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y(if)30 b Ff(F)37 b Fi(is)30 b(geo)r(desible,)f(there)g(is)h(a)f(v)n (ector)f(\014eld)i Fg(V)49 b Fi(tangen)n(t)29 b(to)g Ff(F)38 b Fi(and)29 b(a)h(one{form)648 3604 y Fg(\013)e Fi(suc)n(h)f(that)1543 3753 y Fg(i)1572 3765 y Fc(V)1629 3753 y Fg(\013)c Fi(=)g(1)k(and)h Fg(i)2053 3765 y Fc(V)2110 3753 y Fg(d\013)23 b Fi(=)g(0)k Fg(:)789 b Fi(\(1\))715 3901 y(a\))41 b(Sho)n(w)22 b(that)g Ff(F)30 b Fi(has)21 b(exactly)g(t)n(w)n(o)g(closed)g(orbits.)h(Describ)r(e)g Ff(F)29 b Fi(and)22 b(the)g(asym-)830 4001 y(ptotic)28 b(b)r(eha)n(viour)e(of)i(the)g(\015o)n(w)f(of)g Fg(X)7 b Fi(.)710 4120 y(b\))42 b(Sho)n(w)27 b(that)h Fg(\013)19 b Ff(^)g Fg(d\013)24 b Fi(=)e(0.)830 4229 y(Y)-7 b(ou)39 b(can)f(pro)r(ceed)g(as)g(follo)n(ws:)g(Sho)n(w)g(that)g(the)h(\015o)n (w)f Fg(')h Fi(of)g Fg(V)57 b Fi(preserv)n(es)830 4329 y Fg(\013)19 b Ff(^)g Fg(d\013)p Fi(.)28 b(Fix)g(a)f(v)n(olume)g(form)g Fg(\026)h Fi(on)f Fg(S)2072 4299 y Fe(3)2137 4329 y Fi(suc)n(h)g(that) 1728 4477 y Ff(\000)p Fg(\026)22 b(<)h(\013)c Ff(^)g Fg(d\013)k(<)g(\026)28 b(:)830 4626 y Fi(No)n(w)f(w)n(e)h(argue)e(b)n (y)h(con)n(tradicton.)f(Assume)1870 4774 y Fg(\013)19 b Ff(^)g Fg(d\013)k Ff(6)p Fi(=)g(0)830 4922 y(at)k Fg(p)c Ff(2)g Fg(S)1130 4892 y Fe(3)1194 4922 y Fi(and)j Fg(p)h Fi(do)r(es)f(not)h(lie)f(in)h(a)f(closed)g(orbit)h(of)f Fg(X)7 b Fi(.)26 b(Let)h Fg(U)35 b Fi(b)r(e)27 b(a)g(small)830 5022 y(neigh)n(b)r(ourho)r(o)r(d)g(of)g Fg(p)p Fi(.)h(Use)f Fg(\026)h Fi(to)g(sho)n(w)1686 5208 y(lim)1665 5258 y Fc(t)p Fd(!1)1836 5095 y Fb(Z)1882 5284 y Fc(')1926 5292 y Fa(t)1954 5284 y Fe(\()p Fc(U)6 b Fe(\))2075 5208 y Fg(\013)19 b Ff(^)g Fg(d\013)k Fi(=)g(0)830 5413 y(and)28 b(conlude)f(that)h Fg(\013)19 b Ff(^)g Fg(d\013)k Fi(=)g(0)k(ev)n (erywhere.)p eop %%Page: 2 2 2 1 bop 720 531 a Fi(c\))41 b(Use)21 b(\(1\))g(to)g(sho)n(w)f(that)i Fg(d\013)i Fi(=)e(0.)f(Find)g(a)g(con)n(tradiction)f(to)h(the)g(geo)r (desibilit)n(y)830 631 y(of)28 b Ff(F)8 b Fi(.)541 814 y(4.)42 b(Let)34 b Fg(\032)f Fi(:)g Fg(\031)982 826 y Fe(1)1020 814 y Fi(\()p Fg(B)t Fi(\))h Ff(!)g Fi(Di\013)7 b(\()p Fg(F)12 b Fi(\))34 b(b)r(e)h(a)e(homomorphism.)g(Let)h Ff(F)42 b Fi(b)r(e)34 b(the)g(horizon)n(tal)648 913 y(foliation)j(of)g (the)h(asso)r(ciated)e(\015at)i(bundle)g Fg(M)48 b Fi(=)39 b(\()2370 892 y(~)2350 913 y Fg(B)29 b Ff(\002)c Fg(F)12 b Fi(\))p Fg(=\031)2718 925 y Fe(1)2755 913 y Fi(\()p Fg(B)t Fi(\))41 b Ff(!)e Fg(B)t Fi(.)f(\(As)648 1013 y(usual,)908 992 y(~)888 1013 y Fg(B)32 b Fi(is)27 b(the)h(univ)n (ersal)e(co)n(v)n(ering)g(of)h Fg(B)t Fi(.\))715 1179 y(a\))41 b(Pro)n(v)n(e)33 b(that)j(the)g(horizon)n(tal)e(foliation)h (on)g Fg(M)44 b Fi(is)35 b(geo)r(desible)g(and)g(exhibit)830 1279 y(Riemannian)28 b(metrics)f(for)g(whic)n(h)g(it)i(is)e(totally)g (geo)r(desic.)710 1411 y(b\))42 b(No)n(w)31 b(assume)g(Im\()p Fg(\032)p Fi(\))f Ff(\032)f Fi(Isom)o(\()p Fg(F)r(;)14 b(g)s Fi(\))32 b(for)f(some)g(Riemmanian)g(metric)h Fg(g)i Fi(on)830 1511 y Fg(F)12 b Fi(.)27 b(What)h(can)f(b)r(e)g(said)g(ab)r (out)g(the)h(geo)r(desic)e(and)h(Riemannian)g(prop)r(erties)830 1611 y(of)h(the)g(horizon)n(tal)e(and)h(v)n(ertical)g(foliations)f(on)i Fg(M)9 b Fi(.)565 1793 y(Please)26 b(hand)h(in)h(y)n(our)e(solutions)h (during)g(the)h(lecture)f(on)g(W)-7 b(ednesda)n(y)g(,)27 b(18)p Fg(:)p Fi(12)p Fg(:)p Fi(2002.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF