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(isometry)f(group)g(of)h(\()p Fj(H)2054 4779 y Fi(n)2105 4809 y Fh(;)14 b(g)s Fl(\))37 b(is)g(isomorphic)f(to)g Fh(O)2952 4779 y Fg(+)3008 4809 y Fl(\(1)p Fh(;)14 b(n)p Fl(\))39 b(=)440 4909 y Ff(f)p Fh(\036)23 b Ff(2)g Fh(E)5 b(nd)p Fl(\()p Fj(R)878 4879 y Fi(n)p Fg(+1)1013 4909 y Fl(\))23 b Ff(j)h Fh(Q)p Fl(\()p Fh(\036v)s(;)14 b(\036w)r Fl(\))25 b(=)d Fh(Q)p Fl(\()p Fh(v)s(;)14 b(w)r Fl(\))p Fh(;)29 b(\036)p Fl(\()p Fh(H)2076 4879 y Fi(n)2122 4909 y Fl(\))23 b Ff(\032)g Fh(H)2341 4879 y Fi(n)2386 4909 y Ff(g)p Fl(,)k(with)h Fh(Q)f Fl(as)g(ab)r(o)n(v)n(e.)440 5257 y(4.)d(a\))h(Let)g Fh(M)t(;)14 b(N)33 b Fl(b)r(e)25 b(di\013eren)n(tiable)f(manifolds)h(and)f Fh(\036)g Fl(:)f Fh(N)32 b Ff(!)23 b Fh(M)33 b Fl(a)25 b(surjectiv)n(e)f(di\013eren-)440 5357 y(tiable)k(map)f(of)h(constan)n(t)f(rank.)f(Sho)n(w)h(that)h Fh(\036)g Fl(is)g(a)f(submersion.)p eop %%Page: 2 2 2 1 bop 565 531 a Fl(b\))38 b(Let)h Fh(G)26 b Ff(\002)f Fh(M)50 b Ff(!)41 b Fh(M)47 b Fl(b)r(e)39 b(a)f(smo)r(oth)g(Lie)h (group)e(action.)h(Supp)r(ose)h(that)f Fh(M)47 b Fl(is)440 631 y(connected)39 b(and)g(the)g(action)g(is)g(transitiv)n(e.)f(Sho)n (w)g(that)i(the)f(induced)g(action)g(of)g(the)440 731 y(iden)n(tit)n(y)28 b(comp)r(onen)n(t)f Fh(G)1236 743 y Fg(0)1301 731 y Fl(is)h(still)g(transitiv)n(e.)440 930 y(Please)e(hand)i(in)g(y)n(our)e(solutions)h(on)g Fa(Monda)m(y)p Fl(,)i(the)f(24th)f(of)g(Ma)n(y)-7 b(,)27 b(in)h(the)g(lecture.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF