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b(shall)g(do)g(this)h(in)f(the)g(ne)o(xt)g(lecture.)1430 b Fh(\003)p eop %%Page: 3 3 3 2 bop 2614 -89 a Fg(3)1130 46 y Fr(26)17 b(April)f(2004)67 179 y(\(10\))29 b(W)-5 b(e)16 b(still)g(ha)o(v)o(e)g(to)g(pro)o(v)o(e)g (\(*\))q(.)21 b(T)-5 b(o)16 b(do)g(so,)h(we)f(\002rst)i(pro)o(v)o(e:) 208 272 y Fs(Lemma)13 b(12.)21 b Fl(If)13 b Fq(f)h Fo(:)23 b Fq(G)c Fn(!)g Fq(H)f Fl(is)13 b(a)f(homomorphism)e(of)j(Lie)g(gr)m (oups)f(and)g Fq(X)23 b Fn(2)c Fk(g)p Fl(,)14 b(then)d Fo(\()p Fq(D)2348 282 y Fp(g)2375 272 y Fq(f)c Fo(\)\()p Fq(X)e Fo(\()p Fq(g)r Fo(\)\))17 b(=)208 351 y(\()p Fq(f)267 361 y Fj(\003)293 351 y Fq(X)5 b Fo(\)\()p Fq(f)i Fo(\()p Fq(g)r Fo(\)\))p Fl(.)275 443 y Fr(After)24 b(this,)f(the)g(f)o(act)g (that)g(we)f(are)i(dealing)d(with)h(a)h(homomorphism)g(of)h(Lie)f (groups)f(plays)h(no)208 522 y(further)17 b(role)g(in)f(the)g(proof.)21 b(W)-5 b(e)16 b(pro)o(v)o(e)g(the)h(follo)n(wing:)208 614 y Fs(Lemma)22 b(13.)30 b Fl(Let)22 b Fq(f)15 b Fo(:)24 b Fq(M)36 b Fn(!)29 b Fq(N)g 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Fn(D)20 b(\032)g Fq(T)9 b(M)23 b Fr(or)17 b(rank)g Fq(k)r Fr(.)275 2100 y(The)d(commutator)g(of)g(v)o(ector)g(\002elds)g (with)f(v)n(alues)g(in)h Fn(D)i Fr(de\002nes)e(a)g(bilinear)f(map)h Fo(\000\()p Fn(D)r Fo(\))5 b Fn(\002)g Fo(\000\()p Fn(D)r Fo(\))18 b Fn(!)208 2179 y(X)10 b Fo(\()p Fq(M)d Fo(\))p Fr(,)15 b(which)h(usually)f(does)h(not)g(tak)o(e)g(v)n(alues)g(in)g Fo(\000\()p Fn(D)r Fo(\))g Fr(only)l(.)j(The)e(cases)f(when)g(when)g (it)g(does)g(are)208 2258 y(characterized)g(by)g(the)g(follo)n(wing)g (result:)208 2350 y Fs(Theor)o(em)i(16)f Fr(\(Frobenius)h(Theorem\))p Fs(.)30 b Fl(F)-7 b(or)18 b(a)f(r)o(ank)h Fq(k)i Fl(distrib)o(ution)c Fn(D)22 b(\032)g Fq(T)9 b(M)24 b Fl(the)18 b(following)e(two)208 2429 y(conditions)f(ar)n(e)h(equivalent:)230 2507 y Fr(\(a\))29 b Fl(the)16 b(space)h(of)f(sections)g Fo(\000\()p Fn(D)r Fo(\))h Fl(is)f(closed)g(under)h Fo([)g Fq(;)28 b Fo(])p Fl(,)17 b(and)227 2586 y Fr(\(b\))28 b Fl(for)f(e)o(very)f Fq(p)38 b Fn(2)f Fq(M)c Fl(ther)n(e)26 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y(lea)o(v)o(es)g(are)i(the)f(equi)n (v)n(alence)f(classes)h(of)h(this)e(equi)n(v)n(alence)g(relation,)h(i.) g(e.)g(the)f(maximal)i(connected)208 3544 y(inte)o(gral)e(manifolds.)p eop %%Page: 4 4 4 3 bop 0 -89 a Fg(4)1130 46 y Fr(29)17 b(April)f(2004)67 203 y(\(13\))29 b(W)-5 b(e)16 b(no)n(w)h(return)g(to)f(the)h(proof)g (of)g(Theorem)g(14.)k(Suppose)c(we)g(are)g(gi)n(v)o(en)f(a)h(Lie)g (subalgebra)f Fk(h)k Fn(\032)f Fk(g)p Fr(.)208 281 y(By)26 b(e)n(v)n(aluation)e(at)i(all)g(points)f(of)h Fq(G)g Fr(this)g(de\002nes)g(a)g(left-in)m(v)n(ariant)g(distrib)o(ution)e Fn(D)k Fr(on)d Fq(G)p Fr(.)49 b(The)208 360 y(assumption)16 b(that)h Fk(h)j Fn(\032)g Fk(g)d Fr(is)h(a)f(subalgebra,)f(i.)h(e.)h (it)f(is)g(closed)g(under)f(commutation,)h(implies)g(that)f Fn(D)208 439 y Fr(satis\002es)f(condition)d(\(a\))k(in)e(the)g (Frobenius)g(theorem.)20 b(By)14 b(the)g(Frobenius)h(theorem)f Fn(D)j Fr(is)d(completely)208 517 y(inte)o(grable.)35 b(Let)22 b Fq(H)27 b Fr(be)22 b(the)g(leaf)g(of)g(the)g(corresponding)f (foliation)g(through)g(the)g(neutral)h(element)208 596 y Fq(e)j Fn(2)h Fq(G)p Fr(.)31 b(The)21 b(leaf)f(through)g Fq(a)25 b Fn(2)h Fq(G)20 b Fr(is)h(then)e(obtained)g(by)h(left)g (translation)g(with)f Fq(a)h Fr(applied)f(to)h Fq(H)5 b Fr(.)208 675 y(It)19 b(turns)h(out)e(that)h Fq(H)24 b Fr(is)c(then)e(a)i(Lie)f(subgroup,)g(and)g(is)g(the)g(only)f (connected)g(such)h(group)g(with)f(Lie)208 753 y(algebra)e Fk(h)j Fn(\032)g Fk(g)p Fr(.)i(This)c(completes)f(the)g(proof)h(of)g (Theorem)g(14.)67 832 y(\(14\))29 b(Lie)16 b(subgroups)f(are)h(not)f (usually)g(closed)g(subsets.)20 b(The)c(easiest)g(e)o(xample)f(of)h (this)g(phenomenon)e(is)i(a)208 911 y(densely)g(immersed)h(cop)o(y)g (of)g Fm(R)h Fr(in)e Fq(T)1116 886 y Fi(2)1143 911 y Fr(.)208 1009 y Fs(Pr)o(oposition)j(17.)28 b Fl(If)19 b(the)g(immer)o(sion)e(of)i(a)f(Lie)h(subgr)m(oup)f Fq(H)27 b Fn(\032)c Fq(G)c Fl(is)g(pr)m(oper)-7 b(,)18 b(then)g Fq(H)23 b Fl(is)c(a)g(closed)208 1087 y(subset)d(of)h Fq(G)p Fl(.)67 1186 y Fr(\(15\))29 b(The)16 b(abo)o(v)o(e)g (proposition)f(has)i(a)g(strong)f(con)m(v)o(erse:)208 1284 y Fs(Theor)o(em)k(18.)30 b Fl(If)21 b Fq(G)g Fl(is)g(a)f(Lie)h(gr) m(oup)f(and)g Fq(H)31 b Fn(\032)26 b Fq(G)21 b Fl(is)f(a)h(closed)f (subset)g(that)g(is)h(also)f(an)g(abstr)o(act)208 1363 y(subgr)m(oup,)15 b(then)h Fq(H)22 b Fl(is)17 b(a)f(Lie)h(subgr)m(oup)f (whose)g(inclusion)f(is)i(pr)m(oper)-7 b(.)275 1461 y Fr(Note)21 b(that)h Fq(H)k Fr(is)d(only)d(assumed)j(to)e(be)h(an)g (abstract)f(subgroup,)h(not)g(a)g(Lie)g(subgroup.)35 b(F)o(or)23 b(the)208 1539 y(proof,)17 b(see)f([1)q(],)h(p.)f (139\226141.)1157 1696 y(3)g(May)h(2004)67 1853 y(\(16\))29 b(W)-5 b(e)20 b(no)n(w)h(be)o(gin)f(the)h(discussion)f(of)i(Lie)f (group)f(actions)h(on)f(manifolds.)34 b(All)21 b(our)g(actions)f(will)g (be)208 1932 y(smooth:)208 2030 y Fs(De\002nition)e(19.)28 b Fr(A)17 b(\(left\))i(action)d(of)i(a)g(Lie)f(group)g Fq(G)h Fr(on)f(a)h(smooth)f(manifold)g Fq(M)24 b Fr(is)18 b(a)g(smooth)f(map)208 2109 y Fq(\026)7 b Fo(:)23 b Fq(G)15 b Fn(\002)g Fq(M)26 b Fn(!)19 b Fq(M)24 b Fr(such)16 b(that)271 2188 y Fn(\017)28 b Fq(\026)p Fo(\()p Fq(e;)11 b(p)p Fo(\))18 b(=)h Fq(p)e Fr(for)h(all)e Fq(p)j Fn(2)g Fq(M)7 b Fr(,)16 b(and)271 2266 y Fn(\017)28 b Fq(\026)p Fo(\()p Fq(g)431 2276 y Fi(1)457 2266 y Fq(;)11 b(\026)p Fo(\()p Fq(g)584 2276 y Fi(2)610 2266 y Fq(;)g(p)p Fo(\)\))19 b(=)g Fq(\026)p Fo(\()p Fq(g)911 2276 y Fi(1)937 2266 y Fq(g)969 2276 y Fi(2)995 2266 y Fq(;)11 b(p)p Fo(\))17 b Fr(for)h(all)e Fq(p)j Fn(2)f Fq(M)7 b Fr(,)17 b Fq(g)1532 2276 y Fi(1)1558 2266 y Fq(;)11 b(g)1619 2276 y Fi(2)1665 2266 y Fn(2)19 b Fq(G)p Fr(.)275 2365 y(When)13 b(it)h(is)g(clear)h (which)e(action)g(of)h Fq(G)g Fr(on)g Fq(M)21 b Fr(we)14 b(ha)o(v)o(e)f(in)h(mind,)g(we)g(often)f(simplify)h(the)g(notation)208 2443 y(and)e(write)h Fq(g)r Fo(\()p Fq(p)p Fo(\))f Fr(or)h Fq(g)s Fn(\001)q Fq(p)g Fr(instead)f(of)h Fq(\026)p Fo(\()p Fq(g)r(;)e(p)p Fo(\))p Fr(.)19 b(Then)13 b(the)f(second)h(de\002ning)f (property)g(can)h(be)f(e)o(xpressed)208 2522 y(as)17 b Fq(g)313 2532 y Fi(1)339 2522 y Fo(\()p Fq(g)397 2532 y Fi(2)423 2522 y Fo(\()p Fq(p)p Fo(\)\))h(=)h(\()p Fq(g)680 2532 y Fi(1)706 2522 y Fq(g)738 2532 y Fi(2)764 2522 y Fo(\)\()p Fq(p)p Fo(\))p Fr(.)208 2610 y Fl(Remark)e Fr(20)p Fl(.)26 b Fr(When)16 b(nothing)e(further)j(is)f(said,)g(all)g (our)g(actions)f(will)g(be)h(left)g(actions.)k(Right)15 b(actions)208 2689 y(are)i(de\002ned)f(analogously)l(,)e(so)j(that)f Fq(e)p Fo(\()p Fq(p)p Fo(\))i(=)h Fq(p)e Fr(and)f Fq(g)1472 2699 y Fi(1)1499 2689 y Fo(\()p Fq(g)1557 2699 y Fi(2)1582 2689 y Fo(\()p Fq(p)p Fo(\)\))i(=)h(\()p Fq(g)1839 2699 y Fi(2)1865 2689 y Fq(g)1897 2699 y Fi(1)1924 2689 y Fo(\)\()p Fq(p)p Fo(\))p Fr(.)275 2777 y(An)i(action)g(is)h(said)f(to)h (be)f Fs(effecti)o(v)o(e)i Fr(if)f(for)h(all)e Fq(g)30 b Fn(6)p Fo(=)e Fq(e)g Fn(2)g Fq(G)22 b Fr(there)f(e)o(xists)h(a)g Fq(p)28 b Fn(2)g Fq(M)g Fr(such)22 b(that)208 2856 y Fq(g)r Fo(\()p Fq(p)p Fo(\))c Fn(6)p Fo(=)h Fq(p)p Fr(.)h(An)13 b(action)f(is)h Fs(transiti)o(v)o(e)i Fr(if)f(for)g(all)f Fq(p;)e(q)22 b Fn(2)c Fq(M)j Fr(there)13 b(e)o(xists)g(a)h Fq(g)20 b Fn(2)f Fq(G)14 b Fr(such)f(that)f Fq(g)r Fo(\()p Fq(p)p Fo(\))18 b(=)h Fq(q)r Fr(.)208 2935 y(The)d Fs(orbit)g Fr(of)h(a)f(point)f Fq(p)k Fn(2)f Fq(M)24 b Fr(under)15 b(the)h(action)f(consists)h(of)g(all)g(the)f(points)g Fq(g)r Fo(\()p Fq(p)p Fo(\))p Fr(.)20 b(P)o(artitioning)15 b Fq(M)208 3013 y Fr(into)f(the)h(orbits)g(of)g(the)g(action)f (de\002nes)h(an)g(equi)n(v)n(alence)f(relation)g(on)h Fq(M)7 b Fr(.)20 b(T)n(ransiti)n(vity)14 b(is)i(equi)n(v)n(alent)208 3092 y(to)g(the)g(requirement)h(that)f(there)h(is)f(only)g(one)g (orbit.)67 3171 y(\(17\))29 b(Examples:)19 b(the)12 b(tri)n(vial)h (action,)f(the)h(action)f(of)h Fm(R)i Fr(de\002ned)e(by)g(a)g(\003o)n (w)h(on)e Fq(M)7 b Fr(,)14 b(the)f(action)f(of)h Fq(GL)p Fo(\()p Fq(n;)e Fm(R)p Fo(\))208 3249 y Fr(on)16 b Fm(R)339 3225 y Fp(n)373 3249 y Fr(,)g(the)g(action)g(of)h Fq(S)t(O)r Fo(\()p Fq(n)p Fo(\))g Fr(on)f Fm(R)1096 3225 y Fp(n)1129 3249 y Fr(,)h(its)g(restriction)f(to)g Fq(S)1648 3225 y Fp(n)p Fj(\000)p Fi(1)1740 3249 y Fr(,.)10 b(.)g(.)67 3328 y(\(18\))29 b(F)o(or)21 b(e)n(v)o(ery)f Fq(p)26 b Fn(2)f Fq(M)i Fr(we)20 b(de\002ne)g Fq(G)1044 3338 y Fp(p)1096 3328 y Fn(\032)26 b Fq(G)20 b Fr(to)g(consist)g(of)g(the)g (elements)g Fq(g)27 b Fn(2)e Fq(G)c Fr(with)e(the)h(property)208 3407 y Fq(g)r Fo(\()p Fq(p)p Fo(\))e(=)h Fq(p)p Fr(.)i(This)c(is)g(a)f (subgroup)g(of)h Fq(G)g Fr(called)f(the)g Fs(isotr)o(opy)h(subgr)o(oup) h Fr(of)f Fq(p)p Fr(.)208 3505 y Fs(Lemma)g(21.)27 b Fl(F)-7 b(or)17 b(e)o(very)g Fq(p)h Fn(2)h Fq(M)24 b Fl(the)16 b(isotr)m(opy)g(subgr)m(oup)f Fq(G)1688 3515 y Fp(p)1732 3505 y Fl(is)i(a)g(closed)f(subgr)m(oup)f(of)h Fq(G)p Fl(.)275 3603 y Fr(Note)g(that)g(Theorem)h(18)g(no)n(w)f(tells)g (us)h(that)f Fq(G)1393 3613 y Fp(p)1437 3603 y Fr(is)g(a)h(properly)f (embedded)g(Lie)h(subgroup)f(of)h Fq(G)p Fr(.)p eop %%Page: 5 5 5 4 bop 2614 -89 a Fg(5)67 46 y Fr(\(19\))29 b(Let)18 b Fq(G)g Fr(be)g(a)g(Lie)g(group,)g(and)g Fq(H)26 b Fn(\032)21 b Fq(G)e Fr(a)f(closed)f(subgroup.)24 b(The)18 b(set)g Fq(G=H)24 b Fr(of)18 b(left)g(cosets)g(of)g Fq(H)24 b Fr(in)208 124 y Fq(G)17 b Fr(is)g(a)f(topological)f(space)h(with)g(the) g(quotient)f(topology)l(,)f(and)i(is)h(called)f(a)h Fs(homogeneous)g (space)p Fr(.)208 229 y Fs(Theor)o(em)j(22.)29 b Fl(F)-7 b(or)21 b(e)o(very)f(closed)f(subgr)m(oup)g Fq(H)30 b Fn(\032)c Fq(G)20 b Fl(the)g(corr)n(esponding)e(homo)o(g)o(eneous)h (space)208 308 y Fq(G=H)i Fl(has)c(a)f(natur)o(al)f(structur)n(e)h(as)g (a)g(smooth)g(manifold)e(of)i(dimension)f Fq(dim)p Fo(\()p Fq(G)p Fo(\))e Fn(\000)h Fq(dim)p Fo(\()p Fq(H)5 b Fo(\))p Fl(,)15 b(suc)o(h)208 386 y(that)h(the)g(pr)m(ojection)f Fq(\031)10 b Fo(:)23 b Fq(G)c Fn(!)g Fq(G=H)j Fl(is)16 b(smooth)g(and)g(admits)g(smooth)f(local)h(sections.)275 491 y Fr(In)i(f)o(act)g(the)f(projection)g(will)f(turn)i(out)e(to)i(be) f(a)h(submersion.)23 b(In)18 b(the)f(proof)h(we)f(shall)g(use)h(the)f (no-)208 569 y(tation)f Fq(l)399 579 y Fp(a)434 569 y Fo(:)23 b Fq(G=H)i Fn(!)20 b Fq(G=H)i Fr(for)c(the)f(continuous)f(maps) h(induced)f(on)h Fq(G=H)22 b Fr(by)17 b(the)g(left)g(translations)208 648 y(on)j Fq(G)p Fr(.)31 b(These)21 b(are)g(all)f(homeomorphisms,)h (and)f(will)f(turn)h(into)g(dif)n(feomorphisms)h(once)f(we)g(ha)o(v)o (e)208 727 y(de\002ned)c(the)g(smooth)h(structure)f(on)h Fq(G=H)5 b Fr(.)208 868 y Fl(Pr)m(oof)o(.)27 b Fr(By)19 b(de\002nition,)e(a)i(subset)f Fq(U)29 b Fn(\032)23 b Fq(G=H)h Fr(is)19 b(open)e(if)i(and)f(only)g(if)h(its)f(preimage)h Fq(\031)2322 844 y Fj(\000)p Fi(1)2387 868 y Fo(\()p Fq(U)7 b Fo(\))21 b Fn(\032)i Fq(G)208 947 y Fr(is)d(open.)28 b(Therefore,)22 b(a)d(countable)f(basis)i(for)g(the)g(topology)d(of)j Fq(G)g Fr(induces)f(a)h(countable)e(basis)h(for)208 1026 y(the)d(topology)f(of)i Fq(G=H)5 b Fr(.)275 1104 y(The)21 b(assumption)f(that)h Fq(H)31 b Fn(\032)c Fq(G)22 b Fr(is)f(a)g(closed) f(subset)h(implies)f(that)h(its)g(preimage)g Fq(R)27 b Fn(\032)g Fq(G)18 b Fn(\002)h Fq(G)208 1183 y Fr(under)g(the)g(map)h Fo(\()p Fq(g)677 1193 y Fi(1)703 1183 y Fq(;)11 b(g)764 1193 y Fi(2)791 1183 y Fo(\))24 b Fn(7!)h Fq(g)967 1155 y Fj(\000)p Fi(1)965 1200 y(1)1031 1183 y Fq(g)1063 1193 y Fi(2)1109 1183 y Fr(is)20 b(also)f(closed.)29 b(If)21 b Fq(aH)29 b Fn(6)p Fo(=)24 b Fq(bH)5 b Fr(,)21 b(then)e Fo(\()p Fq(a;)11 b(b)p Fo(\))32 b Fq(=)-41 b Fn(2)24 b Fq(R)q Fr(,)c(so)g(we)f(can)208 1262 y(\002nd)c(open)f (neighbourhoods)f Fq(U)22 b Fr(and)14 b Fq(V)31 b Fr(of)15 b Fq(a)g Fr(respecti)n(v)o(ely)f Fq(b)h Fr(in)g Fq(G)g Fr(so)g(that)g Fo(\()p Fq(U)g Fn(\002)9 b Fq(V)16 b Fo(\))9 b Fn(\\)g Fq(R)18 b Fo(=)i Fn(;)p Fr(.)f(Then)208 1340 y Fq(\031)r Fo(\()p Fq(U)7 b Fo(\))21 b Fr(and)f Fq(\031)r Fo(\()p Fq(V)c Fo(\))21 b Fr(are)g(disjoint)e(open)h(neighbourhoods)f (of)i Fq(aH)k Fr(respecti)n(v)o(ely)c Fq(bH)26 b Fr(in)20 b Fq(G=H)5 b Fr(.)33 b(Thus)208 1419 y Fq(G=H)22 b Fr(has)16 b(the)h(Hausdorf)n(f)g(property)l(.)275 1498 y(It)i(remains)h(to)e (construct)g(an)g(atlas)h(with)f(smooth)g(transition)g(maps.)27 b(First,)20 b(let)e Fo(\()p Fq(V)t(;)11 b(\036)p Fo(\))19 b Fr(be)f(a)h(chart)208 1576 y(for)e Fq(G)f Fr(around)f Fq(e)p Fr(.)20 b(As)c Fq(H)24 b Fn(\032)19 b Fq(G)d Fr(is)g(closed,)g (we)g(may)g(assume)g(that)g Fq(H)h Fn(\\)12 b Fq(V)32 b Fr(is)16 b(connected.)i(W)-5 b(e)16 b(choose)208 1655 y Fq(\036)21 b Fr(so)h(that)e(it)h(maps)h(the)f(intersections)f(of)h (the)g(cosets)g(of)h Fq(H)k Fr(with)20 b Fq(V)37 b Fr(to)21 b(the)g(subsets)g Fm(R)2344 1630 y Fp(k)2393 1655 y Fn(\002)e(f)p Fq(c)p Fn(g)27 b(\032)208 1734 y Fm(R)255 1709 y Fp(k)304 1734 y Fn(\002)19 b Fm(R)422 1709 y Fp(n)p Fj(\000)p Fp(k)545 1734 y Fo(=)28 b Fm(R)671 1709 y Fp(n)732 1734 y Fo(=)f Fq(\036)p Fo(\()p Fq(V)15 b Fo(\))p Fr(.)34 b(By)21 b(shrinking)g Fq(V)15 b Fr(,)22 b(we)f(\002nd)h(smaller)f(open) g(sets)h Fq(U)28 b Fr(and)21 b Fq(V)2426 1744 y Fi(1)2474 1734 y Fr(which)208 1812 y(ha)o(v)o(e)16 b(the)h(same)h(properties)f (and)g(in)f(addition)g(satisfy)h Fq(U)1561 1788 y Fj(\000)p Fi(1)1640 1812 y Fn(\001)e Fq(U)27 b Fn(\032)20 b Fq(V)1856 1822 y Fi(1)1901 1812 y Fr(and)d Fq(V)2054 1822 y Fi(1)2096 1812 y Fn(\001)e Fq(V)2168 1822 y Fi(1)2215 1812 y Fn(\032)20 b Fq(V)15 b Fr(.)23 b(It)17 b(follo)n(ws)208 1891 y(that)24 b(e)n(v)o(ery)i(coset)f(of)h Fq(H)k Fr(either)25 b(does)g(not)g(meet)g Fq(V)1472 1901 y Fi(1)1525 1891 y Fr(at)g(all,)i(or)f(meets)f(it)g(in)g (a)h(connected)d(subset)208 1970 y(dif)n(feomorphic)17 b(via)f Fq(\036)h Fr(to)f Fm(R)880 1945 y Fp(k)911 1970 y Fr(.)275 2052 y(De\002ne)490 2034 y Fo(~)481 2052 y Fq(\036)520 2028 y Fj(\000)p Fi(1)592 2052 y Fo(:)26 b Fm(R)683 2028 y Fp(n)p Fj(\000)p Fp(k)816 2052 y Fn(!)37 b Fq(\031)r Fo(\()p Fq(U)7 b Fo(\))26 b Fr(by)g Fq(\031)e Fn(\016)e Fq(\036)1339 2028 y Fj(\000)p Fi(1)1403 2052 y Fr(,)29 b(where)d Fm(R)1687 2028 y Fp(n)p Fj(\000)p Fp(k)1820 2052 y Fo(=)36 b Fn(f)p Fo(0)p Fn(g)22 b(\002)g Fm(R)2151 2028 y Fp(n)p Fj(\000)p Fp(k)2284 2052 y Fn(\032)37 b Fm(R)2420 2028 y Fp(n)2453 2052 y Fr(.)49 b(This)208 2131 y(map)23 b(is)g(bijecti)n(v)o(e,)g(continuous)e(and)i(an)g(open)f (map,)i(and)f(so)g(is)g(a)g(homeomorphism.)40 b(W)-5 b(e)22 b(de\002ne)216 2196 y Fo(~)208 2214 y Fq(\036)7 b Fo(:)24 b Fq(\031)r Fo(\()p Fq(U)7 b Fo(\))21 b Fn(!)h Fm(R)596 2189 y Fp(n)p Fj(\000)p Fp(k)711 2214 y Fr(by)17 b(in)m(v)o(ersion,)h(and)g(this)f(is)h(then)g(also)g(a)g (homeomorphism.)25 b(W)-5 b(e)17 b(shall)g(use)i(this)208 2292 y(as)e(a)g(chart)f(for)i Fq(G=H)k Fr(around)16 b(the)g(point)f Fq(eH)23 b Fo(=)d Fq(H)5 b Fr(.)275 2377 y(F)o(or)26 b(an)o(y)f Fq(a)34 b Fn(2)g Fq(G)25 b Fr(we)g(obtain)f(a)h(chart)g(for) h Fq(G=H)k Fr(around)25 b Fq(aH)k Fr(by)c(taking)f Fo(\()p Fq(\031)r Fo(\()p Fq(l)2257 2387 y Fp(a)2285 2377 y Fo(\()p Fq(U)7 b Fo(\)\))p Fq(;)2452 2360 y Fo(~)2444 2377 y Fq(\036)2483 2387 y Fp(aH)2588 2377 y Fo(=)216 2445 y(~)208 2462 y Fq(\036)20 b Fn(\016)h Fq(l)342 2474 y Fp(a)367 2461 y Fe(\000)p Fd(1)425 2462 y Fo(\))p Fr(.)43 b(Thus,)26 b(we)d(ha)o(v)o(e)h(the)f(structure)h(of)h(a)f(topological)e(manifold)h (on)h Fq(G=H)5 b Fr(.)43 b(T)-5 b(o)24 b(see)g(that)208 2541 y(this)19 b(atlas)h(actually)f(de\002nes)h(a)g(smooth)g (structure,)h(it)e(remains)i(to)f(check)f(that)g(the)h(transition)f (maps)208 2620 y(between)14 b(dif)n(ferent)h(charts)h(are)f(smooth.)20 b(Suppose)15 b(that)f Fq(\031)r Fo(\()p Fq(l)1648 2630 y Fp(a)1677 2620 y Fo(\()p Fq(U)7 b Fo(\)\))i Fn(\\)g Fq(\031)r Fo(\()p Fq(l)1955 2630 y Fp(b)1978 2620 y Fo(\()p Fq(U)e Fo(\)\))17 b Fn(6)p Fo(=)i Fn(;)p Fr(.)h(Then)15 b(we)g(ha)o(v)o(e)208 2705 y(to)h(pro)o(v)o(e)g(that)572 2687 y Fo(~)563 2705 y Fq(\036)602 2715 y Fp(bH)683 2705 y Fn(\016)741 2687 y Fo(~)732 2705 y Fq(\036)771 2677 y Fj(\000)p Fi(1)771 2723 y Fp(aH)859 2705 y Fr(is)h(smooth)f(on)g Fq(X)24 b Fo(=)1378 2687 y(~)1369 2705 y Fq(\036)1408 2715 y Fp(aH)1479 2705 y Fo(\()p Fq(\031)r Fo(\()p Fq(l)1590 2715 y Fp(a)1618 2705 y Fo(\()p Fq(U)7 b Fo(\)\))14 b Fn(\\)h Fq(\031)r Fo(\()p Fq(l)1907 2715 y Fp(b)1930 2705 y Fo(\()p Fq(U)7 b Fo(\)\)\))p Fr(.)20 b(Note)c(that)567 2825 y Fo(~)558 2843 y Fq(\036)597 2853 y Fp(bH)678 2843 y Fn(\016)735 2825 y Fo(~)727 2843 y Fq(\036)766 2815 y Fj(\000)p Fi(1)766 2861 y Fp(aH)855 2843 y Fo(=)j(\()960 2825 y(~)951 2843 y Fq(\036)c Fn(\016)g Fq(l)1074 2854 y Fp(b)1094 2841 y Fe(\000)p Fd(1)1153 2843 y Fo(\))f Fn(\016)h Fo(\()1277 2825 y(~)1268 2843 y Fq(\036)g Fn(\016)f Fq(l)1390 2854 y Fp(a)1415 2841 y Fe(\000)p Fd(1)1474 2843 y Fo(\))1500 2815 y Fj(\000)p Fi(1)1582 2843 y Fo(=)1661 2825 y(~)1652 2843 y Fq(\036)h Fn(\016)g Fq(l)1775 2854 y Fp(b)1795 2841 y Fe(\000)p Fd(1)1851 2854 y Fp(a)1894 2843 y Fn(\016)1951 2825 y Fo(~)1943 2843 y Fq(\036)1982 2815 y Fj(\000)p Fi(1)2063 2843 y Fq(:)208 2981 y Fr(Let)f Fq(p)k Fn(2)h Fq(X)5 b Fr(.)20 b(Then)13 b Fq(l)696 2992 y Fp(b)716 2979 y Fe(\000)p Fd(1)772 2992 y Fp(a)804 2981 y Fn(\016)851 2963 y Fo(~)842 2981 y Fq(\036)881 2956 y Fj(\000)p Fi(1)945 2981 y Fo(\()p Fq(p)p Fo(\))18 b Fn(2)h Fq(\031)r Fo(\()p Fq(U)7 b Fo(\))p Fr(,)14 b(so)g(there)g(e)o (xists)f(a)h Fq(g)21 b Fn(2)d Fq(H)h Fr(such)14 b(that)f Fq(l)2182 2992 y Fp(b)2202 2979 y Fe(\000)p Fd(1)2258 2992 y Fp(a)2290 2981 y Fn(\016)t Fq(\036)2367 2956 y Fj(\000)p Fi(1)2430 2981 y Fo(\()p Fq(p)p Fo(\))t Fn(\001)t Fq(g)20 b Fn(2)208 3059 y Fq(U)7 b Fr(.)20 b(There)c(is)g(a)f (neighbourhood)e Fq(W)25 b Fr(of)16 b Fq(p)g Fr(in)f Fq(X)21 b Fr(such)15 b(that)g Fq(l)1628 3071 y Fp(b)1648 3058 y Fe(\000)p Fd(1)1704 3071 y Fp(a)1743 3059 y Fn(\016)c Fq(\036)1827 3035 y Fj(\000)p Fi(1)1890 3059 y Fo(\()p Fq(W)e Fo(\))i Fn(\001)g Fq(g)20 b Fn(\032)f Fq(U)7 b Fr(.)20 b(On)15 b Fq(W)25 b Fr(we)15 b(can)208 3144 y(re)n(write)422 3127 y Fo(~)414 3144 y Fq(\036)453 3154 y Fp(bH)534 3144 y Fn(\016)591 3127 y Fo(~)583 3144 y Fq(\036)622 3116 y Fj(\000)p Fi(1)622 3163 y Fp(aH)709 3144 y Fr(as)981 3249 y Fq(\031)1018 3259 y Fi(2)1060 3249 y Fn(\016)g Fq(\036)g Fn(\016)g Fq(r)1242 3259 y Fp(g)1283 3249 y Fn(\016)g Fq(l)1352 3260 y Fp(b)1372 3248 y Fe(\000)p Fd(1)1428 3260 y Fp(a)1471 3249 y Fn(\016)g Fq(\036)1559 3221 y Fj(\000)p Fi(1)1640 3249 y Fq(;)208 3367 y Fr(where)f Fq(\031)424 3377 y Fi(2)466 3367 y Fr(is)g(the)g(projection)f(from)j Fm(R)1103 3343 y Fp(k)1140 3367 y Fn(\002)7 b Fm(R)1246 3343 y Fp(n)p Fj(\000)p Fp(k)1358 3367 y Fr(to)14 b(the)g(second)g(f)o (actor)l(.)21 b(This)14 b(composition)f(is)i(clearly)208 3446 y(smooth.)275 3524 y(Thus,)22 b(we)f(ha)o(v)o(e)f(\002nally)h (check)o(ed)f(that)h Fq(G=H)26 b Fr(is)21 b(a)h(smooth)e(manifold.)34 b(In)21 b(the)g(charts)g(we)g(ha)o(v)o(e)208 3603 y(constructed)g Fq(\031)k Fr(corresponds)d(to)f Fq(\031)1062 3613 y Fi(2)1112 3603 y Fr(and)g(is)h(therefore)h(smooth.)36 b(The)22 b(smooth)g(local)f(sections)h(are)p eop %%Page: 6 6 6 5 bop 0 -89 a Fg(6)208 46 y Fr(pro)o(vided)24 b(by)i(the)f(left)h (translations)e(of)i(the)g(restriction)f(of)h Fq(\036)1737 21 y Fj(\000)p Fi(1)1827 46 y Fr(to)f(the)g(second)h(f)o(actor)g(in)f Fm(R)2534 21 y Fp(k)2587 46 y Fn(\002)208 124 y Fm(R)255 100 y Fp(n)p Fj(\000)p Fp(k)351 124 y Fr(.)2219 b Fh(\003)1131 347 y Fr(R)t Fc(E)t(F)t(E)t(R)t(E)s(N)t(C)s(E)s(S)28 454 y Fb(1.)28 b(L.)15 b(Conlon,)f Fa(Dif)o(ferentiable)d(Manifolds)i (\227)h(A)g(First)f(Course)p Fb(,)h(Birkh)s(\250)-22 b(auser)14 b(V)-6 b(erlag)12 b(1993.)28 521 y(2.)28 b(F)l(.)14 b(W)-5 b(.)14 b(W)l(arner)n(,)f Fa(F)o(oundations)h(of)g(Dif)o (ferentiable)c(Manifolds)j(and)i(Lie)f(Groups)p Fb(,)f(Springer)h(V)-6 b(erlag)12 b(1983.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF