(classes)f(for)h(the)681 3248 y(de)l(gener)l (ations)e(of)f(two{plane)h(\014elds)f(in)g(four)g(dimensions)p Fx(,)h(P)n(aci\014c)c(J.)h(of)g(Math.)681 3348 y(179)26 b(\(1997\),)h(no.)g(2,)g(355{370.)386 3448 y([Mar])98 b(J.)20 b(Martinet,)h Fc(F)-6 b(ormes)23 b(de)h(c)l(ontact)e(sur)h(les) g(vari)n(\023)-40 b(et)n(\023)g(es)23 b(de)h(dimension)g(3)p Fx(,)e(Pro)r(ceedings)681 3547 y(of)42 b(the)f(Liv)n(erp)r(o)r(ol)g (Singularities)f(Symp)r(osium)i(I)r(I,)g(Lect.)g(Notes)f(in)h(Math.)f (209)681 3647 y(\(1971\),)26 b(Springer,)h(142{164.)386 3746 y([Mo1])89 b(R.)31 b(Mon)n(tgomery)-7 b(,)30 b Fc(Generic)k (Distributions)e(and)h(Lie)h(A)n(lgebr)l(as)f(of)g(V)-6 b(e)l(ctor)32 b(Fields)p Fx(,)681 3846 y(J.)c(of)f(Di\013.)i(Equ.)e (103)f(\(1993\),)g(no.)i(2,)f(387{393.)386 3946 y([Mo2])89 b(R.)32 b(Mon)n(tgomery)-7 b(,)31 b Fc(Engel)j(deformations)h(and)f(c)l (ontact)f(structur)l(es)p Fx(,)e(North.)h(Calif.)681 4045 y(Sympl.)c(Geom.)g(Sem.,)g(AMS)g(T)-7 b(ransl.)27 b(Ser.)g(2,)g(196)f(\(1999\),)h(103{117.)386 4145 y([Mor])98 b(J.)38 b(W.)g(Morgan,)g Fc(Nonsingular)h(Morse{Smale)i(\015ows)f(on)f Fx(3)p Fc(-dimensional)h(mani-)681 4245 y(folds)p Fx(,)29 b(T)-7 b(op)r(ology)26 b(18)h(\(1979\),)f(no.)i(1,)f(41{53.)386 4344 y([Ste])134 b(N.)28 b(Steenro)r(d,)g Fb(The)f(top)r(ology)g(of)g (\014bre)g(bundles)p Fx(,)h(Princeton)f(Univ.)h(Press)e(\(1999\).)386 4444 y([W)-7 b(ei])111 b(A.)45 b(W)-7 b(einstein,)50 b Fc(Contact)c(sur)l(gery)g(and)g(symple)l(ctic)h(hand)t(le)g(b)l(o)l (dies)p Fx(,)k(Hokk)-5 b(aido)681 4543 y(Math.)28 b(Jour.)e(20)h (\(1991\),)f(no.)i(2,)f(241{251.)486 4660 y Fc(E-mail)j(addr)l(ess)7 b Fx(:)38 b Fa(tvogel@mathematik)o(.u)o(ni-)o(mu)o(en)o(che)o(n.)o(de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF