%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: geom30.dvi %%Pages: 2 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: XYATIP10 XYBTIP10 XYDASH10 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips -o geom30.ps geom30.dvi %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 2006.11.03:1824 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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{dup dup 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup 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 /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 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rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /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 %%BeginFont: XYATIP10 % This is a font description converted from raw PostScript % by Xy-pstoAdobe1 running on revision 333 of Aladdin Ghostscript. % purpose: above-portion of arrow-tips 10 dict begin /FontInfo 9 dict dup begin /version(001.001) def /Notice (Copyright \(C\) 1995 Ross Moore) readonly def /FullName(XYATIP10) def /FamilyName 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9b5fadebbda555c6e9ec5201336b2159836bb904f055a9d164 8fd09b283bd3a3599a1e0a937a7bea0c46d1fe72f058e22094 2f30db1b6c |- /d114 55 -! 9b5fadf69a0c7715d08201bc69b257e0d2363c5fa8ea6cb7e1 bef93e5fd91b3f3711413c6ae5e416853708e217759e7b3a4b 44d2f8be3a |- /d14 55 -! 9b5fad0b08b42d851d9ade35c8dee9449462c78e726abc3a14 4f9d9de2ad77dec6e4ef627e6d07599ac7e69547c2c6760168 f66637e4a3 |- /d27 55 -! 9b5fad06feeff8a4b93e75fe568049eccd5973a94866719568 9d1c57d94582f25f14232a43149d47f76f18a345422ce94638 c689868024 |- /d38 55 -! 9b5fadfca09bc4d9f3e142805c1e5dc9ff1aff70c5c81eb93f 82320df8126c883d9ac6859578578b8c2e61381bbb4794a69e a68fd53c23 |- /d51 55 -! 9b5fadf18a3aaefc42ed15bf8c9cfa7f1c1cb544fda9e24a8f 16812f97ec1114754afcdf96ade7c33b319e863821b95627c8 755d793d50 |- /d64 53 -! 9b5fade533deb7943e797335dea3832d23e39bdd4940f1ca9a 85393a8694f856aa36a96649791bea843aa021354d72c1cfeb b452fe |- /d77 55 -! 9b5fade262f40bd51624ba1d7d2fa2970c2bf816d7bf24714d 73b85b15935c23758f186be290d604b408c9ceecfd40b83683 675f765348 |- /d90 55 -! 9b5fade0d2e93c21178ddf3a012511cb51ae5a698bcccc59c0 659eb85d64d1f5207af9985026f7f6d37dc0689f0cfb0ddd9c 8f835c3590 |- /d103 56 -! 9b5fadeac4af62932102c2f0126d52fa35095418e5324ce35a 7fe716bb2fd96573cb4b06fd190a3e7889845c77dc0954095f 4f772e8dfacc |- /d116 55 -! 9b5fadfb92bf28f1f738328ab2bad4c66ddc33b67f3b82cc86 bf4aa7e65248b45d970dcab9c04d838b124c1bbad8842630cd 4a43f29d8b |- /d16 55 -! 9b5fad0b0b19f29735f2047e80bc4b039ea59800d95dd5a8bf 3aa3f5d6710330a69cda009072b008fcbddd9988c7e499ea3a daaf07b0c1 |- /d29 55 -! 9b5fad07320070f1816fb11753eb630caf8140fdd55cf52f74 524d901d27e2d0ecd6db31ea90ed1bac137770b2ea656d171d 139fcd0845 |- /d40 57 -! 9b5fadfdd0c85f74d88b309b32c9c2ed919a4cf71e6ca90ab7 1ce6fc006e846f7baacc04f27090c2e6e8d0f8bc5cbe15ce04 dcdeae36ad7b9b |- /d53 55 -! 9b5fadec8cd258492dee9d4ab9f1b46e281e04e38afd11c1fa 61f810aa0c7e6022dd9668bf4c21b4205d15b249ff4a685d1d dae502876d |- /d66 53 -! 9b5fade6630c6d00011f5537dea19cedc4f114a1cb95e8c4c8 160a18cc319a0ce8282eb200682bf58910080927411f7b6752 83a54c |- /d79 55 -! 9b5fade33a2e75c7b1b306b2df9cba5defe8852b1d48575c76 762776eedb5c365cde02e1a70780d4650628cecac743adf8d5 f6c8e9ecc6 |- /d92 55 -! 9b5fade77a557de15e164e3f5425d884237ef46b1992b1f702 6d4262de52d252c4cb4dcdebdbf19482437965ebe5d7eff71e e26a866c24 |- /d105 55 -! 9b5fade8260b5ef7c934e98ea632f3dc99e47891407a77be18 2758341475cfa71481113b473cfd95f2aba92936a3636124d5 2e47a71ae5 |- /d118 53 -! 9b5fadf9eeb1e5c8a9bcb00640b2660032c7f4bfe9be2065ec b0bf1e9bd18423e2a4d95e5e8ba674256c6a165ccd7668c14f 4f1e89 |- /d5 55 -! 9b5fad07da984c72eb6975a910d32eecdea2b19686f271b90d a9e2fbeaac684eae425e8506cd657ab531101a47fd9ab6fd9e 4e6952f50e |- /d0 53 -! 9b5fad01ec67373d1539976ded5250bd52dffef0decd71d129 63a33ad6cc0b76817936128c40877a2e96303156e079575854 091535 |- /d2 53 -! 9b5fad00311be0d6c679963177b8f180ef5e73d4761060d658 2b3e7b9b1b9f58dea80d783f7e0ccc4026d99534c0c1efe499 a24460 |- /d4 55 -! 9b5fad07d590d5e4182e8ad834320dc1c4d9877953b6a0bdcd ca064debc1821dcf55e0cc010dbbabd9dd646dcac7a35c01be c388895cd9 |- /d1 51 -! 9b5fad01efc838474c1c07682291aaff04ccadc7904e68eaf0 4d32da4b39d0d4195e0fc298ae98f9a1f6cbbdfa8bd4216071 07 |- /d3 55 -! 9b5fad00302a6646256a12f2f06396b7d7e5f0b6f47c12e48e 685120dea1663827d6ad9d22328a7929314716e3091b3a6f0c 858038374e |- end end readonly put noaccess put dup /FontName get exch definefont pop %%EndFont %%BeginFont: XYBTIP10 % This is a font description converted from raw PostScript % by Xy-pstoAdobe1 running on revision 333 of Aladdin Ghostscript. % purpose: below-portion of arrow-tips 10 dict begin /FontInfo 9 dict dup begin /version(001.001) def /Notice (Copyright \(C\) 1995 Ross Moore) readonly def /FullName(XYBTIP10) def /FamilyName (XYBTIP) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def /UnderlinePosition -276 def /UnderlineThickness 138 def end readonly def /FontName(XYBTIP10) def /PaintType 0 def /FontType 1 def /FontMatrix[ 0.001 0 0 0.001 0 0] readonly def /FontBBox{ -552 -552 552 552} readonly def /UniqueID 4345435 def /Encoding 256 array 0 1 255{1 index exch /.notdef put} for dup 0 /d0 put dup 1 /d1 put dup 2 /d2 put dup 3 /d3 put dup 4 /d4 put dup 5 /d5 put dup 6 /d6 put dup 7 /d7 put dup 8 /d8 put dup 9 /d9 put dup 10 /d10 put dup 11 /d11 put dup 12 /d12 put dup 13 /d13 put dup 14 /d14 put dup 15 /d15 put dup 16 /d16 put dup 17 /d17 put dup 18 /d18 put dup 19 /d19 put dup 20 /d20 put dup 21 /d21 put dup 22 /d22 put dup 23 /d23 put dup 24 /d24 put dup 25 /d25 put dup 26 /d26 put dup 27 /d27 put dup 28 /d28 put dup 29 /d29 put dup 30 /d30 put dup 31 /d31 put dup 32 /d32 put dup 33 /d33 put dup 34 /d34 put dup 35 /d35 put dup 36 /d36 put dup 37 /d37 put dup 38 /d38 put dup 39 /d39 put dup 40 /d40 put dup 41 /d41 put dup 42 /d42 put dup 43 /d43 put dup 44 /d44 put dup 45 /d45 put dup 46 /d46 put dup 47 /d47 put dup 48 /d48 put dup 49 /d49 put dup 50 /d50 put dup 51 /d51 put dup 52 /d52 put dup 53 /d53 put dup 54 /d54 put dup 55 /d55 put dup 56 /d56 put dup 57 /d57 put dup 58 /d58 put dup 59 /d59 put dup 60 /d60 put dup 61 /d61 put dup 62 /d62 put dup 63 /d63 put dup 64 /d64 put dup 65 /d65 put dup 66 /d66 put dup 67 /d67 put dup 68 /d68 put dup 69 /d69 put dup 70 /d70 put dup 71 /d71 put dup 72 /d72 put dup 73 /d73 put dup 74 /d74 put dup 75 /d75 put dup 76 /d76 put dup 77 /d77 put dup 78 /d78 put dup 79 /d79 put dup 80 /d80 put dup 81 /d81 put dup 82 /d82 put dup 83 /d83 put dup 84 /d84 put dup 85 /d85 put dup 86 /d86 put dup 87 /d87 put dup 88 /d88 put dup 89 /d89 put dup 90 /d90 put dup 91 /d91 put dup 92 /d92 put dup 93 /d93 put dup 94 /d94 put dup 95 /d95 put dup 96 /d96 put dup 97 /d97 put dup 98 /d98 put dup 99 /d99 put dup 100 /d100 put dup 101 /d101 put dup 102 /d102 put dup 103 /d103 put dup 104 /d104 put dup 105 /d105 put dup 106 /d106 put dup 107 /d107 put dup 108 /d108 put dup 109 /d109 put dup 110 /d110 put dup 111 /d111 put dup 112 /d112 put dup 113 /d113 put dup 114 /d114 put dup 115 /d115 put dup 116 /d116 put dup 117 /d117 put dup 118 /d118 put dup 119 /d119 put dup 120 /d120 put dup 121 /d121 put dup 122 /d122 put dup 123 /d123 put dup 124 /d124 put dup 125 /d125 put dup 126 /d126 put dup 127 /d127 put readonly def currentdict end dup /Private 9 dict dup begin /|{ readonly put} readonly def /-!{ string currentfile exch readhexstring pop} readonly def /|-{ readonly def} readonly def /-|{ string currentfile exch readstring pop} readonly def /UniqueID 4345435 def /BlueValues[] def /MinFeature{ 16 16} readonly def /password 5839 def /lenIV 0 def 2 index /CharStrings 129 dict dup begin /d18 49 -! 6ada4d35409d29ab5c164cdbc324c51f496e0215100f2c8080 a0a60261e5f7e4ec533c0ad6d9ecb1107720e4637bc2b6ee |- /d31 56 -! 6ada4d2af98b7f22fddd1dee14163c2f2cd4179b5c46e39fca 6ce0a8925da2b9355aca1c4bf5cbc2ee6f07031d012be7ac5a 858221d54c27 |- /d42 54 -! 6ada4d284a59dfeb42abdf9561584aeb2401ceb23ddef7a580 4bf408da46abb5fb9dda3a958dfa0aa8fd1fdd62cdbe273990 88cbbc1f |- /d55 56 -! 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6ada4d292f346dc9d0ea1615cb9d485f00bd7b192b07284641 bf973f29335d680da854a0d1b05a8e0cd8e47499f187d712b5 1da3fc87549d |- /d16 49 -! 6ada4d354099f07acac864776a87f7f5006b1bf725a3ca453c 0f4e21ca7c2c3a2ac92970007e11b8195ba990dcb489fc10 |- /d29 56 -! 6ada4d2af98b7888f636480aa8218d7730c8493f73f12b86aa d7924a03ed19ca39b90896e3bb21359e8f8b5eab2f984f7dc2 a27e10a7a9d2 |- /d40 56 -! 6ada4d28553efd9a74c8eab1f731c5e3dcd2825db62bf3297a 4da1ae3f0dd1e846b68f611796bd268340398ab1c6e94af36c 050e5fc5b344 |- /d53 56 -! 6ada4d2fef87818eaae169442bb77c28ce8936ae134e450c99 42e75014cd9a81e479836910a985131c54971e4be9bde6e091 e6ca989acd7e |- /d66 56 -! 6ada4d228a5c3f0571ae95ce52aa2e9d3b4d7b22d559693341 ad27877beabbb16244d3ee0057deb3ec8a98e185bac79b0c66 3d2c06165418 |- /d79 50 -! 6ada4d235e7c4d51a331f3897e262d74027290e56cd0af99ce a84098d8b6eb79b26c88756f70594664e9b0506ebc30c1c6c0 |- /d92 56 -! 6ada4d228cf3a53e246d7f113e544a2dca4e043e3401dc1966 57be07fc9e580d944c876e95fd9824d9a619f687ee8edc504a ee33fe69f2b9 |- /d105 54 -! 6ada4d2c93864e7c3d678d4e6af87d6628be5ce5fc38ef58d1 512fc8119767a071a63d36b19a47c21ac35f7022d5e950db69 d2db63e7 |- /d118 56 -! 6ada4d292f346a7e2aa6a98010d83f23d1aad4a6b541ba60e1 d326e8acfca89121dce6677769369a29b0ae1a16bb597c7535 d4a5e2403adc |- /d5 56 -! 6ada4d2aff615ea8c759748cdb8add71345d80df11afe2497e c59d9c16d05325739003c29d3253346e804b1c73cf14307aec ecc873383c40 |- /d0 56 -! 6ada4d2bc105f0a39ec541c3929fca1bfce24248b77177cbb4 28d648b21a168633e061ea40d95ff589322963418ced47cdad 5f743f161855 |- /d2 56 -! 6ada4d2aff6158fdeb0ca694a6548f7ff6d9fe781a59e63b0b f78a26e6afc08edb573cc92c7245ebdbc1bae6f3f98a7465a2 58ea1eac1b38 |- /d4 56 -! 6ada4d2aff615f9bb8ad6a67b45b6ab88d53bd541995afb00b dcb5bb54cfdf84f2ad5ea0aac72ec519e7faca83a82d5f99c6 426d88beba72 |- /d1 56 -! 6ada4d2af049c302e9e8dcdcd5fb3f8b6be1fd45ff0113828a c72cde4e3acb0a117986e1c3493bb15596d7e37e011850501b 0e683f1bd6c2 |- /d3 56 -! 6ada4d2aff615f9bb8a731b24edd06d3446dde64bf5974c989 d24bd22aeb0bfe9dcc4e6371e1be979d7d1ab34259d3f9a1d8 f0457f2781bb |- end end readonly put noaccess put dup /FontName get exch definefont pop %%EndFont %%BeginFont: XYDASH10 % This is a font description converted from raw PostScript % by Xy-pstoAdobe1 running on revision 333 of Aladdin Ghostscript. % purpose: % straight segments, dashes 10 dict begin /FontInfo 9 dict dup begin /version(001.001) def /Notice (Copyright \(C\) 1995 Ross Moore) readonly def /FullName(XYDASH10) def /FamilyName (XYDASH) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def /UnderlineThickness 150 def /UnderlinePosition -300 def end readonly def /FontName(XYDASH10) def /PaintType 0 def /FontType 1 def /FontMatrix[ 0.001 0 0 0.001 0 0] readonly def /FontBBox{ -600 -600 600 600} readonly def /UniqueID 4345440 def /Encoding 256 array 0 1 255{1 index exch /.notdef put} for dup 0 /a0 put dup 1 /a1 put dup 2 /a2 put dup 3 /a3 put dup 4 /a4 put dup 5 /a5 put dup 6 /a6 put dup 7 /a7 put dup 8 /a8 put dup 9 /a9 put dup 10 /a10 put dup 11 /a11 put dup 12 /a12 put dup 13 /a13 put dup 14 /a14 put dup 15 /a15 put dup 16 /a16 put dup 17 /a17 put dup 18 /a18 put dup 19 /a19 put dup 20 /a20 put dup 21 /a21 put dup 22 /a22 put dup 23 /a23 put dup 24 /a24 put dup 25 /a25 put dup 26 /a26 put dup 27 /a27 put dup 28 /a28 put dup 29 /a29 put dup 30 /a30 put dup 31 /a31 put dup 32 /a32 put dup 33 /a33 put dup 34 /a34 put dup 35 /a35 put dup 36 /a36 put dup 37 /a37 put dup 38 /a38 put dup 39 /a39 put dup 40 /a40 put dup 41 /a41 put dup 42 /a42 put dup 43 /a43 put dup 44 /a44 put dup 45 /a45 put dup 46 /a46 put dup 47 /a47 put dup 48 /a48 put dup 49 /a49 put dup 50 /a50 put dup 51 /a51 put dup 52 /a52 put dup 53 /a53 put dup 54 /a54 put dup 55 /a55 put dup 56 /a56 put dup 57 /a57 put dup 58 /a58 put dup 59 /a59 put dup 60 /a60 put dup 61 /a61 put dup 62 /a62 put dup 63 /a63 put dup 64 /a64 put dup 65 /a65 put dup 66 /a66 put dup 67 /a67 put dup 68 /a68 put dup 69 /a69 put dup 70 /a70 put dup 71 /a71 put dup 72 /a72 put dup 73 /a73 put dup 74 /a74 put dup 75 /a75 put dup 76 /a76 put dup 77 /a77 put dup 78 /a78 put dup 79 /a79 put dup 80 /a80 put dup 81 /a81 put dup 82 /a82 put dup 83 /a83 put dup 84 /a84 put dup 85 /a85 put dup 86 /a86 put dup 87 /a87 put dup 88 /a88 put dup 89 /a89 put dup 90 /a90 put dup 91 /a91 put dup 92 /a92 put dup 93 /a93 put dup 94 /a94 put dup 95 /a95 put dup 96 /a96 put dup 97 /a97 put dup 98 /a98 put dup 99 /a99 put dup 100 /a100 put dup 101 /a101 put dup 102 /a102 put dup 103 /a103 put dup 104 /a104 put dup 105 /a105 put dup 106 /a106 put dup 107 /a107 put dup 108 /a108 put dup 109 /a109 put dup 110 /a110 put dup 111 /a111 put dup 112 /a112 put dup 113 /a113 put dup 114 /a114 put dup 115 /a115 put dup 116 /a116 put dup 117 /a117 put dup 118 /a118 put dup 119 /a119 put dup 120 /a120 put dup 121 /a121 put dup 122 /a122 put dup 123 /a123 put dup 124 /a124 put dup 125 /a125 put dup 126 /a126 put dup 127 /a127 put readonly def currentdict end dup /Private 9 dict dup begin /|{ readonly put} readonly def /|-{ readonly def} readonly def /-|{ string currentfile exch readstring pop} readonly def /-!{ string currentfile exch readhexstring pop} readonly def /UniqueID 4345440 def /BlueValues[] def /MinFeature{ 16 16} readonly def /password 5839 def /lenIV 0 def 2 index /CharStrings 129 dict dup begin /a9 45 -! 6ada4d3e185576cac1ea873b2078a65a5d7d9e86071ada0469 acf58d623c01743b642b03be18e5c7e65a9d1441 |- /a121 45 -! 6ada4d3db2c6bdf10502067edccc3e9cc0236982bd9b2772c6 7656e8c315cb536d650f178787aaec4e97bf2e44 |- /.notdef 4 -! 9b5fad72 |- /a123 45 -! 6ada4d3db52ab2a44f872a574e54ad298a1542eed49410faf2 d9dda6c016252d0edd0f98ac0522dd2dea8c7093 |- /a0 45 -! 6ada4d3c86ab4ec08268e41ab8aa5fa47fd8cbfaa779848412 e260e88c809ad370457f0ac87b753c1f238e8f56 |- /a125 45 -! 6ada4d3c8798f9da790d9da4717a28793245497b2a52dd3bdf b84265bbc1d4ed027982fe99b98059f745838021 |- /a2 45 -! 6ada4d3f18ceb8c94b21455dd519838a00e2bc29b09a9dc283 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Fk(b)r(e)f(a)e(metric)h(on)g (E)g(for)f Fi(F)37 b Fc(\032)25 b Fi(E)35 b Fk(a)28 b(subbundle)i(of)f (E,)f(de\014ne)i Fi(F)2801 501 y Fh(?)2882 531 y Fk(:=)25 b Fc(f)p Fi(v)j Fc(2)e Fi(E)34 b Fc(j)440 631 y Fi(g)s Fk(\()p Fi(v)s(;)14 b(w)r Fk(\))25 b(=)f(0,)k(for)g(all)g Fi(w)f Fc(2)d Fi(F)41 b Fk(with)28 b Fi(\031)s Fk(\()p Fi(w)r Fk(\))e(=)e Fi(\031)s Fk(\()p Fi(v)s Fk(\))p Fc(g)p Fk(.)29 b(Sho)n(w)f(using)g(part)f(a\))i(that)f Fi(F)3094 601 y Fh(?)3179 631 y Fk(is)g(a)440 731 y(v)n(ector)e(bundle)i(o)n(v)n (er)e(M.)i(F)-7 b(urthermore)27 b(sho)n(w)f(that)i Fi(E)5 b(=F)2329 708 y Fc(\030)2329 735 y Fk(=)2416 731 y Fi(F)2481 700 y Fh(?)2565 731 y Fk(as)27 b(bundles.)440 930 y(5.)j(Using)f(the)i (follo)n(wing)e(description)g(of)h(the)g(tangen)n(t)g(bundle)g(of)g(a)g (sphere)f(as)h Fi(T)12 b(S)3169 900 y Fe(n)3240 930 y Fk(=)440 1029 y Fc(f)p Fk(\()p Fi(x;)i(v)s Fk(\))31 b Fc(2)f Fi(S)845 999 y Fe(n)911 1029 y Fc(\002)21 b Fb(R)1051 999 y Fe(n)p Fa(+1)1218 1029 y Fc(j)32 b Fi(x:v)i Fk(=)c(0)p Fc(g)p Fk(,)h(construct)g(a)g(non-v)-5 b(anishing)31 b(section)h(on)f Fi(T)12 b(S)3143 999 y Fa(2)p Fe(n)p Fa(+1)440 1129 y Fk(for)27 b Fi(n)c Fc(\025)g Fk(0.)440 1229 y(F)-7 b(or)27 b(extra)g(credit)g(sho)n(w)g(that)h Fi(T)12 b(S)1542 1199 y Fa(3)1601 1207 y Fc(\030)1601 1233 y Fk(=)1688 1229 y Fi(S)1744 1199 y Fa(3)1800 1229 y Fc(\002)18 b Fb(R)1937 1199 y Fa(3)1980 1229 y Fi(:)440 1328 y Fk(\(Hin)n(t:)28 b(since)g Fi(n)f Fk(is)h(o)r(dd)g(y)n(ou)e(ma)n (y)h(iden)n(tify)h Fb(R)1905 1298 y Fa(2)p Fe(n)q Fa(+2)2101 1328 y Fk(with)g Fb(C)2344 1298 y Fe(n)p Fa(+1)2480 1328 y Fk(\))p Fi(:)440 1677 y Fk(Please)e(hand)h(in)h(y)n(our)e(solutions)g (on)h(F)-7 b(rida)n(y)26 b(the)i(10th)f(of)g(No)n(v)n(em)n(b)r(er)f(b)n (y)h(4)g(PM,)f(in)i(the)440 1777 y(appropriate)e(b)r(o)n(x.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF