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Fk(\010)p Fj(;)456 4140 y Fk(where)36 b Fj(z)k Fk(denotes)d(the)e(zero)h(section)f(of)g Fj(T)14 b(M)46 b Fk(and)35 b Fj(x)h Fk(runs)g(through)f Fj(M)10 b Fk(.)52 b(It)36 b(is)456 4256 y(kno)m(wn)k(that)e(the)h(plane)f(\014elds)h Fj(E)1762 4220 y Fg(u)1846 4256 y Fk(and)g Fj(E)2120 4220 y Fg(s)2195 4256 y Fk(are)g(\(in)f(general)g(not)g(uniquely\))456 4372 y(in)m(tegrable.)49 b(In)35 b(the)g(case)h(in)m(tegral)d (submanifolds)h(are)h(uniquely)f(determined)456 4488 y(w)m(e)d(call)d(them)i Fh(F)1102 4452 y Fg(u)1176 4488 y Fk(and)h Fh(F)1446 4452 y Fg(s)1512 4488 y Fk(resp)s(ectiv)m(ely)-8 b(.)43 b(One)31 b(can)f(get)g(unique)h(in)m(tegrabilit)m(y)456 4605 y(assuming)h(for)g(example)h(that)g(the)g(plane)g(\014elds)g Fj(E)2381 4569 y Fg(u)2459 4605 y Fk(and)g Fj(E)2727 4569 y Fg(s)2797 4605 y Fk(are)g(of)g(class)g Fj(C)3378 4569 y Fi(1)3417 4605 y Fk(.)456 4721 y(The)g(follo)m(wing)d(result)j (is)f(due)h(to)f(No)s(da.)456 4867 y Fd(Theorem)45 b(1.)g Fc(A)c Fj(C)1261 4831 y Ff(1)1377 4867 y Fc(\015ow)g(on)g Fj(T)1807 4831 y Fi(2)1887 4867 y Fc(without)h(R)-5 b(e)g(eb)41 b(c)-5 b(omp)g(onents)39 b(c)-5 b(an)41 b(b)-5 b(e)41 b(ex-)456 4983 y(tende)-5 b(d)36 b(to)i(a)f(pr)-5 b(oje)g(ctively)37 b(A)n(nosov)f(\015ow)h(on)g Fj(T)2251 4947 y Fi(2)2314 4983 y Fh(\002)24 b Fk(\()p Fh(\000)p Fj(\017;)17 b(\017)p Fk(\))39 b Fc(so)d(that)i(invariant)456 5099 y(foliations)33 b(ar)-5 b(e)34 b(of)g(class)g Fj(C)1471 5063 y Fi(1)1545 5099 y Fc(and)g Fj(T)1805 5063 y Fi(2)1878 5099 y Fc(is)h(a)f(c)-5 b(omp)g(act)33 b(le)-5 b(af)34 b(of)g(the)h(unstable)f(foli-)456 5216 y(ation.)1931 5315 y Fb(1)p eop %%Page: 2 2 2 1 bop 456 251 a Fb(2)555 450 y Fk(T)-8 b(o)25 b(pro)m(v)m(e)h(this)f (theorem)g(w)m(e)h(will)c(need)k(one)g(criterion)d(of)i(b)s(eing)f(pro) 5 b(jectiv)m(ely)456 566 y(Anoso)m(v,)33 b(whic)m(h)g(w)m(e)h(no)m(w)f (state.)456 711 y Fd(Prop)s(osition)44 b(1.)h Fc(Supp)-5 b(ose)41 b Fk(\010)1650 675 y Fg(t)1722 711 y Fc(is)h(a)g(pA)g(\015ow)f (with)h Fj(C)2599 675 y Fi(1)2638 711 y Fc(-smo)-5 b(oth)42 b(stable)f(and)456 827 y(unstable)35 b(foliations)g Fh(F)1344 791 y Fg(u)1424 827 y Fc(and)g Fh(F)1696 791 y Fg(s)1732 827 y Fc(.)47 b(Then)35 b Fh(F)2146 791 y Fg(u)2226 827 y Fc(and)g Fh(F)2498 791 y Fg(s)2570 827 y Fc(c)-5 b(an)35 b(b)-5 b(e)36 b(de\014ne)-5 b(d)34 b(by)i Fj(C)3405 791 y Fi(1)456 944 y Fc(1-forms)e Fj(\013)878 959 y Fg(u)957 944 y Fc(and)h Fj(\013)1209 959 y Fg(s)1280 944 y Fc(such)g(that)456 1117 y Fk(\(1\))558 b Fj(<)28 b(\013)1305 1132 y Fg(u)1350 1117 y Fj(;)17 b(\013)1456 1132 y Fg(s)1520 1117 y Fj(>)p Fk(=)27 b Fj(\013)1761 1132 y Fg(u)1828 1117 y Fh(^)c Fj(d\013)2030 1132 y Fg(s)2089 1117 y Fk(+)f Fj(\013)2249 1132 y Fg(s)2308 1117 y Fh(^)g Fj(d\013)2509 1132 y Fg(u)2582 1117 y Fj(>)27 b Fk(0)p Fj(:)456 1291 y Fc(Conversely,)32 b(supp)-5 b(ose)33 b Fj(C)1413 1254 y Fi(1)1486 1291 y Fc(forms)g Fj(\013)1822 1306 y Fg(s)1892 1291 y Fc(and)g Fj(\013)2142 1306 y Fg(u)2221 1291 y Fc(satisfy)g(the)h(ine)-5 b(quality)42 b Fk(\(1\))33 b Fc(and)456 1407 y(de\014ne)c(the)h (foliations)g Fh(F)1396 1371 y Fg(u)1470 1407 y Fc(and)g Fh(F)1737 1371 y Fg(s)1773 1407 y Fc(,)i(then)e(any)g(non-vanishing)e (ve)-5 b(ctor)30 b(\014eld)g(on)456 1523 y Fh(F)538 1487 y Fg(u)604 1523 y Fh(\\)23 b(F)775 1487 y Fg(s)846 1523 y Fc(de\014nes)34 b(a)g(pA)i(\015ow.)555 1668 y(Pr)-5 b(o)g(of)38 b Fk(of)g(the)h(theorem.)60 b(Assume)40 b(that)e(the)g (\015o)m(w)h(\010)g(on)f Fj(T)2833 1632 y Fi(2)2911 1668 y Fk(has)g(no)h(Reeb)456 1784 y(comp)s(onen)m(ts.)59 b(Then)39 b(the)f(1-foliation)33 b Fh(F)2029 1748 y Fg(s)2103 1784 y Fk(it)k(de\014nes)i(on)e Fj(T)2742 1748 y Fi(2)2819 1784 y Fk(can)h(b)s(e)g(de\014ned)456 1900 y(b)m(y)33 b(the)g(smo)s(oth)f(1-form)456 2074 y(\(2\))948 b Fj(\013)28 b Fk(=)g Fj(dy)d Fk(+)d Fj( )t Fk(\()p Fj(x;)17 b(y)t Fk(\))p Fj(dx;)456 2247 y Fk(for)32 b(some)g(co)s(ordinate)g(system)h (\()p Fj(x;)17 b(y)t Fk(\))32 b(on)h Fj(T)2118 2211 y Fi(2)2189 2247 y Fk(and)g(some)f(smo)s(oth)g(function)g Fj( )t Fk(.)456 2363 y(Consider)43 b(the)h(manifold)c Fj(T)1537 2327 y Fi(2)1605 2363 y Fh(\002)30 b Fk(\()p Fh(\000)p Fj(\017;)17 b(\017)p Fk(\))44 b(and)f(let)g(the)g(last)g (factor)f(carry)i(the)456 2480 y(natural)33 b(co)s(ordinate)g Fj(z)t Fk(.)50 b(F)-8 b(rom)33 b(no)m(w)i(on)f(the)h(co)s(ordinates)f (\()p Fj(x;)17 b(y)t(;)g(z)t Fk(\))34 b(are)g(\014xed.)456 2596 y(W)-8 b(e)27 b(let)f(the)h(foliation)c Fh(F)1373 2560 y Fg(s)1437 2596 y Fk(on)j Fj(T)1637 2560 y Fi(2)1687 2596 y Fh(\002)10 b Fk(\()p Fh(\000)p Fj(\017;)17 b(\017)p Fk(\))28 b(b)s(e)f(de\014ned)h(b)m(y)g(the)f(smo)s(oth)f(1-form)456 2712 y(whic)m(h)32 b(satis\014es)i(\(2\))d(on)i(the)f(whole)h Fj(T)1901 2676 y Fi(2)1961 2712 y Fh(\002)23 b Fk(\()p Fh(\000)p Fj(\017;)17 b(\017)p Fk(\).)44 b(T)-8 b(o)32 b(simplify)e(the)j(notation)456 2828 y(this)43 b(1-form)g(is)g(again)g (denoted)i(b)m(y)g Fj(\013)q Fk(.)78 b(Let)45 b(the)f(smo)s(oth)g (foliation)c Fh(F)3255 2792 y Fg(u)3344 2828 y Fk(b)s(e)456 2945 y(de\014ned)34 b(b)m(y)f(the)g(smo)s(oth)f(1-form)1616 3118 y Fj(\014)h Fk(=)28 b Fj(Az)t(dx)23 b Fk(+)f Fj(dz)t(;)456 3292 y Fk(where)42 b(the)g(p)s(ositiv)m(e)e(constan)m(t)i Fj(A)f Fk(is)g(not)g(\014xed)h(for)f(the)h(momen)m(t.)68 b(Then)42 b(it)456 3408 y(holds)32 b(that)1245 3541 y Fj(<)27 b(\013)q(;)17 b(\014)33 b(>)p Fk(=)28 b(\()p Fj(A)22 b Fh(\000)g Fj( )2018 3556 y Fg(y)2060 3541 y Fk(\))p Fj(dx)g Fh(^)h Fj(dy)i Fh(^)e Fj(dz)t(:)456 3694 y Fk(This)30 b(is)g(p)s(ositiv)m(e)g(for)g(su\016cien)m(tly)h(large)e Fj(A)p Fk(.)43 b(The)31 b(application)d(of)i(Prop)s(osition)456 3810 y(1)i(with)g Fj(\013)821 3825 y Fg(s)886 3810 y Fk(=)27 b Fj(\013)q Fk(,)32 b Fj(\013)1173 3825 y Fg(u)1246 3810 y Fk(=)c Fj(\014)38 b Fk(completes)32 b(the)h(pro)s(of)f(of)g(the) h(theorem.)555 3926 y(No)m(w)53 b(w)m(e)g(discuss)h(sev)m(eral)e (examples)h(of)e(Anoso)m(v)j(pro)5 b(jectiv)m(ely)52 b(Anoso)m(v)456 4042 y(\015o)m(ws.)100 b(Consider)52 b(the)f(simple)f(Lie)g(algebra)g Fj(psl)r Fk(\(2)p Fj(;)17 b Fe(R)11 b Fk(\))51 b(and)g(\014x)h(the)f(basis)456 4220 y(\()p Fj(h;)17 b(l)r(;)g(k)s Fk(\))29 b(for)h(it:)41 b Fj(h)28 b Fk(=)1269 4181 y Fi(1)p 1269 4197 36 4 v 1269 4254 a(2)1331 4080 y Fa(\022)1445 4160 y Fk(1)122 b(0)1445 4276 y(0)83 b Fh(\000)p Fk(1)1745 4080 y Fa(\023)1818 4220 y Fk(,)31 b Fj(l)f Fk(=)2049 4181 y Fi(1)p 2049 4197 V 2049 4254 a(2)2110 4080 y Fa(\022)2225 4160 y Fk(0)122 b(1)2225 4276 y(1)83 b Fh(\000)p Fk(0)2525 4080 y Fa(\023)2598 4220 y Fk(,)31 b Fj(k)g Fk(=)2851 4181 y Fi(1)p 2851 4197 V 2851 4254 a(2)2913 4080 y Fa(\022)3028 4160 y Fk(0)83 b Fh(\000)p Fk(1)3028 4276 y(1)g Fh(\000)p Fk(0)3327 4080 y Fa(\023)3417 4220 y Fj(:)456 4398 y Fk(One)39 b(c)m(hec)m(ks)k(b)m(y)d(hand)g(the)g(follo)m(wing)d(comm)m (utator)h(relations:)56 b([)p Fj(h;)17 b(l)r Fk(])39 b(=)h Fh(\000)p Fj(l)r Fk(,)456 4516 y([)p Fj(h;)17 b(k)s Fk(])53 b(=)h Fj(k)s Fk(.)89 b(T)-8 b(ak)m(e)49 b(a)e(lattice)g (\(co-compact)g(discrete)h(subgroup\))g(\000)g(in)f Fe(R)3405 4474 y Fi(3)456 4632 y Fk(view)m(ed)d(as)g(the)g(univ)m(ersal)g(co)m(v) m(er)h(of)e Fj(S)6 b(L)p Fk(\(2)p Fj(;)17 b Fe(R)11 b Fk(\))43 b(and)h(consider)g(the)g(quotien)m(t)456 4751 y Fj(M)38 b Fk(=)28 b Fe(R)764 4708 y Fi(3)803 4751 y Fj(=)p Fk(\000.)43 b(Let)32 b Fj(X)8 b Fk(,)33 b Fj(e)1351 4714 y Fg(s)1420 4751 y Fk(and)f Fj(e)1654 4714 y Fg(u)1731 4751 y Fk(b)s(e)h(the)f(v)m(ector)i(\014elds)e(coming)f(from)g(the)h (left)456 4867 y(in)m(v)-5 b(arian)m(t)38 b(v)m(ector)j(\014elds)e (corresp)s(onding)h(to)f(the)h(Lie)f(algebra)g(elemen)m(ts)h Fj(h)p Fk(,)h Fj(l)456 4983 y Fk(and)d Fj(k)k Fk(resp)s(ectiv)m(ely)-8 b(.)63 b(Then)39 b(the)h(ab)s(o)m(v)m(e)f(comm)m(utator)e(relations)g (mean)i(that)456 5099 y Fj(X)45 b Fk(is)38 b(an)g(Anoso)m(v)h(v)m (ector)g(\014eld)e(and)h Fj(e)1931 5063 y Fg(s)2006 5099 y Fk(and)g Fj(e)2246 5063 y Fg(u)2329 5099 y Fk(span)h(the)f (\(strongly\))g(stable)456 5216 y(and)32 b(the)h(unstable)g (distributions)e(resp)s(ectiv)m(ely)-8 b(.)p eop %%Page: 3 3 3 2 bop 3406 251 a Fb(3)555 452 y Fk(T)-8 b(ak)m(e)28 b(ab)s(elian)c(Lie)i(groups)h Fe(R)1656 410 y Fi(2)1721 452 y Fk(and)g Fe(R)37 b Fk(together)27 b(with)f(their)g(lattices)f Fe(Z)3221 410 y Fi(2)3287 452 y Fk(and)456 568 y Fe(Z)38 b Fk(resp)s(ectiv)m(ely)-8 b(.)70 b(T)-8 b(ak)m(e)42 b(a)f(unimo)s(dular)e(h)m(yp)s(erb)s(olic)h(matrix)g Fj(A)i Fh(2)g Fj(S)6 b(L)p Fk(\(2)p Fj(;)17 b Fe(Z)m Fk(\),)456 685 y Fj(tr)s(aceA)28 b(>)f Fk(2,)h(and)e(consider)h(the)g (semi-direct)e(pro)s(ducts)i Fj(S)2654 700 y Fg(A)2737 685 y Fk(of)f(the)h(Lie)f(groups)456 801 y(and)37 b(\000)711 816 y Fg(A)804 801 y Fk(of)g(their)f(lattices)g(de\014ned)i(b)m(y)g (the)f(action)f Fj(exp)p Fk(\()p Fj(tl)r(og)t(A)p Fk(\))h(of)f Fj(t)f Fh(2)h Fe(R)47 b Fk(on)456 919 y Fe(R)528 877 y Fi(2)567 919 y Fk(.)53 b(Consider)37 b(the)f(Lie)f(algebra)g Fj(s)1787 934 y Fg(a)1864 919 y Fk(asso)s(ciated)h(to)f(the)i(Lie)e (group)h Fj(S)3133 934 y Fg(A)3190 919 y Fk(.)53 b(It)36 b(is)456 1036 y(easy)30 b(to)g(compute)g(that)f(the)h(Lie)g(algebra)e Fj(s)2096 1051 y Fg(a)2167 1036 y Fk(is)i(a)f(semi-direct)g(pro)s(duct) h(of)f(the)456 1152 y(Lie)f(groups)h Fe(R)998 1109 y Fi(2)1067 1152 y Fk(and)f Fe(R)40 b Fk(de\014ned)30 b(b)m(y)f(the)h (action)d Fj(\034)11 b(l)r(og)t Fk(\()p Fj(A)p Fk(\))29 b(of)f Fj(\034)39 b Fh(2)28 b Fe(R)40 b Fk(on)28 b Fe(R)3215 1109 y Fi(2)3255 1152 y Fk(.)42 b(By)456 1270 y(c)m(hanging)32 b(the)h(basis)g(of)g Fe(R)1457 1228 y Fi(2)1529 1270 y Fk(according)f(to)h(the)g(eigen)m(v)-5 b(alue)33 b(decomp)s(osition)e (of)456 1386 y Fj(A)k Fk(and)h(c)m(hanging)f(the)h(basis)g(of)f Fe(R)46 b Fk(b)m(y)37 b(scaling)d(with)h(the)h(log)f(of)g(the)h (greatest)456 1503 y(eigen)m(v)-5 b(alue)31 b(of)h Fj(A)h Fk(w)m(e)g(see)g(that)g(the)f(Lie)g(algebra)f Fj(s)2368 1518 y Fg(a)2442 1503 y Fk(is,)h(indep)s(enden)m(tly)h(of)f(the)456 1619 y(c)m(hoice)26 b(of)g Fj(A)p Fk(,)i(isomorphic)c(to)i(the)h(solv) -5 b(able)25 b(Lie)h(algebra)f Fj(sol)r(v)t Fk(,)i(the)g(semi-direct) 456 1791 y(pro)s(duct)34 b(of)g Fe(R)1008 1749 y Fi(2)1082 1791 y Fk(and)h Fe(R)11 b Fk(,)35 b(de\014ned)h(b)m(y)f(the)g(action)f Fj(\034)2418 1650 y Fa(\022)2533 1731 y Fk(1)121 b(0)2533 1847 y(0)83 b Fh(\000)p Fk(1)2832 1650 y Fa(\023)2940 1791 y Fk(of)34 b Fj(\034)43 b Fh(2)31 b Fe(R)45 b Fk(on)456 1979 y Fe(R)528 1937 y Fi(2)567 1979 y Fk(.)51 b(If)35 b(w)m(e)h(tak)m(e)g(the)g(basis)f(\()p Fj(k)s(;)17 b(l)r(;)g(h)p Fk(\))35 b(of)f Fj(sol)r(v)t Fk(,)i(according)e(to)h(the)h(ab)s(o)m(v)m (e)g(semi-)456 2095 y(direct)30 b(pro)s(duct)h(decomp)s(osition,)f(it)f (is)i(easy)g(to)g(calculate)e(that)i(the)g(follo)m(wing)456 2211 y(comm)m(utator)i(relations)g(hold:)46 b([)p Fj(h;)17 b(l)r Fk(])31 b(=)g Fh(\000)p Fj(l)r Fk(,)36 b([)p Fj(h;)17 b(k)s Fk(])31 b(=)g Fj(k)s Fk(.)49 b(T)-8 b(ak)m(e)36 b(the)f(lattice)e(\000)3403 2226 y Fg(a)456 2328 y Fk(the)40 b(Lie)g(group)g Fj(S)1147 2343 y Fg(A)1244 2328 y Fk(and)h(consider)f (the)h(quotien)m(t)g Fj(M)51 b Fk(=)41 b Fj(S)2720 2343 y Fg(A)2777 2328 y Fj(=)p Fk(\000)2887 2343 y Fg(a)2928 2328 y Fk(.)67 b(Let)40 b Fj(X)8 b Fk(,)42 b Fj(e)3407 2292 y Fg(s)456 2444 y Fk(and)33 b Fj(e)691 2408 y Fg(u)770 2444 y Fk(b)s(e)g(the)h(v)m(ector)g(\014elds)g(coming)d(from)i(the)g (left)g(in)m(v)-5 b(arian)m(t)32 b(v)m(ector)i(\014elds)456 2560 y(corresp)s(onding)25 b(to)h(the)g(Lie)f(algebra)g(elemen)m(ts)h Fj(h)p Fk(,)h Fj(l)h Fk(and)e Fj(k)j Fk(resp)s(ectiv)m(ely)-8 b(.)42 b(Then)456 2676 y(the)29 b(ab)s(o)m(v)m(e)g(comm)m(utator)e (relations)g(mean)h(that)h Fj(X)36 b Fk(is)28 b(an)g(Anoso)m(v)i(v)m (ector)g(\014eld)456 2793 y(and)e Fj(e)686 2756 y Fg(s)752 2793 y Fk(and)g Fj(e)982 2756 y Fg(u)1056 2793 y Fk(span)h(the)g (\(strongly\))f(stable)g(and)g(the)h(unstable)g(distributions)456 2909 y(resp)s(ectiv)m(ely)-8 b(.)555 3025 y(This)50 b(last)e(example)h (giv)m(es)h(rise)f(to)g(an)g(EpA)h(\015o)m(w,)k(i.e.)94 b(a)49 b(pro)5 b(jectiv)m(ely)456 3141 y(Anoso)m(v)44 b(\015o)m(w,)k(whic)m(h)c(is)f(not)g(Anoso)m(v.)78 b(It)44 b(is)f(w)m(ell-kno)m(wn)h(that)f(the)i(mani-)456 3258 y(fold)38 b Fj(M)49 b Fk(w)m(e)41 b(ha)m(v)m(e)f(just)g(constructed)h (admits)c(another)j(description,)g(namely)456 3374 y(it)f(is)g(a)h Fj(T)826 3338 y Fi(2)865 3374 y Fk(-bundle)g(o)m(v)m(er)h Fj(S)1507 3338 y Fi(1)1587 3374 y Fk(obtained)e(as)i(a)f(susp)s(ension) h(of)e(the)i(h)m(yp)s(erb)s(olic)456 3490 y(linear)36 b(di\013eomorphism)g Fj(A)i Fk(of)g Fj(T)1723 3454 y Fi(2)1762 3490 y Fk(.)61 b(And)38 b(the)h(v)m(ector)g(\014eld)f Fj(X)46 b Fk(is)38 b(the)h(canon-)456 3606 y(ical)e(horizon)m(tal)g(v)m (ector)i(\014eld.)62 b(Because)40 b(the)f(map)f Fj(A)h Fk(has)g(\014xed)h(p)s(oin)m(ts,)g(the)456 3722 y(v)m(ector)35 b(\014eld)f Fj(X)42 b Fk(has)34 b(closed)h(orbits.)47 b(Then)36 b(the)e(\015o)m(w)h(of)f Fj(X)42 b Fk(can)34 b(b)s(e)g(deformed)456 3839 y(in)k(a)h(small)f(neigh)m(b)s(ourho)s(o)s (d)g(of)h(the)h(a)f(closed)h(orbit)e(in)g(suc)m(h)j(a)e(w)m(a)m(y)i (that)e(it)456 3955 y(is)34 b(con)m(tracting)g(there)i(\(i.e.)49 b(is)34 b(no)h(longer)e(Anoso)m(v\),)k(but)e(it)e(con)m(tracts)j(in)e (one)456 4071 y(direction)25 b(m)m(uc)m(h)h(faster)h(than)f(in)f(the)i (other,)g(i.e.)41 b(is)26 b(a)g(pA)g(\015o)m(w.)42 b(Next)27 b(example)456 4187 y(sho)m(ws)42 b(ho)m(w)f(to)f(construct)h(an)g(EpA)g (\015o)m(w)g(on)f(an)m(y)h Fj(T)2513 4151 y Fi(2)2552 4187 y Fk(-bundle)f(o)m(v)m(er)i Fj(S)3195 4151 y Fi(1)3234 4187 y Fk(,)g(not)456 4304 y(necessarily)33 b(with)f(h)m(yp)s(erb)s (olic)g(mono)s(drom)m(y)-8 b(.)555 4420 y(A)43 b(pair)e(\005)k(=)g(\()p Fj(\030)1201 4435 y Fi(0)1240 4420 y Fj(;)17 b(\021)1332 4435 y Fi(0)1371 4420 y Fk(\))42 b(of)g(con)m(tact)i(plane)e(\014elds)h (is)f(called)f(pre-bi-con)m(tact)456 4536 y(structure)35 b(\(pbC\))h(if)d(there)i(exists)h(a)e(common)f(smo)s(oth)h (non-singular)e(Legen-)456 4652 y(drian)h(v)m(ector)i(\014eld)f Fj(X)1300 4667 y Fi(\005)1356 4652 y Fk(.)48 b(Its)35 b(singular)d(set)j(\006)2171 4667 y Fi(\005)2262 4652 y Fk(is)f(de\014ned)h(to)f(b)s(e)g Fj(x)p Fh(j)p Fj(\030)3080 4667 y Fi(0)p Fg(x)3186 4652 y Fk(=)28 b Fj(\021)3338 4667 y Fi(0)p Fg(x)3417 4652 y Fk(,)456 4769 y(whic)m(h)49 b(easily)g(turns)h(out)f(to)g(b)s(e)g(smo)s(oth)f(tori,)k(tangen)m(t)e (to)f Fj(X)2966 4784 y Fi(\005)3023 4769 y Fk(.)93 b(A)50 b(pbC)456 4885 y(structure)43 b(with)e(non-empt)m(y)h(singular)e(set)i (is)g(called)e(essen)m(tial)i(pbC)g(\(EpbC)456 5001 y(for)36 b(short\))g(structure.)57 b(An)m(y)37 b(EpbC)h(structures)g(can)f(b)s (e)f(mo)s(di\014ed)g(in)m(to)f(a)i(bi-)456 5117 y(con)m(tact)g (structure)g(as)g(follo)m(ws.)54 b(Let)36 b Fj(\013)1959 5132 y Fi(0)2035 5117 y Fk(and)g Fj(\014)2283 5132 y Fi(0)2359 5117 y Fk(b)s(e)h(con)m(tact)g(1-forms)e(whic)m(h)p eop %%Page: 4 4 4 3 bop 456 251 a Fb(4)456 450 y Fk(de\014ne)45 b(\()p Fj(\030)830 465 y Fi(0)868 450 y Fk(\))f(and)g(\()p Fj(\021)1237 465 y Fi(0)1276 450 y Fk(\))g(resp)s(ectiv)m(ely)-8 b(.)78 b(and)43 b Fj(\015)49 b Fk(b)s(e)44 b(nonsingular)e(1-form)g(suc)m(h) 456 566 y(that)31 b Fj(\015)5 b Fk(\()p Fj(X)841 581 y Fi(\005)898 566 y Fk(\))28 b Fj(>)f Fk(0.)43 b(Then)33 b(it)e(is)h(easy)g(to)g(see)h(that)f(for)f(su\016cien)m(tly)h(small)e Fj(\017)e(>)g Fk(0)456 683 y(a)f(new)i(pair)d(of)i(con)m(tact)g (1-forms)e(\()p Fj(\013)q(;)17 b(\014)6 b Fk(\))27 b(=)g(\()p Fj(\013)2188 698 y Fi(0)2228 683 y Fj(;)17 b(\014)2327 698 y Fi(0)2378 683 y Fk(+)12 b Fj(\017\015)5 b Fk(\))28 b(de\014nes)i(a)d(bi-con)m(tact)456 799 y(structure.)83 b(On)45 b Fj(T)1196 763 y Fi(3)1281 799 y Fk(with)g(its)f(standard)i (co)s(ordination)d(\()p Fj(x;)17 b(y)t(;)g(z)t Fk(\),)48 b(a)e(pair)e(of)456 915 y(con)m(tact)33 b(1-forms)599 1072 y(\()p Fj(\013)699 1087 y Fi(0)738 1072 y Fj(;)17 b(\014)837 1087 y Fi(0)877 1072 y Fk(\))27 b(=)h(\()p Fj(cos)p Fk(\(2)p Fj(\031)t(z)t Fk(\))p Fj(dx)22 b Fk(+)g Fj(sin)p Fk(\(2)p Fj(\031)t(z)t Fk(\))p Fj(dy)t(;)17 b(cos)p Fk(\(2)p Fj(\031)t(z)t Fk(\))p Fj(dx)22 b Fh(\000)g Fj(sin)p Fk(\(2)p Fj(\031)t(z)t Fk(\))p Fj(dy)t Fk(\))456 1229 y(de\014nes)29 b(an)e(EpbC)h(structure)h(with)e(its)f(singular)g (set)i Fh(f)p Fj(z)k Fh(2)2676 1190 y Fi(1)p 2676 1206 36 4 v 2676 1263 a(4)2721 1229 y Fe(Z)m Fh(g)p Fj(;)27 b Fk(and)h(the)f(com-)456 1346 y(mon)k(Legendrian)i(v)m(ector)g (\014eld)g(is)f(giv)m(en)h(b)m(y)g Fj(@)5 b(=@)g(z)t Fk(.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF