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Fi(Hin)n(t:)35 b(F)-7 b(or)34 b(the)h(')p Ff(\()p Fi(')g(direction)f(use)h(2)f(b\))h(ab)r(o)n (v)n(e)e(and)i(the)g(fact)g(that)f([)p Fg(X)r(;)14 b(Y)19 b Fi(])35 b(=)f Fg(L)3152 5014 y Fd(X)3215 5002 y Fg(Y)18 b Fi(.)440 5101 y(F)-7 b(or)35 b(the)h(other)f(direction)g(y)n(ou)g (should)g(consider)g(for)g(\014xed)h Fg(p)g Ff(2)g Fg(M)45 b Fi(the)36 b(curv)n(e)e Fg(c)p Fi(\()p Fg(t)p Fi(\))j(:)440 5201 y(\()p Ff(\000)p Fg(\017;)14 b(\017)p Fi(\))23 b Ff(!)g Fg(T)852 5213 y Fd(p)889 5201 y Fg(M)1554 5383 y(c)p Fi(\()p Fg(t)p Fi(\))g(=)g(\(\(\010)1919 5349 y Fd(X)1919 5404 y(t)1983 5383 y Fi(\))2015 5395 y Fa(\003)2053 5383 y Fg(Y)c Fi(\))2152 5395 y Fd(p)p eop %%Page: 2 2 2 1 bop 440 531 a Fi(and)27 b(sho)n(w)g(that)h(this)g(is)f(constan)n(t) g(for)g(all)h Fg(t)p Fi(.)440 631 y(Remark:)d(The)h(pro)r(of)f(of)g (the)h(fact)g(that)g(the)g(\015o)n(ws)f(\010)2136 601 y Fd(X)2136 651 y(t)2225 631 y Fi(and)h(\010)2445 601 y Fd(Y)2445 651 y(s)2528 631 y Fi(comm)n(utes)f(if)h(and)g(only)440 731 y(if)j(the)h(Lie)f(brac)n(k)n(et)e([)p Fg(X)r(;)14 b(Y)k Fi(])26 b(=)e(0)29 b(is)g(of)g(a)f(purely)h(lo)r(cal)f(nature)g (so)h(the)g(same)f(conclusion)440 830 y(holds)36 b(without)h(the)g (completeness)f(assumption.)g(Note)g(that)h(in)g(this)g(case)e(the)i (\015o)n(ws)440 930 y(no)n(w)24 b(are)g(only)g(de\014ned)h(for)f (certain)g(v)-5 b(alues)24 b(of)h(\()p Fg(t;)14 b(x)p Fi(\))p Fg(;)g Fi(\()p Fg(s;)g(y)s Fi(\))24 b Ff(2)f Fc(R)13 b Ff(\002)g Fg(M)38 b Fi(so)24 b(the)h(statemen)n(t)440 1029 y(\010)500 999 y Fd(X)500 1050 y(t)583 1029 y Ff(\016)20 b Fi(\010)705 999 y Fd(Y)705 1050 y(s)790 1029 y Fi(=)27 b(\010)942 999 y Fd(Y)942 1050 y(s)1020 1029 y Ff(\016)20 b Fi(\010)1142 999 y Fd(X)1142 1050 y(t)1235 1029 y Fi(will)31 b(on)f(mak)n(e)f(sense)h(lo)r(cally)-7 b(,)30 b(ie)g(if)h(one)f (restricts)f(the)i(\015o)n(ws)e(to)440 1129 y(certain)c(neib)r(ourho)r (o)r(ds)g(of)h(a)f(c)n(hosen)g(p)r(oin)n(t,)g(then)i(for)e(su\016cien)n (tly)g(small)h Fg(t;)14 b(s)p Fi(,)25 b(the)h(\015o)n(ws)440 1229 y(and)g(the)g(comp)r(ositions)f(will)i(b)r(e)f(w)n(ell)g (de\014ned.)g(Th)n(us)g(the)g(conlusion)g(will)g(hold)g(in)g(some)440 1328 y(neib)r(ourho)r(o)r(d)h(of)h(ev)n(ery)e(p)r(oin)i Fg(x)23 b Ff(2)g Fg(M)9 b Fi(.)28 b(Y)-7 b(ou)27 b(ma)n(y)g(use)g(this) h(more)f(general)f(result)h(in)h(the)440 1428 y(\014nal)g(problem.)440 1677 y(4.)i(Let)h Fg(M)40 b Fi(b)r(e)31 b(a)f(2-dimensional)f (di\013eren)n(tiable)i(manifold.)g(Supp)r(ose)f(there)h(are)f(v)n (ector)440 1777 y(\014elds)21 b Fg(e)685 1789 y Fe(1)722 1777 y Fg(;)14 b(e)798 1789 y Fe(2)856 1777 y Fi(de\014ned)21 b(in)g(the)h(neigh)n(b)r(ourho)r(o)r(d)e(of)h(a)f(p)r(oin)n(t)h Fg(p)i Ff(2)h Fg(M)29 b Fi(whic)n(h)21 b(are)f(ev)n(erywhere)440 1876 y(linearly)k(indep)r(enden)n(t)i(and)f(suc)n(h)g(that)g(the)h (comm)n(utator)e([)p Fg(e)2393 1888 y Fe(1)2430 1876 y Fg(;)14 b(e)2506 1888 y Fe(2)2543 1876 y Fi(])25 b(v)-5 b(anishes)25 b(iden)n(tically)-7 b(.)440 1976 y(Sho)n(w)30 b(that)g(there)g(is)h(a)e(c)n(hart)h(\()p Fg(U;)14 b(\036)p Fi(\))31 b(around)e Fg(p)h Fi(suc)n(h)g(that)g Fg(e)2440 1988 y Fe(1)2477 1976 y Fg(;)14 b(e)2553 1988 y Fe(2)2621 1976 y Fi(restricted)29 b(to)h Fg(U)39 b Fi(are)440 2076 y(the)28 b(asso)r(ciated)e(co)r(ordinate)h(v)n(ector)f(\014elds.)440 2524 y(Please)g(hand)i(in)g(y)n(our)e(solutions)h(on)g(Th)n(ursda)n(y) -7 b(,)26 b(the)i(24th)f(of)h(No)n(v)n(em)n(b)r(er,)e(b)n(y)i(4pm.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF