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Fk(\032)29 b Fl(M)41 b Fn(a)31 b(neigh)n(b)r(ourho)r(o)r(d)f (of)h Fl(p)h Fn(di\013eomorphic)f(to)456 525 y Fd(R)510 495 y Fh(2)p Fi(n)594 525 y Fn(.)43 b(By)30 b(linear)f(algebra,)f(w)n (e)h(can)h(c)n(ho)r(ose)e(a)i(basis)f Fl(e)2203 537 y Fh(1)2239 525 y Fl(;)14 b(:::;)g(e)2421 537 y Fi(n)2493 525 y Fk(2)27 b(F)2635 537 y Fi(p)2673 525 y Fl(;)h(e)2763 537 y Fi(n)p Fh(+1)2891 525 y Fl(;)14 b(:::;)g(e)3073 537 y Fh(2)p Fi(n)3178 525 y Fk(2)27 b(G)3309 537 y Fi(p)3378 525 y Fn(of)456 624 y Fl(T)505 636 y Fi(p)542 624 y Fl(M)37 b Fn(suc)n(h)27 b(that)1143 765 y Fl(!)s Fn(\()p Fl(e)1269 777 y Fi(i)1296 765 y Fl(;)14 b(e)1372 777 y Fi(n)p Fh(+)p Fi(i)1492 765 y Fn(\))23 b(=)g(1)p Fl(;)k(!)s Fn(\()p Fl(e)1853 777 y Fi(i)1880 765 y Fl(;)14 b(e)1956 777 y Fi(j)1991 765 y Fn(\))23 b(=)g(0)k(otherwise)82 b(\()p Fk(\003)p Fn(\))p Fl(:)456 906 y Fn(Using)18 b(the)h(canonical)e (connection,)i(w)n(e)f(can)g(extend)h(these)f(v)n(ectors)f(to)i(v)n (ector)e(\014elds)h Fl(E)3189 918 y Fh(1)3227 906 y Fl(;)c(:::;)g(E) 3431 918 y Fh(2)p Fi(n)456 1005 y Fn(on)30 b Fl(U)9 b Fn(.)44 b(F)-7 b(act)30 b(1)g(ab)r(o)n(v)n(e)f(sho)n(ws)g(that)i Fl(E)1690 1017 y Fh(1)1727 1005 y Fl(;)14 b(:::;)g(E)1931 1017 y Fi(n)2004 1005 y Fk(2)28 b(F)38 b Fn(and)30 b Fl(E)2410 1017 y Fi(n)p Fh(+1)2540 1005 y Fl(;)14 b(:::;)g(E)2744 1017 y Fh(2)p Fi(n)2850 1005 y Fk(2)28 b(G)5 b Fn(.)45 b(Because)29 b Fk(r)456 1105 y Fn(is)f(parallel,)g(\()p Fk(\003)p Fn(\))h(still)g(holds)g(in)g(ev)n(ery)e(p)r(oin)n(t,)i(and)g (b)r(ecause)f(the)h(torsion)f(v)-5 b(anishes)28 b(w)n(e)h(ha)n(v)n(e) 456 1205 y([)p Fl(E)540 1217 y Fi(i)568 1205 y Fl(;)14 b(E)666 1217 y Fi(j)701 1205 y Fn(])26 b(=)h(0)i(for)g(all)h Fl(i;)14 b(j)5 b Fn(.)42 b(Hence)30 b(w)n(e)g(can)f(in)n(tegrate)f(the) j(v)n(ector)d(\014elds)i(to)f(get)h(lo)r(cal)f(co)r(or-)456 1304 y(dinates)f(whic)n(h)h(giv)n(e)f(the)h(isomorphism)e(with)j(the)f (standard)f(bi-Lagrangian)e(structure)i(on)456 1404 y Fd(R)510 1374 y Fh(2)p Fi(n)594 1404 y Fn(.)p 3384 1404 4 57 v 3388 1351 50 4 v 3388 1404 V 3437 1404 4 57 v 456 1562 a Fg(R)l(emark)h(5.)43 b Fn(The)37 b(canonical)f(connection)h (is)g(not)h(necessarily)d(complete.)66 b(Therefore,)39 b(w)n(e)456 1661 y(cannot)18 b(conclude)g(in)g(general)f(that)i(a)f (\015at)g(bi-Lagrangian)e(structure)i(on)g(a)g(simply-connected)456 1761 y(manifold)27 b(is)h(globally)e(isomorphic)g(to)i(the)g(standard)f (structure)g(on)g Fd(R)2731 1731 y Fh(2)p Fi(n)2815 1761 y Fn(.)1054 1947 y(2.)46 b Fm(Geodesic)31 b(flo)n(w)h(of)f(manif)n (olds)g(with)h Fl(K)d(<)22 b Fn(0)555 2097 y(W)-7 b(e)27 b(need)f(some)f(preparations.)35 b(Let)26 b Fl(X)32 b Fn(b)r(e)27 b(a)e(Riemannian)h(manifold)g(and)g Fl(S)5 b(X)32 b Fn(the)27 b(unit)456 2196 y(sphere)32 b(bundle.)55 b(The)33 b(Liouville)g(1-form)f Fl(\025)i Fn(on)f Fl(T)12 b(X)2206 2174 y Fk(\030)2206 2200 y Fn(=)2304 2196 y Fl(T)2365 2166 y Ff(\003)2402 2196 y Fl(X)40 b Fn(induces)33 b(a)g(1-form)f Fl(\022)j Fn(=)d Fl(i)3358 2166 y Ff(\003)3396 2196 y Fl(\025)456 2296 y Fn(on)37 b Fl(S)5 b(X)44 b Fn(where)37 b Fl(i)h Fn(denotes)g(the)g(em)n(b)r(edding)g Fl(i)i Fn(:)g Fl(S)5 b(X)46 b Fk(!)40 b Fl(T)12 b(X)7 b Fn(.)66 b(W)-7 b(e)39 b(ha)n(v)n(e)d(the)j(follo)n(wing)456 2396 y(prop)r(osition.)456 2548 y Fp(Prop)s(osition)30 b(6.)41 b Fl(\022)g Fg(is)f(a)g(c)l(ontact)f(form)h(on)g Fl(S)5 b(X)45 b Fg(and)40 b(the)f(ge)l(o)l(desic)i(spr)l(ay)f(is)g(the) g(R)l(e)l(eb)456 2648 y(ve)l(ctor)30 b(\014eld)g(for)g Fl(\022)r Fg(.)555 2801 y Fn(Here)j(the)g Fg(R)l(e)l(eb)h(ve)l(ctor)g (\014eld)g Fn(of)e(a)g(con)n(tact)g(form)h Fl(\022)i Fn(is)d(the)h(unique)g(v)n(ector)f(\014eld)h(whic)n(h)456 2900 y(satis\014es)26 b Fl(\022)r Fn(\()p Fl(X)7 b Fn(\))24 b(=)e(1)p Fl(;)14 b(i)1161 2912 y Fi(X)1223 2900 y Fl(d\022)26 b Fn(=)d(0.)456 3058 y Fg(Pr)l(o)l(of.)43 b Fn(W)-7 b(e)28 b(only)g(sk)n(etc)n(h)f(the)i(pro)r(of)e(of)h(the)h(prop)r(osition.)37 b(T)-7 b(o)28 b(sho)n(w)f(that)h Fl(\022)j Fn(is)d(con)n(tact)f(one)456 3158 y(can)e(do)g(a)h(direct)f(computation)h(in)g(lo)r(cal)f(co)r (ordinates.)35 b(T)-7 b(o)25 b(pro)n(v)n(e)f(that)i(the)g(geo)r(desic)f (spra)n(y)456 3258 y(is)36 b(the)i(Reeb)f(v)n(ector)e(\014eld)i(one)g (can)f(use)h(that)g(the)g(geo)r(desic)f(spra)n(y)g(is)g(the)i (Hamiltonian)456 3357 y(v)n(ector)23 b(\014eld)i(for)f(the)h(function)g Fl(H)30 b Fn(=)1662 3325 y Fh(1)p 1662 3339 34 4 v 1662 3386 a(2)1705 3357 y Fk(h)p Fl(v)s(;)14 b(v)s Fk(i)25 b Fn(on)g Fl(T)2091 3327 y Ff(\003)2128 3357 y Fl(X)31 b Fn(and)24 b(the)h(standard)f(symplectic)h(form)456 3457 y Fl(d\025)p Fn(.)37 b(\(This)28 b(in)g(turn)g(can)f(b)r(e)h(used) f(to)h(pro)n(v)n(e)e(the)i(\014rst)f(part)g(of)h(the)g(assertion.\))p 3384 3457 4 57 v 3388 3404 50 4 v 3388 3457 V 3437 3457 4 57 v 555 3615 a(Let)h Fl(M)37 b Fn(b)r(e)29 b(a)f(closed)g (Riemannian)g(manifold)g(with)h(sectional)f(curv)-5 b(ature)27 b(strictly)i(nega-)456 3714 y(tiv)n(e,)e(without)h(loss)f(of)g (generalit)n(y)f Fl(K)j Fk(\024)23 b(\000)p Fn(1.)36 b(Then)27 b(the)h(geo)r(desic)f(\015o)n(w)g(on)g Fl(S)5 b(M)36 b Fn(admits)28 b(an)456 3814 y(Anoso)n(v)e(splitting,)i(giv)n (en)f(b)n(y)g(the)h(horospheric)e(foliations.)456 4013 y Fp(Assumption:)34 b Fn(The)27 b(Anoso)n(v)g(splitting)h(on)f Fl(S)5 b(M)36 b Fn(is)27 b(smo)r(oth.)555 4213 y(W)-7 b(e)24 b(consider)f(the)h(univ)n(ersal)e(co)n(v)n(ering)g Fl(X)29 b Fn(=)2039 4192 y(~)2008 4213 y Fl(M)j Fn(and)24 b(the)g(unit)g(sphere)g(bundle)g Fl(V)42 b Fn(=)22 b Fl(S)5 b(X)i Fn(.)456 4312 y(Let)27 b Fd(R)e Fk(\002)18 b Fl(V)42 b Fk(!)23 b Fl(V)46 b Fn(b)r(e)28 b(the)g(geo)r(desic)f (\015o)n(w.)456 4465 y Fp(Lemma)i(7.)41 b Fg(The)24 b(quotient)f Fl(P)35 b Fn(=)23 b Fl(V)9 b(=)p Fd(R)29 b Fg(is)24 b(a)f(manifold)j (and)e Fl(V)41 b Fk(!)24 b Fl(P)35 b Fg(is)23 b(a)h(smo)l(oth)g Fd(R)p Fg(-bund)t(le.)456 4618 y(Pr)l(o)l(of.)43 b Fn(T)-7 b(o)29 b(sho)n(w)g(that)h Fl(P)42 b Fn(admits)30 b(a)f(smo)r(oth)h (structure,)f(w)n(e)h(giv)n(e)f(a)g(di\013eren)n(t)h(description)456 4717 y(of)37 b(it.)65 b(Let)37 b Fl(R)j Fn(=)e Fk(f)p Fl(r)j Fn(:)e([0)p Fl(;)14 b Fk(1)p Fn(\))39 b Fk(!)g Fl(X)45 b Fk(j)39 b Fl(r)30 b Fn(geo)r(desic)o Fk(g)37 b Fn(b)r(e)g(the)h(set)f(of)g(ra)n(ys.)63 b(W)-7 b(e)38 b(call)e(t)n(w)n(o)456 4817 y(elemen)n(ts)23 b Fl(r)828 4829 y Fh(1)866 4817 y Fl(;)14 b(r)940 4829 y Fh(2)1001 4817 y Fn(of)23 b Fl(R)h Fn(asymptotic,)g Fl(r)1662 4829 y Fh(1)1723 4817 y Fk(\030)f Fl(r)1848 4829 y Fh(2)1885 4817 y Fn(,)i(if)f(and)f(only)g(if)h(the)g(distance)f(dist\()p Fl(r)3073 4829 y Fh(1)3111 4817 y Fn(\()p Fl(t)p Fn(\))p Fl(;)14 b(r)3279 4829 y Fh(2)3318 4817 y Fn(\()p Fl(t)p Fn(\)\))456 4917 y(is)29 b(b)r(ounded)h(for)f Fl(t)e Fk(!)f(1)p Fn(.)43 b(The)30 b(quotien)n(t)f Fl(B)i Fn(=)26 b Fl(R)q(=)f Fk(\030)30 b Fn(is)f(called)g(the)h(imaginary)e(b)r (oundary)456 5016 y(of)34 b Fl(X)41 b Fn(at)35 b(in\014nit)n(y)-7 b(.)59 b(\(Note)35 b(that,)i(as)d Fl(X)1780 4994 y Fk(\030)1780 5020 y Fn(=)1880 5016 y Fd(R)1934 4986 y Fi(n)p Fh(+1)2069 5016 y Fn(,)i Fl(B)j Fn(is)c(homeomorphic)f(to)g Fl(S)3051 4986 y Fi(n)3096 5016 y Fn(,)j(where)d(a)456 5116 y(top)r(ological)22 b(structure)h(on)g Fl(B)28 b Fn(is)c(induced)g(b)n(y)g(the)g(bijection) g Fl(V)2447 5128 y Fi(p)2509 5116 y Fk(!)f Fl(B)t(;)14 b(v)26 b Fk(7!)d Fn(lim)3006 5128 y Fi(t)p Ff(!1)3182 5116 y Fl(exp)3310 5128 y Fi(p)3348 5116 y Fl(tv)s Fn(.)456 5216 y(One)34 b(can)h(sho)n(w)f(that)h(the)g(induced)h(top)r(ological)d (structure)h(on)h Fl(B)k Fn(do)r(es)c(not)g(dep)r(end,)i(up)p eop %%Page: 4 4 4 3 bop 456 226 a Fh(4)1160 b(MARK)29 b(HAMIL)-5 b(TON)456 425 y Fn(to)34 b(homeomorphism,)h(on)f(the)h(p)r(oin)n(t)f Fl(p)g Fk(2)h Fl(X)41 b Fn(c)n(hosen.\))56 b(W)-7 b(e)35 b(claim)f(that)h Fl(P)46 b Fn(=)34 b(\()p Fl(B)27 b Fk(\002)c Fl(B)k Fk(n)456 525 y Fl(diag)s(onal)r Fn(\))p Fl(=)p Fd(Z)909 537 y Fh(2)941 525 y Fn(.)36 b(This)27 b(follo)n(ws)f(b)r (ecause)h(for)f(distinct)i Fl(b)2230 495 y Ff(\000)2285 525 y Fl(;)14 b(b)2358 495 y Fh(+)2436 525 y Fk(2)23 b Fl(B)32 b Fn(there)26 b(exists,)h(due)g(to)g(the)456 624 y(assumption)d Fl(K)k Fk(\024)23 b(\000)p Fn(1,)i(a)f(unique)h(geo) r(desic)f(line)h Fl(l)h Fn(of)f Fl(X)31 b Fn(suc)n(h)25 b(that)g Fl(l)r Fn(\()p Fl(t)p Fn(\))e Fk(!)g Fl(b)2935 594 y Ff(\006)3015 624 y Fn(as)i Fl(t)e Fk(!)g(\0061)p Fn(.)456 724 y(If)31 b(w)n(e)g(divide)h(b)n(y)f(the)g Fd(Z)1246 736 y Fh(2)1277 724 y Fn(-action)f(p)r(erm)n(uting)i Fl(b)1999 694 y Ff(\000)2054 724 y Fl(;)14 b(b)2127 694 y Fh(+)2213 724 y Fn(w)n(e)31 b(get)g(a)g(unique)h(p)r(oin)n(t)f(of)h Fl(P)12 b Fn(.)48 b(This)456 824 y(sho)n(ws)27 b(that)i Fl(P)40 b Fn(admits)29 b(a)f(di\013eren)n(tiable)g(structure.)40 b(W)-7 b(e)28 b(can)h(c)n(ho)r(ose)e(this)i(structure)f(suc)n(h)456 923 y(that)f(the)h(pro)5 b(jection)27 b Fl(V)42 b Fk(!)23 b Fl(P)40 b Fn(is)27 b(smo)r(oth.)p 3384 923 4 57 v 3388 871 50 4 v 3388 923 V 3437 923 4 57 v 456 1088 a(Let)21 b Fl(\022)i Fn(b)r(e)f(the)f(the)h(canonical)d(con)n(tact)i(structure)f (on)h Fl(V)40 b Fn(and)21 b Fl(u)g Fn(the)g(geo)r(desic)f(spra)n(y)-7 b(.)34 b(Because)456 1187 y Fl(u)23 b Fn(is)h(the)h(Reeb)f(v)n(ector)f (\014eld)i(of)f Fl(\022)r Fn(,)h Fl(i)1584 1199 y Fi(u)1627 1187 y Fl(d\022)h Fn(=)c(0,)j(and)f Fl(L)2127 1199 y Fi(u)2170 1187 y Fl(d\022)h Fn(=)e Fl(i)2394 1199 y Fi(u)2437 1187 y Fl(dd\022)14 b Fn(+)e Fl(di)2725 1199 y Fi(u)2768 1187 y Fl(d\022)25 b Fn(=)e(0,)i(hence)f Fl(d\022)j Fn(is)456 1287 y(in)n(v)-5 b(arian)n(t)24 b(under)h(the)h(geo)r(desic)e(\015o)n (w.)36 b(This)25 b(implies)g(that)h Fl(d\022)i Fn(descends)d(to)g(a)g (closed)g(2-form)456 1386 y Fl(!)30 b Fn(on)d Fl(P)12 b Fn(.)37 b(Moreo)n(v)n(er,)25 b Fl(d\022)30 b Fn(is)e(non-degenerate)d (on)j Fl(k)s(er)r(\022)r Fn(,)g(hence)g Fl(!)i Fn(is)d(symplectic.)555 1486 y(Let)36 b Fl(T)12 b(X)41 b Fn(=)35 b Fd(R)p Fl(u)30 b Fk(\010)23 b Fl(E)1269 1456 y Fi(ss)1359 1486 y Fk(\010)g Fl(E)1513 1456 y Fi(uu)1632 1486 y Fn(b)r(e)35 b(the)h(Anoso)n(v)e (splitting.)60 b(Then)36 b Fl(E)2878 1456 y Fi(ss)2980 1486 y Fn(and)f Fl(E)3215 1456 y Fi(uu)3333 1486 y Fn(are)456 1586 y(foliations)22 b(whic)n(h)h(are)f(in)n(v)-5 b(arian)n(t)22 b(under)h(the)g(geo)r(desic)f(\015o)n(w)g(and)h(hence)g(descend)g(to)g (comple-)456 1685 y(men)n(tary)j(foliations)h Fk(F)1209 1655 y Fi(ss)1276 1685 y Fl(;)14 b Fk(F)1381 1655 y Fi(uu)1491 1685 y Fn(on)27 b Fl(P)12 b Fn(.)37 b(By)28 b(assumption,)f(this)h (splitting)g(of)f Fl(T)12 b(P)39 b Fn(is)27 b(smo)r(oth.)456 1841 y Fp(Lemma)i(8.)41 b Fg(The)31 b(foliations)g Fk(F)1510 1811 y Fi(ss)1577 1841 y Fl(;)14 b Fk(F)1682 1811 y Fi(uu)1794 1841 y Fg(ar)l(e)30 b(L)l(agr)l(angian)h(with)g(r)l(esp)l(e)l(ct)e(to)h Fl(!)s Fg(.)456 1996 y(Pr)l(o)l(of.)43 b Fn(W)-7 b(e)28 b(ha)n(v)n(e)e(the)i(inequalities)1143 2141 y Fk(j)p Fl(D)r(\036)1286 2153 y Fi(t)1316 2141 y Fl(\030)t Fk(j)83 b(\024)g Fl(const)18 b Fk(\001)h Fl(e)1904 2107 y Ff(\000)p Fi(t)1985 2141 y Fk(j)p Fl(\030)t Fk(j)83 b Fn(for)27 b Fl(\030)g Fk(2)d Fl(E)2489 2107 y Fi(ss)2555 2141 y Fl(;)14 b(t)23 b(>)g Fn(0)1087 2265 y Fk(j)p Fl(D)r(\036)1230 2277 y Ff(\000)1286 2265 y Fl(t\030)t Fk(j)83 b(\024)g Fl(const)18 b Fk(\001)h Fl(e)1904 2231 y Ff(\000)p Fi(t)1985 2265 y Fk(j)p Fl(\030)t Fk(j)83 b Fn(for)27 b Fl(\030)g Fk(2)d Fl(E)2489 2231 y Fi(uu)2571 2265 y Fl(;)14 b(t)23 b(>)g Fn(0)p Fl(:)456 2410 y Fn(Supp)r(ose)k Fl(\030)817 2422 y Fh(1)855 2410 y Fl(;)14 b(\030)928 2422 y Fh(2)988 2410 y Fk(2)24 b Fl(E)1133 2380 y Fi(ss)1200 2410 y Fn(.)37 b(Then,)27 b(since)h Fl(d\022)i Fn(is)d(in)n(v)-5 b(arian)n(t)27 b(under)g(the)h(geo)r(desic)f(\015o)n(w,)981 2555 y Fk(j)p Fl(d\022)r Fn(\()p Fl(\030)1156 2567 y Fh(1)1194 2555 y Fl(;)14 b(\030)1267 2567 y Fh(2)1305 2555 y Fn(\))p Fk(j)83 b Fn(=)f Fk(j)p Fn(\()p Fl(\036)1694 2521 y Ff(\003)1694 2576 y Fi(t)1733 2555 y Fl(d\022)r Fn(\)\()p Fl(\030)1917 2567 y Fh(1)1956 2555 y Fl(;)14 b(\030)2029 2567 y Fh(2)2067 2555 y Fn(\))p Fk(j)23 b Fn(=)g Fk(j)p Fl(d\022)r Fn(\()p Fl(\036)2421 2567 y Fi(t)2451 2575 y Ff(\003)2489 2555 y Fl(\030)2525 2567 y Fh(1)2563 2555 y Fl(;)14 b(\036)2649 2567 y Fi(t)2678 2575 y Ff(\003)2717 2555 y Fl(\030)2753 2567 y Fh(2)2790 2555 y Fn(\))p Fk(j)1443 2680 y(\024)82 b Fl(const)19 b Fk(\001)f(j)p Fl(d\022)r Fk(j)h(\001)g(j)p Fl(\036)2108 2692 y Fi(t)2138 2700 y Ff(\003)2176 2680 y Fl(\030)2212 2692 y Fh(1)2249 2680 y Fk(j)g(\001)f(j)p Fl(\036)2404 2692 y Fi(t)2434 2700 y Ff(\003)2472 2680 y Fl(\030)2508 2692 y Fh(2)2546 2680 y Fk(j)1443 2805 y(\024)82 b Fl(const)19 b Fk(\001)f(j)p Fl(d\022)r Fk(j)h(\001)g(j)p Fl(\030)2095 2817 y Fh(1)2133 2805 y Fk(j)f(\001)h(j)p Fl(\030)2275 2817 y Fh(2)2312 2805 y Fk(j)g(\001)f Fl(e)2434 2771 y Ff(\000)p Fh(2)p Fi(t)2548 2805 y Fl(;)97 b Fk(8)p Fl(t)22 b(>)h Fn(0)p Fl(:)456 2950 y Fn(Letting)37 b Fl(t)i Fk(!)g(1)e Fn(w)n(e)g(get)g Fl(d\022)r Fn(\()p Fl(\030)1500 2962 y Fh(1)1538 2950 y Fl(;)14 b(\030)1611 2962 y Fh(2)1648 2950 y Fn(\))40 b(=)e(0.)65 b(Similarly)-7 b(,)39 b Fl(d\022)r Fk(j)2441 2962 y Fi(E)2493 2946 y Fe(uu)2611 2950 y Fn(=)g(0.)65 b(This)37 b(sho)n(ws)f(that)456 3050 y Fk(F)524 3020 y Fi(ss)590 3050 y Fl(;)14 b Fk(F)695 3020 y Fi(uu)805 3050 y Fn(are)27 b(Lagrangian.)p 3384 3050 V 3388 2997 50 4 v 3388 3050 V 3437 3050 4 57 v 456 3214 a(W)-7 b(e)23 b(conclude)f(that)h(\()p Fl(P)r(;)14 b(!)s(;)g Fk(F)1389 3184 y Fi(ss)1456 3214 y Fl(;)g Fk(F)1561 3184 y Fi(uu)1643 3214 y Fn(\))23 b(is)f(a)h(bi-Lagrangian)c(manifold.) 36 b(The)22 b(splitting)h(of)g Fl(T)12 b(P)456 3314 y Fn(is)31 b(smo)r(oth,)i(hence)f(the)g(Kanai)f(connection)h(is)f (de\014ned.)51 b(This)32 b(construction)f(can)g(b)r(e)i(used)456 3413 y(to)27 b(pro)n(v)n(e)f(the)i(follo)n(wing)f(theorem)g(for)g(the)h (geo)r(desic)e(\015o)n(w)h(of)h Fl(M)36 b Fn(\(see)28 b([5)o(],)g([3],)f([2]\).)456 3569 y Fp(Theorem)j(9)i Fn(\(F)-7 b(eres,)27 b(F)-7 b(eres-Katok,)25 b(Kanai\))p Fp(.)40 b Fg(L)l(et)25 b Fl(M)34 b Fg(b)l(e)25 b(as)h(ab)l(ove.)38 b(Assume)25 b(further)g(that)555 3691 y Fn(1.)41 b Fk(\000)p Fn(4)22 b Fl(<)h(K)29 b Fk(\024)22 b(\000)p Fn(1)29 b Fg(or)555 3791 y Fn(2.)41 b Fl(dimM)d Fg(is)30 b(o)l(dd.)456 3913 y(Then)39 b(the)g(ge)l(o)l(desic)i(\015ow)e(of)h Fl(M)48 b Fg(is)39 b Fl(C)1740 3883 y Ff(1)1811 3913 y Fg(-isomorphic)i(to)e(the)g(ge)l(o)l(desic)i(\015ow)e(of)h(a)f(close) l(d)456 4012 y(R)n(iemannian)30 b(manifold)h Fl(M)1359 3982 y Ff(0)1412 4012 y Fg(of)g(c)l(onstant)e(ne)l(gative)h(curvatur)l (e.)555 4168 y Fn(Here)d(isomorphic)g(means)g(conjugate)g(under)g(a)h Fl(C)2172 4138 y Ff(1)2242 4168 y Fn(-di\013eomorphism)f Fl(S)5 b(M)31 b Fk(!)23 b Fl(S)5 b(M)3274 4138 y Ff(0)3297 4168 y Fn(.)456 4323 y Fg(R)l(emark)29 b(10.)43 b Fn(1.)81 b(Note)42 b(that)g Fl(M)51 b Fn(is)42 b(homotop)n(y)f(equiv)-5 b(alen)n(t)42 b(to)g Fl(M)2717 4293 y Ff(0)2740 4323 y Fn(:)67 b(This)42 b(is)g(clear)f(for)456 4423 y Fl(dimM)31 b Fn(=)23 b(2.)36 b(Assume)28 b Fl(n)23 b Fn(=)f Fl(dimM)32 b(>)22 b Fn(2.)37 b(The)27 b(exact)g(homotop)n(y)g(sequence)1136 4568 y Fl(\031)1183 4580 y Fh(1)1221 4568 y Fn(\()p Fl(S)1309 4533 y Fi(n)p Ff(\000)p Fh(1)1439 4568 y Fn(\))c Fk(!)g Fl(\031)1647 4580 y Fh(1)1685 4568 y Fn(\()p Fl(S)5 b(M)k Fn(\))23 b Fk(!)g Fl(\031)2071 4580 y Fh(1)2109 4568 y Fn(\()p Fl(M)9 b Fn(\))23 b Fk(!)g Fl(\031)2439 4580 y Fh(0)2476 4568 y Fn(\()p Fl(S)2564 4533 y Fi(n)p Ff(\000)p Fh(1)2695 4568 y Fn(\))p Fl(;)456 4717 y Fn(together)36 b(with)i(the)g(dimension)g(assumption)f(sho)n(ws)f(that)i Fl(\031)2482 4729 y Fh(1)2520 4717 y Fn(\()p Fl(M)9 b Fn(\))2714 4695 y Fk(\030)2714 4722 y Fn(=)2818 4717 y Fl(\031)2865 4729 y Fh(1)2902 4717 y Fn(\()p Fl(S)c(M)k Fn(\).)67 b(Hence,)456 4817 y Fl(\031)503 4829 y Fh(1)540 4817 y Fn(\()p Fl(M)9 b Fn(\))718 4795 y Fk(\030)718 4821 y Fn(=)806 4817 y Fl(\031)853 4829 y Fh(1)890 4817 y Fn(\()p Fl(M)1012 4787 y Ff(0)1035 4817 y Fn(\).)38 b(Because)27 b Fl(M)t(;)14 b(M)1658 4787 y Ff(0)1709 4817 y Fn(are)27 b(aspherical)f(\(as)h(their)h(univ)n(ersal)e(co)n(v)n (ers)g(are)h Fd(R)3361 4787 y Fi(n)3412 4817 y Fn(\))456 4917 y(it)e(follo)n(ws)f(that)h Fl(M)32 b Fk(')22 b Fl(M)1273 4887 y Ff(0)1296 4917 y Fn(.)36 b(A)25 b(conjecture)g(of)g(F)-7 b(eres-Katok)22 b(\([3]\))j(sa)n(ys)f(that)h Fl(M)33 b Fn(is)25 b(isometric)456 5016 y(to)i Fl(M)647 4986 y Ff(0)670 5016 y Fn(.)555 5116 y(2.)36 b(The)26 b(\014rst)g (assumption)f(in)h(the)g(theorem)f(is)h(sharp,)f(as)h(one)f(can)g(sho)n (w)g(b)n(y)h(considering)456 5216 y(Hermitian)h(h)n(yp)r(erb)r(olic)g (space)g Fd(C)15 b Fl(H)1608 5185 y Fi(n)1659 5216 y Fn(.)p eop %%Page: 5 5 5 4 bop 1510 226 a Fh(THE)29 b(KANAI)g(CONNECTION)1021 b(5)456 425 y Fp(Idea)39 b(of)g(pro)s(of)g(of)g(Theorem)f(9:)48 b Fn(F)-7 b(or)34 b Fl(M)42 b Fn(as)34 b(in)g(the)g(h)n(yp)r(otheses)f (of)h(the)g(theorem)g(w)n(e)456 525 y(consider)39 b(the)i(asso)r (ciated)e(bi-Lagrangian)f(manifold)j(\()p Fl(P)r(;)14 b(!)s(;)g Fk(F)2542 495 y Fi(ss)2609 525 y Fl(;)g Fk(F)2714 495 y Fi(uu)2797 525 y Fn(\))40 b(with)i(the)f(Kanai)456 624 y(connection)34 b Fk(r)p Fn(.)61 b(W)-7 b(e)36 b(then)f(try)g(to)h (sho)n(w)e(that,)k(under)d(the)h(conditions)f(on)g(the)g(sectional)456 724 y(curv)-5 b(ature)21 b(of)g(the)h(Levi-Civita)f(connection)h(on)f Fl(M)9 b Fn(,)23 b(the)f(Kanai)f(connection)g(is)h(a\016ne)f(lo)r (cally)456 824 y(symmetric,)34 b(i.e.)53 b Fk(r)p Fl(R)33 b Fk(\021)f Fn(0,)i(where)f Fl(R)h Fn(denotes)f(the)g(curv)-5 b(ature)32 b(of)h Fk(r)p Fn(.)54 b(Then)33 b(w)n(e)g(use)g(the)456 923 y(follo)n(wing)26 b(theorem.)456 1078 y Fp(Theorem)k(11)h Fn(\(Kanai\))p Fp(.)41 b Fg(If)29 b Fn(\()p Fl(P)r(;)14 b Fk(r)p Fn(\))29 b Fg(is)g(lo)l(c)l(al)t(ly)i(symmetric,)e(then)g(the) f(ge)l(o)l(desic)i(\015ow)f(of)h Fl(M)456 1178 y Fg(is)g(isomorphic)i (to)e(the)g(\015ow)g(of)g(some)h(close)l(d)f Fl(M)2025 1148 y Ff(0)2078 1178 y Fg(with)g(c)l(onstant)f(ne)l(gative)i(curvatur) l(e.)555 1332 y Fn(This)c(theorem)f(is)g(pro)n(v)n(ed)f(b)n(y)h(using)g (an)h(algebraic)d(classi\014cation)h(of)i(a\016ne)f(lo)r(cally)g(sym-) 456 1432 y(metric)d(spaces)f(with)i(additional)f(structures)f(giv)n(en) h(b)n(y)g(the)h(complemen)n(tary)e(foliations)h(and)456 1532 y(the)28 b(prop)r(erties)e(of)i Fk(r)g Fn(with)g(resp)r(ect)f(to)h (these)f(foliations.)555 1631 y(The)19 b(pro)r(of)f(that)h(the)f(Kanai) g(connection)g(is)g(lo)r(cally)g(symmetric)g(is)h(di\016cult)g(in)g (the)g(general)456 1731 y(case.)68 b(W)-7 b(e)39 b(just)g(do)f(it)g(in) h(the)g(follo)n(wing)e(sp)r(ecial)h(situation)g(\([5]\))h(\(whic)n(h)g (w)n(as)e(the)i(\014rst)456 1830 y(application)27 b(of)g(the)h (bi-Lagrangian)d(connection)i(to)g(Anoso)n(v)g(systems\).)456 1985 y Fp(Lemma)i(12)i Fn(\(Kanai\))p Fp(.)41 b Fg(If)30 b Fk(\000)1433 1952 y Fh(9)p 1433 1966 34 4 v 1433 2014 a(4)1499 1985 y Fl(<)22 b(K)29 b Fk(\024)22 b(\000)p Fn(1)29 b Fg(then)h Fn(\()p Fl(P)r(;)14 b Fk(r)p Fn(\))31 b Fg(is)f(lo)l(c)l(al)t(ly)h(symmetric.)456 2145 y(Pr)l(o)l(of.)43 b Fn(Supp)r(ose)31 b(that)h Fk(\000)p Fl(\025)1342 2115 y Fh(2)1408 2145 y Fk(\024)d Fl(K)35 b Fk(\024)29 b(\000)p Fn(1)h(for)h(some)g Fl(\025)e Fk(\025)g Fn(1.)48 b(Let)31 b Fl(\036)2667 2157 y Fi(t)2728 2145 y Fn(denote)h(the)f(geo)r(desic) 456 2244 y(\015o)n(w)c(of)g Fl(X)7 b Fn(.)36 b(W)-7 b(e)28 b(ha)n(v)n(e)f(the)h(follo)n(wing)e(inequalities:)736 2388 y Fl(const)931 2354 y Ff(\000)p Fh(1)1039 2388 y Fk(\001)18 b Fl(e)1119 2354 y Ff(\000)p Fi(\025t)1240 2388 y Fk(j)p Fl(\030)t Fk(j)23 b(\024)111 b(j)p Fl(D)r(\036)1668 2400 y Fi(t)1698 2388 y Fl(\030)t Fk(j)g(\024)23 b Fl(const)18 b Fk(\001)h Fl(e)2254 2354 y Ff(\000)p Fi(t)2335 2388 y Fk(j)p Fl(\030)t Fk(j)83 b Fn(for)27 b Fl(\030)g Fk(2)d Fl(E)2839 2354 y Fi(ss)2906 2388 y Fl(;)14 b(t)22 b(>)h Fn(0)736 2517 y Fl(const)931 2482 y Ff(\000)p Fh(1)1039 2517 y Fk(\001)18 b Fl(e)1119 2482 y Ff(\000)p Fi(\025t)1240 2517 y Fk(j)p Fl(\030)t Fk(j)23 b(\024)83 b(j)p Fl(D)r(\036)1640 2529 y Ff(\000)1696 2517 y Fl(t\030)t Fk(j)g(\024)23 b Fl(const)18 b Fk(\001)h Fl(e)2254 2482 y Ff(\000)p Fi(t)2335 2517 y Fk(j)p Fl(\030)t Fk(j)83 b Fn(for)27 b Fl(\030)g Fk(2)d Fl(E)2839 2482 y Fi(uu)2921 2517 y Fl(;)14 b(t)23 b(>)g Fn(0)p Fl(:)456 2660 y Fn(Using)k(that)h(the)g (splitting)g(is)f(in)n(v)-5 b(arian)n(t)27 b(under)g Fl(\036)2056 2672 y Fi(t)2114 2660 y Fn(w)n(e)g(get)g(the)h(follo)n (wing)f(inequalities:)788 2804 y Fl(const)983 2770 y Ff(\000)p Fh(1)1091 2804 y Fk(\001)18 b Fl(e)1171 2770 y Fi(t)1200 2804 y Fk(j)p Fl(\030)t Fk(j)24 b(\024)82 b(j)p Fl(D)r(\036)1600 2816 y Ff(\000)1657 2804 y Fl(t\030)t Fk(j)h(\024)23 b Fl(const)18 b Fk(\001)h Fl(e)2215 2770 y Fi(\025t)2283 2804 y Fk(j)p Fl(\030)t Fk(j)83 b Fn(for)27 b Fl(\030)g Fk(2)d Fl(E)2787 2770 y Fi(ss)2854 2804 y Fl(;)14 b(t)23 b(>)f Fn(0)788 2933 y Fl(const)983 2898 y Ff(\000)p Fh(1)1091 2933 y Fk(\001)c Fl(e)1171 2898 y Fi(t)1200 2933 y Fk(j)p Fl(\030)t Fk(j)24 b(\024)111 b(j)p Fl(D)r(\036)1629 2945 y Fi(t)1658 2933 y Fl(\030)t Fk(j)h(\024)23 b Fl(const)18 b Fk(\001)h Fl(e)2215 2898 y Fi(\025t)2283 2933 y Fk(j)p Fl(\030)t Fk(j)83 b Fn(for)27 b Fl(\030)g Fk(2)d Fl(E)2787 2898 y Fi(uu)2869 2933 y Fl(;)14 b(t)23 b(>)g Fn(0)p Fl(:)456 3079 y Fn(De\014ne)36 b(the)g(tensor)1148 3058 y(\024)1130 3079 y Fl(R)g Fn(on)g(the)g (manifold)g Fl(P)47 b Fn(b)n(y)2098 3058 y(\024)2080 3079 y Fl(R)q Fn(\()p Fl(X)r(;)14 b(Y)5 b(;)14 b(Z)q(;)g(T)e Fn(\))35 b(=)h Fl(!)s Fn(\()p Fl(R)q Fn(\()p Fl(X)r(;)14 b(Y)k Fn(\))p Fl(Z)q(;)c(T)e Fn(\))36 b(and)456 3181 y(let)c Fl(S)i Fn(=)c Fl(\031)810 3151 y Ff(\003)848 3181 y Fn(\()p Fk(r)968 3160 y Fn(\024)949 3181 y Fl(R)q Fn(\),)k(where)d Fl(\031)i Fn(:)d Fl(V)49 b Fk(!)30 b Fl(P)43 b Fn(is)32 b(the)g(pro)5 b(jection.)48 b(Because)31 b Fk(r)p Fl(!)i Fn(=)c(0)j(it)g(su\016ces)456 3281 y(to)37 b(sho)n(w)g(that)h Fl(S)45 b Fk(\021)40 b Fn(0)d(if)h Fl(\025)j(<)1542 3248 y Fh(3)p 1542 3262 V 1542 3309 a(2)1585 3281 y Fn(.)68 b(Note)38 b(that)g Fk(j)p Fl(S)5 b Fk(j)37 b Fn(is)h(b)r(ounded)g(on)g Fl(V)58 b Fn(=)40 b Fl(S)5 b(X)44 b Fn(b)r(ecause)456 3380 y Fl(\031)503 3392 y Fh(1)540 3380 y Fn(\()p Fl(M)9 b Fn(\))32 b(acts)g(on)f Fl(X)39 b Fn(and)32 b Fl(V)18 b Fn(,)34 b(preserving)c(all)h (structures)h(and)f(hence)h(b)n(y)g(isomorphisms)f(of)456 3480 y(\()p Fl(P)r(;)14 b(!)s(;)g Fk(F)740 3450 y Fi(ss)807 3480 y Fl(;)g Fk(F)912 3450 y Fi(uu)994 3480 y Fn(\).)64 b(Therefore)35 b Fl(S)41 b Fn(is)36 b(in)n(v)-5 b(arian)n(t)35 b(under)h(the)h(action)f(of)g Fl(\031)2843 3492 y Fh(1)2881 3480 y Fn(\()p Fl(M)9 b Fn(\))36 b(and)g(hence)456 3580 y(comes)27 b(from)g(a)g(tensor)g(\014eld)h(on)f(the)h(compact)f (manifold)h Fl(V)9 b(=\031)2476 3592 y Fh(1)2513 3580 y Fn(\()p Fl(M)g Fn(\))24 b(=)e Fl(S)5 b(M)k Fn(.)555 3679 y(Supp)r(ose)23 b(for)e(instance)i Fl(\030)1352 3691 y Fh(1)1389 3679 y Fl(;)14 b(\030)1462 3691 y Fh(2)1500 3679 y Fl(;)g(\030)1573 3691 y Fh(3)1633 3679 y Fk(2)24 b Fl(E)1778 3649 y Fi(ss)1867 3679 y Fn(and)e Fl(\021)2064 3691 y Fh(1)2102 3679 y Fl(;)14 b(\021)2180 3691 y Fh(2)2240 3679 y Fk(2)23 b Fl(E)2384 3649 y Fi(uu)2467 3679 y Fn(.)35 b(Since)23 b Fl(S)k Fn(is)22 b(in)n(v)-5 b(arian)n(t)22 b(under)456 3779 y(the)28 b(geo)r(desic)e(\015o)n(w,)514 3923 y Fk(j)p Fl(S)5 b Fn(\()p Fl(\030)661 3935 y Fh(1)698 3923 y Fl(;)14 b(\030)771 3935 y Fh(2)809 3923 y Fl(;)g(\030)882 3935 y Fh(3)933 3923 y Fl(\021)974 3935 y Fh(1)1012 3923 y Fl(;)g(\021)1090 3935 y Fh(2)1127 3923 y Fn(\))p Fk(j)83 b Fn(=)g Fk(j)p Fn(\()p Fl(\036)1517 3888 y Ff(\003)1517 3943 y Fi(t)1556 3923 y Fl(S)5 b Fn(\)\()p Fl(\030)1712 3935 y Fh(1)1750 3923 y Fl(;)14 b(\030)1823 3935 y Fh(2)1860 3923 y Fl(;)g(\030)1933 3935 y Fh(3)1985 3923 y Fl(\021)2026 3935 y Fh(1)2063 3923 y Fl(;)g(\021)2141 3935 y Fh(2)2178 3923 y Fn(\))p Fk(j)1265 4047 y Fn(=)83 b Fk(j)p Fl(S)5 b Fn(\()p Fl(\036)1573 4059 y Fi(t)1603 4067 y Ff(\003)1641 4047 y Fl(\030)1677 4059 y Fh(1)1714 4047 y Fl(;)14 b(\036)1800 4059 y Fi(t)1830 4067 y Ff(\003)1868 4047 y Fl(\030)1904 4059 y Fh(2)1942 4047 y Fl(;)g(\036)2028 4059 y Fi(t)2057 4067 y Ff(\003)2096 4047 y Fl(\030)2132 4059 y Fh(3)2183 4047 y Fl(\036)2232 4059 y Fi(t)2262 4067 y Ff(\003)2300 4047 y Fl(\021)2341 4059 y Fh(1)2378 4047 y Fl(;)g(\036)2464 4059 y Fi(t)2494 4067 y Ff(\003)2532 4047 y Fl(\021)2573 4059 y Fh(2)2611 4047 y Fn(\))p Fk(j)1265 4172 y(\024)83 b Fl(const)18 b Fk(\001)h(j)p Fl(S)5 b Fk(j)18 b(\001)h(j)p Fl(\036)1902 4184 y Fi(t)1931 4192 y Ff(\003)1970 4172 y Fl(\030)2006 4184 y Fh(1)2043 4172 y Fk(j)g(\001)f(j)p Fl(\036)2198 4184 y Fi(t)2228 4192 y Ff(\003)2266 4172 y Fl(\030)2302 4184 y Fh(2)2340 4172 y Fk(j)g(\001)h(j)p Fl(\036)2495 4184 y Fi(t)2524 4192 y Ff(\003)2563 4172 y Fl(\030)2599 4184 y Fh(3)2636 4172 y Fk(j)g(\001)f(j)p Fl(\036)2791 4184 y Fi(t)2821 4192 y Ff(\003)2859 4172 y Fl(\021)2900 4184 y Fh(1)2938 4172 y Fk(j)g(\001)h(j)p Fl(\036)3093 4184 y Fi(t)3122 4192 y Ff(\003)3161 4172 y Fl(\021)3202 4184 y Fh(2)3239 4172 y Fk(j)1265 4303 y(\024)83 b Fl(const)18 b Fk(\001)h(j)p Fl(S)5 b Fk(j)18 b(\001)h(j)p Fl(\030)1889 4315 y Fh(1)1926 4303 y Fk(j)g(\001)f(j)p Fl(\030)2068 4315 y Fh(2)2106 4303 y Fk(j)g(\001)h(j)p Fl(\030)2248 4315 y Fh(3)2286 4303 y Fk(j)f(\001)h(j)p Fl(\021)2433 4315 y Fh(1)2470 4303 y Fk(j)g(\001)f(j)p Fl(\021)2617 4315 y Fh(2)2655 4303 y Fk(j)g(\001)h Fl(e)2777 4269 y Fh(\(2)p Fi(\025)p Ff(\000)p Fh(3\))p Fi(t)3015 4303 y Fl(;)97 b Fk(8)p Fl(t)22 b(>)h Fn(0)p Fl(:)456 4572 y Fn(Letting)28 b Fl(t)d Fk(!)g(1)k Fn(this)g(implies)f Fl(S)5 b Fn(\()p Fl(\030)1592 4584 y Fh(1)1630 4572 y Fl(;)14 b(\030)1703 4584 y Fh(2)1740 4572 y Fl(;)g(\030)1813 4584 y Fh(3)1865 4572 y Fl(\021)1906 4584 y Fh(1)1943 4572 y Fl(;)g(\021)2021 4584 y Fh(2)2058 4572 y Fn(\))26 b(=)e(0.)39 b(The)29 b(other)f(cases)g(can)g(b)r(e)h(done)f(in)456 4671 y(the)g(same)f(w)n(a)n(y)-7 b(,)26 b(hence)i(w)n(e)f(get)g Fl(S)h Fk(\021)23 b Fn(0)k(and)g Fk(r)p Fl(R)d Fk(\021)f Fn(0.)p 3384 4671 4 57 v 3388 4619 50 4 v 3388 4671 V 3437 4671 4 57 v 760 4867 a(3.)45 b Fm(Anoso)n(v)31 b (symplectomorphisms)g(on)g(lo)n(w-dimensional)g(tori)555 5016 y Fn(Let)f(\()p Fl(M)t(;)14 b(!)s Fn(\))31 b(b)r(e)f(a)g(closed)f (symplectic)h(manifold)g(and)g Fl(\036)e Fn(:)f Fl(M)36 b Fk(!)27 b Fl(M)39 b Fn(an)30 b(Anoso)n(v)f(di\013eo-)456 5116 y(morphism)h(preserving)g Fl(!)s Fn(,)h Fl(\036)1409 5086 y Ff(\003)1448 5116 y Fl(!)g Fn(=)e Fl(!)s Fn(.)47 b(In)31 b(the)g(\014rst)g(talk)g(b)n(y)g(D.)g(Kotsc)n(hic)n(k)f(it)h(w) n(as)f(sho)n(wn)456 5216 y(that)h(the)g(Anoso)n(v)f(prop)r(ert)n(y)g (implies)h(that)h(the)f(stable)g(and)f(unstable)h(foliations,)h Fk(F)3238 5185 y Fi(s)3273 5216 y Fl(;)14 b Fk(F)3378 5185 y Fi(u)3421 5216 y Fn(,)p eop %%Page: 6 6 6 5 bop 456 226 a Fh(6)1160 b(MARK)29 b(HAMIL)-5 b(TON)456 425 y Fn(are)26 b(Lagrangian.)456 624 y Fp(Assumption:)34 b Fn(The)27 b(Anoso)n(v)g(splitting)h(on)f Fl(M)36 b Fn(is)28 b(smo)r(oth.)555 824 y(Then)j(\()p Fl(M)t(;)14 b(!)s(;)g Fk(F)1089 794 y Fi(s)1125 824 y Fl(;)g Fk(F)1230 794 y Fi(u)1273 824 y Fn(\))31 b(is)g(a)g(bi-Lagrangian)d(manifold)j (and)g(the)g(canonical)f(connection)456 923 y Fk(r)d Fn(is)f(de\014ned.)37 b(Note)27 b(that)g Fl(\036)h Fn(preserv)n(es)c (all)j(structures,)f(hence)h Fl(\036)g Fn(is)g(a\016ne)g(with)g(resp)r (ect)g(to)456 1023 y Fk(r)p Fn(,)h Fl(\036)625 1035 y Ff(\003)663 1023 y Fn(\()p Fk(r)764 1035 y Fi(X)828 1023 y Fl(Y)18 b Fn(\))24 b(=)e Fk(r)1106 1035 y Fi(\036)1146 1043 y Fb(\003)1181 1035 y Fi(X)1244 1023 y Fl(\036)1293 1035 y Ff(\003)1332 1023 y Fl(Y)d Fn(.)555 1123 y(The)27 b(structure)e(of)h(Anoso)n(v)g(symplectomorphisms)f(is)h(giv)n(en)f(b)n (y)h(the)h(follo)n(wing)e(theorem)456 1222 y(whic)n(h)i(is)h(a)f (corollary)e(of)i(a)h(result)f(b)n(y)g(Benoist-Lab)r(ourie)f(\(see)h ([1]\).)456 1360 y Fp(Theorem)j(13)h Fn(\(Benoist-Lab)r(ourie\))p Fp(.)40 b Fg(L)l(et)35 b Fl(\036)e Fn(:)g Fl(M)41 b Fk(!)33 b Fl(M)44 b Fg(b)l(e)35 b(an)g(A)n(nosov)g(symple)l(ctomor-)456 1459 y(phism)42 b(such)f(that)h(the)f(A)n(nosov)g(splitting)h(is)g(smo) l(oth.)74 b(Then)41 b Fl(\036)h Fg(is)g Fl(C)2810 1429 y Ff(1)2880 1459 y Fg(-c)l(onjugate)g(to)f(a)456 1559 y(hyp)l(erb)l(olic)32 b(infr)l(anil)f(automorphism.)555 1696 y Fn(If)42 b Fl(G)h Fn(is)e(a)h(nilp)r(oten)n(t)g(Lie)g(group,)i (\000)i Fk(\032)h Fl(Af)9 b(f)g Fn(\()p Fl(G)p Fn(\))42 b(a)f(lattice)h(with)g(smo)r(oth)g(quotien)n(t)456 1796 y Fl(M)h Fn(=)35 b Fl(G=)p Fn(\000,)i(then)e(an)g(infranil)g (automorphism)f(of)g Fl(M)44 b Fn(is)35 b(a)f(map)h(whic)n(h)g(comes)f (from)h(an)456 1895 y(automorphism)g(of)i Fl(G)p Fn(,)i(compatible)d (with)h(the)g(lattice)g(\000.)64 b(W)-7 b(e)37 b(are)e(going)h(to)g (consider)g(a)456 1995 y(sp)r(ecial)27 b(case)g(\(see)g([4]\))h(of)f (this)h(theorem.)456 2132 y Fp(Theorem)i(14)h Fn(\(Flaminio-Katok\))p Fp(.)40 b Fg(L)l(et)33 b Fl(M)38 b Fn(=)30 b Fl(T)2116 2102 y Fh(2)2186 2132 y Fg(or)j Fl(T)2357 2102 y Fh(4)2427 2132 y Fg(and)h Fl(\036)g Fg(and)g(A)n(nosov)g(symple)l(c-)456 2232 y(tomorphism)f(such)f(that)g(the)g(A)n(nosov)g(splitting)g(is)g (smo)l(oth.)46 b(Then)32 b Fl(\036)h Fg(is)f Fl(C)2903 2202 y Ff(1)2973 2232 y Fg(-c)l(onjugate)g(to)456 2332 y(a)e(line)l(ar)g(automorphism)h(of)g Fl(M)9 b Fg(.)456 2469 y Fp(Idea)39 b(of)f(pro)s(of:)48 b Fn(One)34 b(\014rst)f(sho)n(ws) f(that)i(\()p Fl(M)t(;)14 b Fk(r)p Fn(\))34 b(is)g(complete)f(and)h(lo) r(cally)f(symmetric.)456 2569 y(Then)27 b(one)h(uses)f(algebra)f(to)h (sho)n(w)g(that)h Fk(r)g Fn(is)f(in)h(fact)g(\015at.)555 2668 y(Supp)r(ose)h Fl(M)34 b Fn(=)25 b Fl(T)1148 2638 y Fh(2)1213 2668 y Fn(\(the)k(case)f Fl(T)1627 2638 y Fh(4)1692 2668 y Fn(follo)n(ws)g(similarly\).)40 b(Lift)30 b Fl(\036)f Fn(to)g(an)f(Anoso)n(v)g(symplec-)456 2781 y(tomorphism)933 2759 y(~)922 2781 y Fl(\036)33 b Fn(on)f(the)g(univ)n (ersal)f(co)n(v)n(er)1857 2760 y Fc(f)1846 2781 y Fl(M)40 b Fn(=)30 b Fd(R)2116 2750 y Fh(2)2159 2781 y Fn(.)51 b(The)32 b(asso)r(ciated)f(Kanai)g(connection)467 2865 y Fc(e)456 2886 y Fk(r)f Fn(is)f(complete)h(and)f(\015at,)i(hence)f(b)n (y)f(Theorem)g(4)g(w)n(e)h(can)f(c)n(hange)f(co)r(ordinates)h(suc)n(h)g (that)456 2994 y(the)k(bi-Lagrangian)e(structure)i(on)1635 2973 y Fc(f)1625 2994 y Fl(M)42 b Fn(is)33 b(the)h(standard)e(one.)53 b(The)34 b(di\013eomorphism)3317 2972 y(~)3306 2994 y Fl(\036)g Fn(is)456 3093 y(Anoso)n(v,)26 b(a\016ne)h(and)h(preserv)n (es)d(the)j(co)r(ordinate)f(foliations,)g(and)g(therefore)g(has)g(the)h (form)1086 3251 y(~)1075 3273 y Fl(\036)q Fn(\()p Fl(x)p Fn(\))c(=)e Fl(Ax)e Fn(+)e Fl(b;)96 b Fn(with)28 b Fl(A)c Fn(=)2076 3156 y Fc(\022)2179 3222 y Fl(\025)131 b Fn(0)2182 3322 y(0)86 b Fl(\025)2358 3292 y Ff(\000)p Fh(1)2489 3156 y Fc(\023)2564 3273 y Fl(;)14 b(\025)24 b Fk(6)p Fn(=)e(1)p Fl(:)456 3472 y Fn(In)35 b(particular,)994 3450 y(~)983 3472 y Fl(\036)h Fn(has)f(a)g(unique)g(\014xed)h(p)r(oin)n (t,)h(hence)f(w)n(e)f(can)g(c)n(ho)r(ose)f(co)r(ordinates)g(suc)n(h)456 3572 y(that)27 b Fl(b)c Fn(=)g(0.)555 3672 y(Let)f Fl(\034)33 b Fk(2)23 b Fl(\031)892 3684 y Fh(1)930 3672 y Fn(\()p Fl(T)1023 3641 y Fh(2)1059 3672 y Fn(\),)g(acting)e(on)1499 3651 y Fc(f)1489 3672 y Fl(M)30 b Fn(b)n(y)21 b(dec)n(k)g (transformations.)33 b(Then)22 b Fl(\034)31 b Fn(is)21 b(a\016ne,)i(preserv)n(es)460 3778 y Fc(e)-50 b Fl(!)27 b Fn(and)d(the)g(co)r(ordinate)f(foliations,)i(and)f(hence)g(has)g(a)g (similar)f(form)h(as)2784 3756 y(~)2773 3778 y Fl(\036)p Fn(.)36 b(Ho)n(w)n(ev)n(er,)23 b(if)i(it)g(is)456 3878 y(not)f(a)h(pure)f(translation,)g(then)h(it)g(m)n(ust)g(ha)n(v)n(e)f(a) g(\014xed)h(p)r(oin)n(t,)g(whic)n(h)g(is)g(imp)r(ossible.)35 b(There-)456 3977 y(fore)24 b(the)h(fundamen)n(tal)f(group)g(acts)g(b)n (y)g(translations,)g(and)h(in)f(the)i(induced)f(co)r(ordinates)e(on)456 4077 y Fl(T)517 4047 y Fh(2)553 4077 y Fn(,)28 b(the)g (di\013eomorphism)f Fl(\036)h Fn(is)f(linear.)p 3384 4077 4 57 v 3388 4024 50 4 v 3388 4077 V 3437 4077 4 57 v 1708 4246 a Fm(References)561 4379 y Fo([1])35 b(Y.)26 b(Benoist,)h(F.)f(Lab)r(ourie,)h Fa(Sur)f(les)f(di\013eomorphisms)f (d'Anoso)n(v)j(a\016nes)g(a)f(feuilletages)h(stable)g(et)671 4462 y(instable)d(di\013eren)n(tiables)p Fo(,)g(In)n(v)n(en)n(t.math.)h (111)f(\(1993\))i(285-308)561 4545 y([2])35 b(R.)24 b(F)-6 b(eres,)24 b Fa(Geo)r(desic)h(\015o)n(ws)f(on)h(manifolds)d(of)i (negativ)n(e)i(curv)l(ature)f(with)g(smo)r(oth)e(horospheric)i(foli-) 671 4628 y(ations)p Fo(,)f(Ergo)r(d.Th.)f(and)h(Dynam.Sys.)f(11)h (\(1991\))h(653-686)561 4711 y([3])35 b(R.)27 b(F)-6 b(eres,)28 b(A.)f(Katok,)i Fa(In)n(v)l(arian)n(t)f(tensor)g(\014elds)g (of)f(dynamical)g(systems)g(with)g(pinc)n(hed)i(Ly)n(apuno)n(v)671 4794 y(exp)r(onen)n(ts)d(and)e(rigidit)n(y)f(of)h(geo)r(desic)g(\015o)n (ws)p Fo(,)g(Ergo)r(d.Th)f(and)i(Dynam.Sys.)d(9)i(\(1989\))h(427-432) 561 4877 y([4])35 b(L.)21 b(Flaminio,)f(A.)h(Katok,)h Fa(Rigidit)n(y)g(of)f(symplectic)g(Anoso)n(v)h(di\013eomorphisms)d(on)j (lo)n(w)f(dimensional)671 4960 y(tori)p Fo(,)i(Ergo)r(d.Th.)g(and)i (Dynam.Sys.)d(11)i(\(1991\))i(427-440)561 5043 y([5])35 b(M.)22 b(Kanai,)h Fa(Geo)r(desic)h(\015o)n(ws)g(of)e(negativ)n(ely)j (curv)n(ed)f(manifolds)d(with)i(smo)r(oth)g(stable)h(and)f(unstable)671 5126 y(foliations)p Fo(,)g(Ergo)r(d.Th.)g(and)h(Dynam.Sys.)f(8)h (\(1988\))h(215-239)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF