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Fm(h)p Fp(\))g Fl(!)h Fm(R)q Fp(\()p Fm(h)p Fp(\))f(b)r(et)n(w)n(een)h(op)r(en)g(sets)g(\(of)g Fo(R)1973 1841 y Fn(n)2007 1865 y Fp(\))g(satisfying)521 1944 y(the)c(follo)n(wing)h(conditions:)543 2023 y(\(a\))28 b(If)22 b Fm(g)r(;)11 b(h)19 b Fl(2)g(G)26 b Fp(and)21 b Fm(R)q Fp(\()p Fm(h)p Fp(\))d Fl(\032)h Fm(D)r Fp(\()p Fm(g)r Fp(\),)i(then)h Fm(g)17 b Fl(\016)d Fm(h)19 b Fl(2)g(G)t Fp(.)539 2101 y(\(b\))28 b(If)22 b Fm(h)c Fl(2)h(G)t Fp(,)j(then)g Fm(h)1115 2077 y Fj(\000)p Fk(1)1197 2101 y Fl(2)d(G)t Fp(.)547 2180 y(\(c\))28 b(The)23 b(restriction)h(of) h(an)n(y)e Fm(h)g Fl(2)f(G)28 b Fp(to)c(an)g(op)r(en)f(subset)h(con)n (tained)656 2258 y(in)e Fm(D)r Fp(\()p Fm(h)p Fp(\))f(is)h(also)g(in)g Fl(G)t Fp(.)539 2337 y(\(d\))28 b(If)22 b Fm(h)7 b Fp(:)23 b Fm(U)j Fl(!)20 b Fm(V)37 b Fp(is)22 b(a)h(homeomorphism)18 b(b)r(et)n(w)n(een)k(op)r(en)g(sets)g(with)656 2416 y(the)i(prop)r(ert) n(y)g(that)g Fm(U)32 b Fp(can)25 b(b)r(e)g(co)n(v)n(ered)h(b)n(y)f(op)r (en)g(sets)g Fm(U)2160 2426 y 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Fq(tangen)n(t)k(bundle)d Fp(of)f(a)h(smo)r(oth)d(manifold,)521 3519 y(see)j([1])g(Section)f(3.3.)p eop %%Page: 3 3 3 2 bop 2305 159 a Ff(3)1082 294 y Fp(29)22 b(Octob)r(er)g(2001)376 428 y(\(15\))27 b Fq(Deriv)l(ativ)n(es)37 b(of)h(smo)r(oth)g(maps)33 b Fp(b)r(et)n(w)n(een)g(manifolds;)38 b(tangen)n(t)521 507 y(v)n(ectors)22 b(as)g(deriv)l(ativ)n(es)h(of)f(paths.)376 585 y(\(16\))27 b(Existence)21 b(of)g(metrics)f(on)h(v)n(ector)f (bundles;)h Fq(Riemannian)j(metrics)521 664 y Fp(and)d(length)h(of)g (tangen)n(t)g(v)n(ectors.)1070 798 y(5)g(No)n(v)n(em)n(b)r(er)g(2001) 376 931 y(\(17\))27 b Fq(Arclength)i Fp(and)f Fq(distance)h Fp(in)f(a)h(Riemannian)e(manifold;)k(the)e(in-)521 1010 y(duced)22 b(metric)f(space)i(structure.)376 1089 y(\(18\))k(A)18 b(subset)f Fm(N)24 b Fp(of)18 b(an)f Fm(m)p Fp(-dimensional)g(manifold) f Fm(M)25 b Fp(is)18 b(a)f Fq(submanifold)521 1167 y Fp(of)22 b(dimension)f Fm(n)h Fp(if)g(ev)n(ery)g(p)r(oin)n(t)f(in)h Fm(N)29 b Fp(has)21 b(an)g(op)r(en)h(neigh)n(b)r(ourho)r(o)r(d)521 1246 y Fm(U)29 b Fp(in)22 b Fm(M)29 b Fp(whic)n(h)22 b(is)g(the)g(domain)f(of)h(a)g(c)n(hart)f(\()p Fm(U;)11 b(\036)p Fp(\))22 b(suc)n(h)g(that)876 1352 y Fm(\036)p Fp(\()p Fm(U)f Fl(\\)15 b Fm(N)7 b Fp(\))19 b(=)g Fm(\036)p Fp(\()p Fm(U)7 b Fp(\))14 b Fl(\\)h Fo(R)1506 1324 y Fn(n)1558 1352 y Fl(\032)k Fo(R)1676 1324 y Fn(m)1745 1352 y Fm(:)521 1458 y Fp(This)j(is)g(called)h(a)f(\\prop)r(erly)f(em)n (b)r(edded)g(submanifold")g(in)h([1].)521 1537 y Fq(Immersions)k Fp(and)g Fq(em)n(b)r(eddings)g Fp(\(=injectiv)n(e)h(immersions\).)40 b(Ev-)521 1615 y(ery)19 b(\(compact\))f(manifold)h(can)g(b)r(e)g(em)n (b)r(edded)g(in)g(a)g(Euclidean)h(space.)1070 1749 y(8)i(No)n(v)n(em)n (b)r(er)g(2001)376 1883 y(\(19\))27 b(The)15 b(image)f(of)h(an)n(y)g (prop)r(er)f(em)n(b)r(edding)f(is)j(a)f(submanifold.)26 b(Examples)521 1962 y(of)c(non-prop)r(er)f(em)n(b)r(eddings.)376 2040 y(\(20\))27 b Fq(Regular)21 b Fp(and)h Fq(critical)j(v)l(alues)c Fp(of)h(smo)r(oth)f(maps,)g Fq(submersions)p Fp(.)376 2119 y(\(21\))27 b Fq(Co)r(cycles)j Fp(and)g Fm(G)p Fq(-structures)h Fp(on)g(v)n(ector)g(bundles,)h(see)g([1])f(Sec-)521 2197 y(tion)17 b(3.4.)28 b(Examples)17 b(for)g Fm(G)p Fp(:)27 b(the)17 b(trivial)g(group,)g(the)g(group)f Fm(GL)p Fp(\()p Fm(l)q(;)11 b(k)c Fl(\000)521 2276 y Fm(l)q Fp(\))19 b Fl(\032)g Fm(Gl)q Fp(\()p Fm(k)r(;)11 b Fo(R)p Fp(\))24 b(preserving)e(a)g(subspace)g(of)g(dimension)f Fm(l)q Fp(.)1054 2410 y(12)h(No)n(v)n(em)n(b)r(er)f(2001)376 2544 y(\(22\))27 b(F)-6 b(urther)23 b(examples:)33 b(the)23 b(iden)n(tit)n(y)h(comp)r(onen)n(t)e(of)i Fm(Gl)q Fp(\()p Fm(k)r(;)11 b Fo(R)p Fp(\),)27 b(the)c(or-)521 2622 y(thogonal)f (group,)f(the)h(complex)f(linear)h(group.)376 2701 y(\(23\))27 b Fq(In\014nitesimal)e(geometric)f(structures)e Fp(on)e(manifolds;)h (the)g Fq(in)n(te-)521 2779 y(grabilit)n(y)26 b(problem)p Fp(.)1054 2913 y(15)c(No)n(v)n(em)n(b)r(er)f(2001)376 3047 y(\(24\))27 b(In)n(tegration)h(of)h Fq(v)n(ector)k(\014elds)p Fp(;)g Fq(lo)r(cal)c Fp(and)f Fq(global)33 b(\015o)n(ws)p Fp(;)g Fq(com-)521 3126 y(pleteness)p Fp(,)22 b(see)h([1])f(Section)g (4.1.)376 3204 y(\(25\))27 b(V)-6 b(ector)23 b(\014elds)f(as)g Fq(deriv)l(ations)g Fp(and)g(the)f Fq(Lie)k(deriv)l(ativ)n(e)p Fp(.)p eop %%Page: 4 4 4 3 bop 308 159 a Ff(4)1054 294 y Fp(19)22 b(No)n(v)n(em)n(b)r(er)f (2001)376 428 y(\(26\))27 b(All)c(deriv)l(ations)f(are)g(v)n(ector)g (\014elds.)376 507 y(\(27\))27 b Fq(Comm)n(utators)g Fp(and)d(the)i(Lie)g(algebra)g(structure)e(on)h(the)h(space)g(of)521 585 y(v)n(ector)c(\014elds,)h(see)f([1)q(])g(Section)g(4.2.)1054 719 y(22)g(No)n(v)n(em)n(b)r(er)f(2001)376 853 y(\(28\))27 b(Comm)n(uting)19 b(v)n(ector)j(\014elds)g(and)g Fm(k)r Fp(-\015o)n(ws.)1054 986 y(26)g(No)n(v)n(em)n(b)r(er)f(2001)376 1120 y(\(29\))27 b(More)f(on)f Fm(k)r Fp(-\015o)n(ws)g(and)g(the)g(in)n (tegrabilit)n(y)h(condition)f(for)g(\(in\014nitesi-)521 1199 y(mal\))c(parallelizations,)j(see)e([1)q(])g(Section)g(4.3.)1054 1332 y(29)g(No)n(v)n(em)n(b)r(er)f(2001)376 1466 y(\(30\))27 b(The)14 b(in)n(tegrabilit)n(y)h(problem)f(for)g(in\014nitesimal)g Fm(GL)p Fp(\()p Fm(l)q(;)d(n)p Fl(\000)p Fm(l)q Fp(\)-structures,)521 1545 y(see)21 b([1)q(])f(Section)h(4.4.)29 b(In\014nitesimal)19 b Fm(GL)p Fp(\()p Fm(l)q(;)11 b(n)h Fl(\000)g Fm(l)q Fp(\)-structures)19 b(are)i(dis-)521 1623 y(tributions,)27 b(and)g(in)g(this)f(case)i(the)f(in)n(tegrabilit)n(y)g(criterion)g(is)g (giv)n(en)521 1702 y(b)n(y)22 b(the)g Fq(F)-6 b(rob)r(enius)24 b(theorem)p Fp(.)1072 1836 y(3)e(Decem)n(b)r(er)h(2001)376 1969 y(\(31\))k Fq(Lie)e(groups)d Fp(and)f(their)h Fq(Lie)i(algebras)p Fp(:)29 b(de\014nitions,)22 b(examples.)1072 2103 y(6)g(Decem)n(b)r(er) h(2001)376 2237 y(\(32\))k(Completeness)14 b(of)h(left-in)n(v)l(arian)n (t)g(v)n(ector)g(\014elds,)h(and)e(their)g Fq(one-parameter)521 2315 y(subgroups)p Fp(.)30 b(The)22 b Fq(exp)r(onen)n(tial)j(map)p Fp(.)1056 2449 y(13)d(Decem)n(b)r(er)g(2001)376 2583 y(\(33\))27 b Fq(Lie)32 b(subgroups)p Fp(.)49 b(The)28 b(corresp)r(ondence)g(b)r(et)n(w)n(een)g(connected)h(Lie)521 2662 y(subgroups)21 b(and)g(Lie)i(subalgebras.)1056 2795 y(17)f(Decem)n(b)r(er)g(2001)376 2929 y(\(34\))27 b(Actions)c(of)f(Lie) h(groups)e(on)h(manifolds;)f Fq(homogeneous)k(spaces)p Fp(.)1056 3063 y(20)d(Decem)n(b)r(er)g(2001)376 3196 y(\(35\))27 b(The)22 b(Lie)h(groups)e Fm(S)t(O)r Fp(\(3\),)g Fm(S)1247 3172 y Fk(3)1296 3196 y Fp(and)g Fm(S)t(U)7 b Fp(\(2\);)21 b(and)g(their)h(Lie)h(algebra.)p eop %%Page: 5 5 5 4 bop 2305 159 a Ff(5)1099 294 y Fp(7)22 b(Jan)n(uary)f(2002)376 428 y(\(36\))27 b(The)21 b(cotangen)n(t)f(bundle;)i Fq(one-forms)f Fp(and)f(their)h(functorial)g(prop)r(er-)521 507 y(ties,)i(see)g([1])f (Section)g(6.1.)1083 640 y(10)g(Jan)n(uary)e(2002)376 774 y(\(37\))27 b Fq(In)n(tegrals)22 b Fp(of)f(one-forms;)g(exactness)g (and)g(lo)r(cal)h(exactness.)30 b(\(See)21 b([1])521 853 y(Section)h(6.2.\))1083 986 y(14)g(Jan)n(uary)e(2002)376 1120 y(\(38\))27 b(The)20 b(\014rst)f Fq(de)k(Rham)g(cohomology)d Fp(group:)28 b(de\014nition)19 b(and)h(homo-)521 1199 y(top)n(y)h(in)n(v)l(ariance.)31 b(\(See)22 b([1])h(Section)f(6.3.\)) 1083 1332 y(17)g(Jan)n(uary)e(2002)376 1466 y(\(39\))27 b(The)e(calculation)h(of)f Fm(H)1135 1442 y Fk(1)1162 1466 y Fp(\()p Fm(S)1233 1442 y Fk(1)1259 1466 y Fp(\),)h(and)e(some)h (top)r(ological)g(applications,)521 1545 y(including)30 b(the)f Fq(Brou)n(w)n(er)k(\014xed)h(p)r(oin)n(t)g(theorem)c Fp(in)f(the)g(plane.)521 1623 y(\(See)22 b([1)q(])g(Section)g(6.3.\)) 376 1702 y(\(40\))27 b(The)22 b Fq(tensor)j(algebra)d Fp(of)g(a)g(real)g(v)n(ector)g(space.)1083 1836 y(21)g(Jan)n(uary)e (2002)376 1969 y(\(41\))27 b Fq(Exterior)e(algebra)p Fp(.)k(\(See)22 b([1)q(])g(Section)g(7.2.\))1083 2103 y(24)g(Jan)n(uary)e(2002)376 2237 y(\(42\))27 b Fq(Di\013eren)n(tial)f (forms)c Fp(and)f(the)h(exterior)g(deriv)l(ativ)n(e.)1083 2371 y(28)g(Jan)n(uary)e(2002)376 2504 y(\(43\))27 b Fq(Closed)19 b Fp(and)f Fq(exact)h Fm(p)p Fp(-froms;)g(the)g Fq(de)j(Rham)f(cohomology)h(alge-)521 2583 y(bra)p Fp(.)376 2662 y(\(44\))27 b(Existence)21 b(and)e(uniqueness)g(of)h(the)g Fq(in)n(tegral)g Fp(of)g Fm(n)p Fp(-forms)f(with)g(com-)521 2740 y(pact)j(supp)r(ort)e(on)i(orien)n(ted)g Fm(n)p Fp(-manifolds.)29 b(\(See)22 b([1])h(Section)f(8.2.\))1083 2874 y(31)g(Jan)n(uary)e(2002)376 3008 y(\(45\))27 b Fq(Stok)n(es's)e(Theorem)d Fp(for)f(orien)n(ted)h(manifolds)g(with)f(b) r(oundary:)1078 3058 y Fd(Z)1116 3210 y Fn(M)1180 3149 y Fm(d!)h Fp(=)1347 3058 y Fd(Z)1384 3210 y Fn(@)s(M)1477 3149 y Fm(!)i(:)521 3294 y Fp(\(See)e([1)q(])g(Section)g(8.2.\))p eop %%Page: 6 6 6 5 bop 308 159 a Ff(6)1085 294 y Fp(4)22 b(F)-6 b(ebruary)22 b(2002)376 428 y(\(46\))27 b(The)20 b Fq(P)n(oincar)n(\023)-36 b(e)22 b(Lemma)p Fp(.)29 b(It)19 b(implies)i(the)f(homotop)n(y-in)n(v)l (ariance)f(of)521 507 y(de)j(Rham)e(cohomology)-6 b(.)30 b(\(See)22 b([1])h(Section)f(8.3.\))376 585 y(\(47\))27 b(The)22 b(de)f(Rham)f(cohomology)h(of)h Fm(S)1420 561 y Fn(n)1451 585 y Fp(.)30 b(Application)22 b(to)f(the)h Fq(Brou)n(w)n(er)521 664 y(\014xed)j(p)r(oin)n(t)h(theorem)c Fp(in)g(dimension)f Fm(n)16 b Fp(+)f(1.)1085 798 y(7)22 b(F)-6 b(ebruary)22 b(2002)376 931 y(\(48\))27 b(Con)n(tractions)33 b(and)g Fq(Lie)38 b(deriv)l(ativ)n(es)33 b Fp(of)h(di\013eren)n(tial)g (forms;)39 b(the)521 1010 y(Cartan)18 b(form)n(ula)h Fm(L)1027 1020 y Fn(X)1091 1010 y Fp(=)g Fm(i)1184 1020 y Fn(X)1230 1010 y Fm(d)10 b Fp(+)g Fm(di)1392 1020 y Fn(X)1438 1010 y Fp(.)29 b Fq(Div)n(ergence)18 b Fp(of)i(a)g(v)n(ector) f(\014eld.)521 1089 y(\(See)j([2)q(])g(Section)g(2.24.\))376 1167 y(\(49\))27 b(The)21 b Fq(Hopf)j(in)n(v)l(arian)n(t)d Fp(for)g(maps)e Fm(S)1471 1143 y Fk(2)p Fn(n)p Fj(\000)p Fk(1)1606 1167 y Fl(!)g Fm(S)1737 1143 y Fn(n)1769 1167 y Fp(,)i(and)f(its)h(generalisa-)521 1246 y(tion)h(to)g(div)n (ergence-free)h(v)n(ector)f(\014elds)g(on)f Fm(S)1690 1221 y Fk(3)1717 1246 y Fp(.)1123 1448 y Fc(References)336 1555 y Fb(1.)28 b(L.)16 b(Conlon,)g Fa(Di\013eren)n(tiable)i(Manifolds) h(|)d(A)f(First)h(Course)p Fb(,)g(Birkh\177)-28 b(auser)17 b(V)-5 b(erlag)17 b(1993.)336 1623 y(2.)28 b(F.)35 b(W.)g(W)-5 b(arner,)39 b Fa(F)-5 b(oundations)37 b(of)f(Di\013eren)n(tiable)h (Manifolds)h(and)e(Lie)g(Groups)p Fb(,)408 1690 y(Springer)20 b(V)-5 b(erlag)20 b(1983.)336 1757 y(3.)28 b(W.)15 b(P)-5 b(.)15 b(Th)n(urston,)i Fa(Three-Dimensional)g(Geometry)d(and)i(T)-5 b(op)r(ology)p Fb(,)16 b(V)-5 b(ol.)16 b(1,)g(Princeton)408 1825 y(Univ)n(ersit)n(y)21 b(Press)e(1997.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF