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Fu(suc)n(h)h(that)e(at)i(the)f (same)g(time)g(the)h(corresp)r(onding)521 972 y Fr(V)559 982 y Fs(x)586 989 y Fm(j)633 972 y Fu(are)d(con)n(tained)g(in)g(the)g (op)r(en)g(set)g Fr(G)1548 982 y Fs(i)p Fp(+1)1643 972 y Fq(n)p 1692 918 133 3 v 15 w Fr(G)1744 982 y Fs(i)p Fo(\000)p Fp(2)1824 972 y Fu(.)30 b(T)-6 b(aking)22 b(all)h(these)521 1051 y Fr(V)559 1061 y Fs(x)586 1068 y Fm(j)634 1051 y Fu(as)f Fr(i)g Fu(ranges)g(o)n(v)n(er)h(the)f(p)r(ositiv)n(e)h(in)n (tegers)f(w)n(e)h(obtain)f(the)g(desired)521 1130 y(atlas.)1603 b Fj(\003)1082 1332 y Fu(27)22 b(Octob)r(er)g(2003)409 1488 y(\(8\))27 b(Ev)n(ery)17 b(op)r(en)f(co)n(v)n(ering)i(of)f(a)f (smo)r(oth)g(manifold)g(admits)f(a)i(sub)r(ordinate)521 1567 y(smo)r(oth)24 b Fv(partition)31 b(of)f(unit)n(y)p Fu(.)43 b(This)26 b(follo)n(ws)g(from)f(paracompact-)521 1645 y(ness)d(and)g(the)f(existence)i(of)f(smo)r(oth)f(cuto\013)g (functions.)409 1724 y(\(9\))27 b Fv(V)-6 b(ector)26 b(bundles)p Fu(;)d(the)f Fv(tangen)n(t)k(bundle)d Fu(of)f(a)h(smo)r (oth)d(manifold,)521 1802 y(see)j([1])g(Section)f(3.3.)1082 1958 y(30)g(Octob)r(er)g(2003)376 2114 y(\(10\))27 b Fv(Deriv)l(ativ)n(es)37 b(of)h(smo)r(oth)g(maps)33 b Fu(b)r(et)n(w)n(een)g(manifolds;)38 b(tangen)n(t)521 2192 y(v)n(ectors)22 b(as)g(deriv)l(ativ)n(es)h(of)f(curv)n(es.)376 2271 y(\(11\))27 b(In)n(tegration)h(of)h Fv(v)n(ector)k(\014elds)p Fu(;)g Fv(lo)r(cal)c Fu(and)f Fv(global)33 b(\015o)n(ws)p Fu(;)g Fv(com-)521 2350 y(pleteness)p Fu(,)22 b(see)h([1])f(Section)g (4.1.)1070 2505 y(3)g(No)n(v)n(em)n(b)r(er)g(2003)376 2661 y(\(12\))27 b(Ev)n(ery)22 b(v)n(ector)g(\014eld)g(with)f(compact)g (supp)r(ort)g(is)h(complete.)376 2739 y(\(13\))27 b(V)-6 b(ector)23 b(\014elds)f(as)g Fv(deriv)l(ations)g Fu(and)g(the)f Fv(Lie)k(deriv)l(ativ)n(e)p Fu(.)1070 2895 y(6)d(No)n(v)n(em)n(b)r(er)g (2003)376 3050 y(\(14\))27 b Fv(Comm)n(utators)g Fu(and)d(the)i(Lie)g (algebra)g(structure)e(on)h(the)h(space)g(of)521 3129 y(v)n(ector)c(\014elds,)h(see)f([1)q(])g(Section)g(4.2.)1054 3285 y(10)g(No)n(v)n(em)n(b)r(er)f(2003)376 3440 y(\(15\))27 b(The)18 b Fv(tensor)i(algebra)d Fu(and)h(the)f Fv(exterior)j(algebra)e Fu(of)g(a)f(real)i(v)n(ector)521 3519 y(space.)30 b(\(See)22 b([1)q(])g(Section)g(7.2.\).)p eop %%Page: 3 3 3 2 bop 2305 159 a Fk(3)376 294 y Fu(\(16\))27 b(Multilinear)c(algebra) f(of)g(v)n(ector)g(bundles)g(and)f Fv(di\013eren)n(tial)k(forms)p Fu(.)1054 428 y(13)d(No)n(v)n(em)n(b)r(er)f(2003)376 562 y(\(17\))27 b(The)22 b Fv(exterior)j(deriv)l(ativ)n(e)p Fu(.)376 640 y(\(18\))i Fv(Pullbac)n(k)33 b Fu(of)g(forms,)j(con)n (traction)c(and)h(the)g Fv(Lie)k(deriv)l(ativ)n(e)c Fu(on)521 719 y(forms,)21 b(Cartan's)g(form)n(ula.)1054 853 y(17)h(No)n(v)n(em)n (b)r(er)f(2003)376 986 y(\(19\))27 b Fv(Closed)19 b Fu(and)f Fv(exact)h Fr(p)p Fu(-froms;)g(the)g Fv(de)j(Rham)f(cohomology)h(alge-) 521 1065 y(bra)p Fu(.)376 1144 y(\(20\))27 b(Existence)21 b(and)e(uniqueness)g(of)h(the)g Fv(in)n(tegral)g Fu(of)g Fr(n)p Fu(-forms)f(with)g(com-)521 1222 y(pact)j(supp)r(ort)e(on)i (orien)n(ted)g Fr(n)p Fu(-manifolds.)29 b(\(See)22 b([1])h(Section)f (8.2.\))1054 1356 y(20)g(No)n(v)n(em)n(b)r(er)f(2003)589 1490 y Fv(No)i(lecture)g(due)g(to)h(the)f(demonstration)h(against)g (funding)521 1568 y(cuts)i(in)f(univ)n(ersities.)1054 1702 y Fu(24)d(No)n(v)n(em)n(b)r(er)f(2003)376 1836 y(\(21\))27 b Fv(Stok)n(es's)e(Theorem)d Fu(for)f(orien)n(ted)h(manifolds)g(with)f (b)r(oundary:)1078 1886 y Fi(Z)1116 2038 y Fs(M)1180 1977 y Fr(d!)h Fu(=)1347 1886 y Fi(Z)1384 2038 y Fs(@)s(M)1477 1977 y Fr(!)i(:)521 2122 y Fu(\(See)e([1)q(])g(Section)g(8.2.\))376 2201 y(\(22\))27 b(The)22 b Fv(P)n(oincar)n(\023)-36 b(e)24 b(lemma)p Fu(:)30 b(ev)n(ery)22 b(closed)h(form)d(is)j(lo)r (cally)g(exact.)1054 2334 y(27)f(No)n(v)n(em)n(b)r(er)f(2003)376 2468 y(\(23\))27 b(The)22 b(de)g(Rham)e(cohomology)h(of)h Fr(S)1421 2444 y Fs(n)1453 2468 y Fu(.)1072 2602 y(1)g(Decem)n(b)r(er)h (2003)376 2735 y(\(24\))k(F)-6 b(or)15 b(an)n(y)f(closed)i(orien)n(ted) e Fr(n)p Fu(-dimensional)h(manifold)f Fr(M)21 b Fu(without)14 b(b)r(ound-)521 2814 y(ary)-6 b(,)22 b(w)n(e)h(ha)n(v)n(e)f Fr(H)959 2790 y Fs(n)954 2832 y(dR)1017 2814 y Fu(\()p Fr(M)7 b Fu(\))18 b(=)h Ft(R)p Fu(.)376 2893 y(\(25\))27 b(Using)16 b(the)e(ab)r(o)n(v)n(e,)j(w)n(e)e(pro)n(v)n(e)g(Moser's)h (result)e(on)h Fv(isotop)n(y)j(of)f(v)n(olume)521 2971 y(forms)p Fu(:)29 b(if)20 b Fr(M)26 b Fu(is)20 b(closed)g(and)f(orien)n (ted,)h(then)f(an)n(y)g(t)n(w)n(o)g(v)n(olume)g(forms)521 3050 y(with)j(the)f(same)h(total)g(v)n(olume)f(are)h(di\013eomorphic)f (to)h(eac)n(h)h(other)e(b)n(y)521 3129 y(a)h(di\013eomorphism)d (isotopic)k(to)e(the)h(iden)n(tit)n(y)-6 b(.)p eop %%Page: 4 4 4 3 bop 308 159 a Fk(4)1072 294 y Fu(4)22 b(Decem)n(b)r(er)h(2003)376 448 y(\(26\))k(W)-6 b(e)35 b(no)n(w)e(b)r(egin)h(the)f(discussion)h(of) g Fv(connections)g Fu(or)f Fv(co)n(v)l(arian)n(t)521 527 y(deriv)l(ativ)n(es)22 b Fu(on)f(v)n(ector)h(bundles.)589 605 y(Let)30 b Fr(E)37 b Fq(!)c Fr(M)38 b Fu(b)r(e)30 b(a)g(smo)r(oth)f(v)n(ector)h(bundle)g(of)h(rank)e Fr(k)k Fu(o)n(v)n(er)e(a)521 684 y(smo)r(oth)20 b(manifold)i Fr(M)28 b Fu(of)23 b(dimension)e Fr(n)p Fu(.)521 781 y Fv(De\014nition)26 b(1.)i Fu(A)23 b(connection)f(on)f Fr(E)26 b Fu(is)d(an)e Ft(R)p Fu(-linear)j(map)308 897 y(\(1\))639 b Fq(r)7 b Fu(:)24 b(\000\()p Fr(E)t Fu(\))18 b Fq(\000)-11 b(!)19 b Fu(\012)1476 870 y Fp(1)1503 897 y Fu(\()p Fr(E)t Fu(\))521 1014 y(satisfying)j(the)g(Leibniz)h(rule)308 1130 y(\(2\))575 b Fq(r)p Fu(\()p Fr(f)7 b(s)p Fu(\))18 b(=)h Fr(d)-11 b(f)22 b Fq(\012)15 b Fr(s)g Fu(+)h Fr(f)7 b Fq(r)p Fu(\()p Fr(s)p Fu(\))521 1246 y(for)22 b(all)h Fr(f)i Fq(2)19 b Fr(C)888 1221 y Fo(1)938 1246 y Fu(\()p Fr(M)7 b Fu(\))22 b(and)f Fr(s)e Fq(2)g Fu(\000\()p Fr(E)t Fu(\).)589 1343 y(Here)f(\012)788 1319 y Fp(1)814 1343 y Fu(\()p Fr(E)t Fu(\))g(=)h(\000\()p Fr(T)1122 1319 y Fo(\003)1149 1343 y Fr(M)13 b Fq(\012)6 b Fr(E)t Fu(\))17 b(is)h(the)f(space)h(of)g(1-forms)f(on)g Fr(M)24 b Fu(with)521 1422 y(v)l(alues)i(in)f Fr(E)t Fu(.)39 b(One)26 b(can)f(ev)l(aluate)g (the)g(1-form)f(on)h(a)g(v)n(ector)g(\014eld)h Fr(X)521 1501 y Fu(to)c(obtain)308 1617 y(\(3\))506 b Fq(r)955 1627 y Fs(X)1001 1617 y Fu(\()p Fr(s)p Fu(\))18 b(:=)i Fq(hr)p Fu(\()p Fr(s)p Fu(\))p Fr(;)11 b(X)5 b Fq(i)19 b(2)g Fu(\000\()p Fr(E)t Fu(\))i Fr(:)521 1733 y Fu(The)32 b(v)l(alue)i(of)f Fq(r)991 1743 y Fs(X)1036 1733 y Fu(\()p Fr(s)p Fu(\))f(at)h(a)f(p)r(oin)n(t)g Fr(p)37 b Fq(2)g Fr(M)j Fu(dep)r(ends)32 b(on)g Fr(X)38 b Fu(only)521 1812 y(through)15 b Fr(X)5 b Fu(\()p Fr(p)p Fu(\))19 b Fq(2)f Fr(T)1031 1822 y Fs(p)1058 1812 y Fr(M)7 b Fu(.)28 b(The)16 b(dep)r(endence)h(on)f Fr(s)h Fu(is)h(more)d(in)n(teresting.) 521 1909 y Fv(Lemma)20 b(2.)25 b Fn(If)20 b Fr(s)g Fn(vanishes)g(in)g (some)g(op)m(en)f(neighb)m(ourho)m(o)m(d)f(of)i Fr(p)p Fn(,)h(then)521 1988 y(so)i(do)m(es)g Fq(r)p Fu(\()p Fr(s)p Fu(\))p Fn(.)589 2085 y Fu(This)c(sho)n(ws)g(that)f Fq(r)i Fu(do)r(es)f(not)g(increase)h(supp)r(orts.)27 b(It)19 b(is)h(therefore)521 2164 y(a)25 b(lo)r(cal)g(di\013eren)n (tial)g(op)r(erator,)f(whic)n(h)h(can)g(b)r(e)f(describ)r(ed)h(in)f(a)h (lo)r(cal)521 2242 y(trivialisation)j(of)f Fr(E)32 b Fu(b)n(y)27 b(partial)g(deriv)l(ativ)n(es)h(with)e(resp)r(ect)i(to)f (lo)r(cal)521 2321 y(co)r(ordinates)22 b(in)g Fr(M)7 b Fu(.)521 2418 y Fv(Lemma)25 b(3.)50 b Fu(\(a\))27 b Fn(If)22 b Fq(r)1134 2428 y Fp(0)1181 2418 y Fn(and)f Fq(r)1363 2428 y Fp(1)1410 2418 y Fn(ar)m(e)g(c)m(onne)m(ctions)f(on)h Fr(E)t Fn(,)g(then)g(so)g(is)656 2497 y Fu(\(1)14 b Fq(\000)i Fr(t)p Fu(\))p Fq(r)903 2507 y Fp(0)944 2497 y Fu(+)f Fr(t)p Fq(r)1090 2507 y Fp(1)1141 2497 y Fn(for)23 b(al)s(l)g Fr(t)c Fq(2)g Ft(R)p Fn(.)539 2576 y Fu(\(b\))28 b Fn(The)21 b(di\013er)m(enc)m(e)i Fq(r)1140 2586 y Fp(0)1177 2576 y Fq(\000)10 b(r)1295 2586 y Fp(1)1344 2576 y Fn(is)21 b(line)m(ar)g(with)h(r)m(esp)m(e)m(ct)e(to)h(multiplic)m(a-)656 2654 y(tion)d(of)h(se)m(ctions)g(by)g(arbitr)m(ary)f(smo)m(oth)f (functions,)k(and)d(is)h(ther)m(e-)656 2733 y(for)m(e)32 b(a)h Fu(1)p Fn(-form)f(with)g(values)i(in)f Fr(E)t(nd)p Fu(\()p Fr(E)t Fu(\))p Fn(,)i(i.)e(e.)g(a)g(se)m(ction)g(of)656 2812 y Fr(T)704 2787 y Fo(\003)730 2812 y Fr(M)22 b Fq(\012)15 b Fr(E)t(nd)p Fu(\()p Fr(E)t Fu(\))k(=)g Fr(T)1251 2787 y Fo(\003)1277 2812 y Fr(M)j Fq(\012)16 b Fr(E)1484 2787 y Fo(\003)1525 2812 y Fq(\012)g Fr(E)t Fn(.)521 2909 y Fv(De\014nition)23 b(4.)k Fu(A)20 b(\(lo)r(cal\))g(frame)e(for)i Fr(E)j Fu(is)d(a)g(set)g(of)f(smo)r(oth)f(sections)521 2988 y Fr(s)552 2998 y Fp(1)579 2988 y Fr(;)11 b(:)g(:)g(:)i(;)e(s)757 2998 y Fs(k)808 2988 y Fu(de\014ned)21 b(o)n(v)n(er)g(some)g(op)r(en)f (set)i Fr(U)j Fq(\032)19 b Fr(M)7 b Fu(,)22 b(whose)f(v)l(alues)h(are) 521 3066 y(linearly)h(indep)r(enden)n(t)e(at)g(ev)n(ery)i(p)r(oin)n(t)e Fr(p)e Fq(2)f Fr(U)7 b Fu(.)589 3164 y(Th)n(us)22 b(a)h(set)h(of)f Fr(k)j Fu(lo)r(cal)e(smo)r(oth)d(sections)j Fr(s)1724 3174 y Fp(1)1751 3164 y Fr(;)11 b(:)g(:)g(:)i(;)e(s)1929 3174 y Fs(k)1982 3164 y Fu(is)24 b(a)f(frame)f(if)521 3242 y(and)i(only)g(if)h Fr(s)893 3252 y Fp(1)920 3242 y Fu(\()p Fr(p)p Fu(\))p Fr(;)11 b(:)g(:)g(:)i(;)e(s)1183 3252 y Fs(k)1212 3242 y Fu(\()p Fr(p)p Fu(\))24 b(is)h(a)g(basis)f(of)h Fr(E)1739 3252 y Fs(p)1788 3242 y Fu(=)f Fr(\031)1902 3218 y Fo(\000)p Fp(1)1966 3242 y Fu(\()p Fr(p)p Fu(\))g(for)h(ev)n (ery)521 3321 y Fr(p)19 b Fq(2)g Fr(U)7 b Fu(.)28 b(Therefore)19 b(a)f(frame)g(de\014ned)g(o)n(v)n(er)h Fr(U)26 b Fu(de\014nes)18 b(a)h(trivialization)521 3400 y(of)j Fr(E)t Fq(j)668 3410 y Fs(U)708 3400 y Fu(,)g(and,)g(con)n(v)n(ersely)-6 b(,)23 b(ev)n(ery)g(suc)n(h)e(trivialization)308 3519 y(\(4\))597 b Fr( )10 b Fu(:)23 b Fr(\031)1123 3491 y Fo(\000)p Fp(1)1188 3519 y Fu(\()p Fr(U)7 b Fu(\))18 b Fq(\000)-11 b(!)19 b Fr(U)j Fq(\002)15 b Ft(R)1618 3491 y Fs(k)p eop %%Page: 5 5 5 4 bop 2305 159 a Fk(5)521 294 y Fu(de\014nes)15 b(a)f(lo)r(cal)i (frame)e(b)n(y)g(setting)g Fr(s)1429 304 y Fs(i)1448 294 y Fu(\()p Fr(p)p Fu(\))19 b(=)g Fr( )1667 270 y Fo(\000)p Fp(1)1731 294 y Fu(\()p Fr(p;)11 b(e)1850 304 y Fs(i)1869 294 y Fu(\),)16 b(where)e Fr(e)2143 304 y Fp(1)2170 294 y Fr(;)d(:)g(:)g(:)i(;)e(e)2348 304 y Fs(k)521 373 y Fu(is)23 b(the)e(standard)g(basis)h(of)g Ft(R)1261 348 y Fs(k)1291 373 y Fu(.)589 451 y(Using)d(a)g(partition)g(of)g(unit)n(y) g(sub)r(ordinate)e(to)i(an)g(op)r(en)g(co)n(v)n(er)g(of)h Fr(M)521 530 y Fu(b)n(y)g(op)r(en)f(sets)h(o)n(v)n(er)g(whic)n(h)g Fr(E)k Fu(is)c(trivial,)h(w)n(e)f(can)g(pro)n(v)n(e)f(the)h(existence) 521 609 y(part)h(of)h(the)g(follo)n(wing:)521 704 y Fv(Prop)r(osition) 40 b(5.)34 b Fn(Every)h(ve)m(ctor)g(bund)s(le)h Fr(E)i Fn(admits)c(c)m(onne)m(ctions.)521 783 y(The)e(sp)m(ac)m(e)e(of)i(al)s (l)f(c)m(onne)m(ctions)g(on)g Fr(E)36 b Fn(is)31 b(an)g(a\016ne)h(sp)m (ac)m(e)f(for)g(the)521 861 y(sp)m(ac)m(e)23 b Fu(\012)741 837 y Fp(1)767 861 y Fu(\()p Fr(E)t(nd)p Fu(\()p Fr(E)t Fu(\)\))18 b(=)h(\000\()p Fr(T)1253 837 y Fo(\003)1279 861 y Fr(M)j Fq(\012)15 b Fr(E)1485 837 y Fo(\003)1526 861 y Fq(\012)f Fr(E)t Fu(\))23 b Fn(of)h Fu(1)p Fn(-forms)e(with)h (values)521 940 y(in)h Fr(E)t(nd)p Fu(\()p Fr(E)t Fu(\))p Fn(.)589 1035 y Fu(The)g(second)i(part)e(of)h(the)g(Prop)r(osition)g (follo)n(ws)h(from)e(the)h(second)521 1114 y(part)c(of)h(Lemma)f(3.)376 1192 y(\(27\))27 b(W)-6 b(e)24 b(no)n(w)e(extend)g(the)h(di\013eren)n (tial)g(op)r(erator)e(giv)n(en)i(b)n(y)g(a)g(connection)521 1271 y Fq(r)f Fu(to)g(bundle-v)l(alued)g(forms)e(of)j(higher)e(degree.) 521 1366 y Fv(Lemma)37 b(6.)d Fn(F)-5 b(or)33 b(every)h(c)m(onne)m (ction)f Fq(r)h Fn(on)f Fr(E)41 b Fq(!)c Fr(M)j Fn(ther)m(e)33 b(is)h(a)521 1445 y(unique)25 b Ft(R)p Fn(-line)m(ar)g(map)1149 1428 y Fu(\026)1137 1445 y Fq(r)8 b Fu(:)22 b(\012)1289 1420 y Fs(k)1318 1445 y Fu(\()p Fr(E)t Fu(\))c Fq(\000)-11 b(!)19 b Fu(\012)1616 1420 y Fs(k)q Fp(+1)1706 1445 y Fu(\()p Fr(E)t Fu(\))k Fn(which)g(satis\014es)308 1560 y Fu(\(5\))807 1543 y(\026)795 1560 y Fq(r)p Fu(\()p Fr(!)18 b Fq(\012)d Fr(s)p Fu(\))j(=)i Fr(d!)d Fq(\012)f Fr(s)f Fu(+)g(\()p Fq(\000)p Fu(1\))1558 1532 y Fs(k)1587 1560 y Fr(!)i Fq(^)e(r)p Fu(\()p Fr(s)p Fu(\))521 1671 y Fn(for)23 b(al)s(l)h Fr(!)d Fq(2)e Fu(\012)895 1647 y Fs(k)923 1671 y Fu(\()p Fr(M)7 b Fu(\))23 b Fn(and)g Fr(s)c Fq(2)g Fu(\000\()p Fr(E)t Fu(\))p Fn(.)30 b(This)23 b(op)m(er)m(ator)f(satis\014es)308 1783 y Fu(\(6\))722 1766 y(\026)710 1783 y Fq(r)q Fu(\()p Fr(f)7 b Fu(\()p Fr(!)16 b Fq(\012)g Fr(s)p Fu(\)\))i(=)h(\()p Fr(d)-11 b(f)21 b Fq(^)15 b Fr(!)r Fu(\))h Fq(\012)f Fr(s)g Fu(+)g Fr(f)1634 1766 y Fu(\026)1622 1783 y Fq(r)p Fu(\()p Fr(!)i Fq(\012)f Fr(s)p Fu(\))23 b Fr(:)1072 1961 y Fu(8)f(Decem)n(b)r(er)h (2003)376 2106 y(\(28\))k(Considering)d(the)h(op)r(erator)1284 2089 y(\026)1272 2106 y Fq(r)17 b(\016)g(r)7 b Fu(:)24 b(\012)1549 2082 y Fp(0)1576 2106 y Fu(\()p Fr(E)t Fu(\))e Fq(\000)-11 b(!)24 b Fu(\012)1883 2082 y Fp(2)1910 2106 y Fu(\()p Fr(E)t Fu(\))f(asso)r(ciated)521 2185 y(with)f(a)g (connection)g Fq(r)g Fu(on)g Fr(E)t Fu(.)30 b(It)21 b(turns)g(out)h (that)f(this)h(is)g(linear)h(o)n(v)n(er)521 2263 y Fr(C)573 2239 y Fo(1)624 2263 y Fu(\()p Fr(M)7 b Fu(\),)15 b(and)g(is)g (therefore)f(giv)n(en)h(b)n(y)f(an)h(elemen)n(t)f Fr(F)1852 2239 y Fo(r)1914 2263 y Fq(2)k Fu(\012)2025 2239 y Fp(2)2052 2263 y Fu(\()p Fr(E)t(nd)p Fu(\()p Fr(E)t Fu(\)\).)521 2342 y(This)k(is)g(called)h(the)f Fv(curv)l(ature)g Fu(of)g Fq(r)p Fu(.)376 2421 y(\(29\))27 b(Fix)22 b(a)g(lo)r(cal)g(frame)f Fr(s)1065 2431 y Fp(1)1092 2421 y Fr(;)11 b(:)g(:)g(:)i(;)e(s)1270 2431 y Fs(k)1321 2421 y Fu(for)22 b(the)f(restriction)g(of)h Fr(E)k Fu(to)21 b(a)h(trivial-)521 2499 y(ising)k(op)r(en)f(set)g(in)g Fr(M)7 b Fu(.)40 b(With)25 b(resp)r(ect)g(to)g(this)g(frame)f(a)i (connection)521 2578 y Fq(r)f Fu(can)g(b)r(e)f(expressed)h(through)e(a) i(matrix)e Fr(!)j Fu(=)d(\()p Fr(!)1852 2588 y Fs(ij)1893 2578 y Fu(\))i(of)f(1-forms)g(b)n(y)521 2657 y(writing)1037 2803 y Fq(r)p Fr(s)1124 2813 y Fs(i)1162 2803 y Fu(=)1268 2719 y Fs(k)1232 2739 y Fi(X)1240 2881 y Fs(j)s Fp(=1)1341 2803 y Fr(!)1382 2813 y Fs(ij)1438 2803 y Fq(\012)16 b Fr(s)1537 2813 y Fs(j)1583 2803 y Fr(:)521 2968 y Fu(Then)21 b(a)i(calculation)f(giv)n(es)1035 3143 y Fr(F)1087 3115 y Fo(r)1130 3143 y Fr(s)1161 3153 y Fs(i)1198 3143 y Fu(=)1304 3058 y Fs(k)1269 3078 y Fi(X)1276 3221 y Fs(j)s Fp(=1)1377 3143 y Fu(\012)1425 3153 y Fs(ij)1481 3143 y Fq(\012)15 b Fr(s)1579 3153 y Fs(j)521 3322 y Fu(with)942 3455 y(\012)990 3465 y Fs(ij)1049 3455 y Fu(=)k Fr(d!)1194 3465 y Fs(ij)1250 3455 y Fq(\000)1354 3371 y Fs(k)1318 3391 y Fi(X)1329 3535 y Fs(l)q Fp(=1)1427 3455 y Fr(!)1468 3465 y Fs(il)1516 3455 y Fq(^)c Fr(!)1617 3465 y Fs(l)q(j)1679 3455 y Fr(:)p eop %%Page: 6 6 6 5 bop 308 159 a Fk(6)521 294 y Fu(W)-6 b(e)24 b(can)f(write)g(this)f (brie\015y)h(as)g(\012)d(=)h Fr(d!)d Fq(\000)e Fr(!)i Fq(^)e Fr(!)r Fu(,)24 b(where)f(the)g(w)n(edge)521 373 y(pro)r(duct)g(on)g(the)h(righ)n(t-hand-side)f(includes)h(matrix)f(m)n (ultiplication,)521 451 y(and)e(is)i(therefore)f(not)f(trivial)h (unless)g Fr(k)g Fu(=)d(1.)589 530 y(Similarly)c(w)n(e)h(compute)e Fr(d)p Fu(\012)19 b(=)g Fr(!)5 b Fq(^)r Fu(\012)r Fq(\000)r Fu(\012)r Fq(^)r Fr(!)r Fu(.)29 b(This)15 b(is)h(the)g Fv(Bianc)n(hi)521 609 y(iden)n(tit)n(y)p Fu(.)589 687 y(Finally)-6 b(,)18 b(supp)r(ose)d(w)n(e)g(ha)n(v)n(e)h(another)f (frame)g Fr(s)1745 663 y Fo(0)1745 704 y Fp(1)1771 687 y Fr(;)c(:)g(:)g(:)j(;)d(s)1950 663 y Fo(0)1950 705 y Fs(k)1995 687 y Fu(on)k(the)g(same)521 766 y(domain)i(of)h (de\014nition)f(as)h(the)g(original)g(frame.)28 b(Let)18 b Fr(!)1900 742 y Fo(0)1934 766 y Fu(and)g(\012)2107 742 y Fo(0)2140 766 y Fu(denote)521 845 y(the)h(connection)h(and)f (curv)l(ature)g(matrices)g(with)g(resp)r(ect)g(to)h(this)f(new)521 923 y(frame.)29 b(If)1111 1056 y Fr(s)1142 1028 y Fo(0)1142 1072 y Fs(i)1180 1056 y Fu(=)1286 972 y Fs(k)1250 992 y Fi(X)1260 1134 y Fs(i)p Fp(=1)1359 1056 y Fr(g)1391 1066 y Fs(ij)1431 1056 y Fr(s)1462 1066 y Fs(j)1509 1056 y Fr(;)521 1220 y Fu(w)n(e)22 b(\014nd)e(the)h(follo)n(wing)i (relations)1387 1193 y Fp(1)1410 1220 y Fu(:)30 b Fr(!)1501 1195 y Fo(0)1536 1220 y Fu(=)19 b Fr(dg)24 b(g)1730 1195 y Fo(\000)p Fp(1)1807 1220 y Fu(+)15 b Fr(g)r(!)r(g)1984 1195 y Fo(\000)p Fp(1)2069 1220 y Fu(and)21 b(\012)2245 1195 y Fo(0)2279 1220 y Fu(=)521 1298 y Fr(g)r Fu(\012)p Fr(g)637 1274 y Fo(\000)p Fp(1)701 1298 y Fu(,)h(where)g Fr(g)e Fu(=)g(\()p Fr(g)1113 1308 y Fs(ij)1153 1298 y Fu(\).)1056 1441 y(11)i(Decem)n(b)r(er)g(2003)376 1584 y(\(30\))27 b(If)22 b Fr(E)g Fq(!)d Fr(M)29 b Fu(is)21 b(a)h(v)n(ector)f(bundle)g(with)g(a)g(connection)h Fq(r)p Fu(,)g(w)n(e)f(sa)n(y)h(that)521 1663 y(a)28 b(section)g Fr(s)h Fq(2)f Fu(\000\()p Fr(E)t Fu(\))f(is)i Fv(parallel)e Fu(with)g(resp)r(ect)h(to)f Fq(r)h Fu(if)h Fq(r)p Fr(s)f Fu(=)h(0.)521 1741 y(In)k(the)f(sp)r(ecial)i(case)g(that)e Fq(r)h Fu(is)g(the)g(connection)f(giv)n(en)i(b)n(y)e(some)521 1820 y(trivialization,)d(a)e(section)g(is)g(parallel)g(if)h(and)e(only) g(if)i(it)e(is)h(constan)n(t)521 1898 y(in)33 b(the)f(giv)n(en)h (trivialization.)61 b(Th)n(us)32 b(parallel)h(sections)g(should)f(b)r (e)521 1977 y(though)n(t)f(of)h(as)g(the)f(analogs)h(of)h(constan)n(t)e (sections)h(for)g(non)n(trivial)521 2056 y(bundles.)521 2150 y Fv(Prop)r(osition)25 b(7.)j Fn(L)m(et)23 b Fr(\031)10 b Fu(:)23 b Fr(E)f Fq(!)e Fr(M)29 b Fn(b)m(e)24 b(a)e(ve)m(ctor)h(bund) s(le)h(with)f(a)f(c)m(on-)521 2229 y(ne)m(ction)g Fq(r)p Fn(,)h(and)e Fr(c)7 b Fu(:)24 b([0)p Fr(;)11 b Fu(1])20 b Fq(!)f Fr(M)29 b Fn(a)21 b(smo)m(oth)g(curve)i(in)f(the)f(b)m(ase)h (sp)m(ac)m(e.)521 2308 y(Then)29 b(for)g(every)h Fr(v)g Fq(2)f Fr(\031)1164 2283 y Fo(\000)p Fp(1)1228 2308 y Fu(\()p Fr(c)p Fu(\(0\)\))f Fn(ther)m(e)h(is)g(a)f(unique)j(smo)m(oth)c (curve)521 2386 y Fr(s)7 b Fu(:)26 b([0)p Fr(;)11 b Fu(1])29 b Fq(!)g Fr(E)k Fn(with)c Fr(\031)22 b Fq(\016)c Fr(s)29 b Fu(=)g Fr(c)p Fn(,)i Fr(s)p Fu(\(0\))d(=)h Fr(v)i Fn(and)e Fq(r)1903 2396 y Fp(_)-19 b Fs(c)1920 2386 y Fr(s)29 b Fu(=)g(0)p Fn(.)47 b(Mor)m(e-)521 2465 y(over,)21 b(the)e(map)f Fr(v)j Fq(7!)f Fr(s)p Fu(\(1\))e Fn(de\014nes)i(an)f(isomorphism)f(of)h (ve)m(ctor)g(sp)m(ac)m(es)521 2544 y Fr(\031)560 2519 y Fo(\000)p Fp(1)625 2544 y Fu(\()p Fr(c)p Fu(\(0\)\))f Fq(!)h Fr(\031)933 2519 y Fo(\000)p Fp(1)997 2544 y Fu(\()p Fr(c)p Fu(\(1\)\))p Fn(.)589 2638 y Fu(Here)j Fr(s)g Fu(can)f(b)r(e)h(though)n(t)e(of)i(as)g(a)f(section)h(of)g(the)g (pullbac)n(k)f(bundle)521 2717 y Fr(c)549 2693 y Fo(\003)576 2717 y Fr(E)28 b Fq(!)d Fu([0)p Fr(;)11 b Fu(1])26 b(that)e(is)i (parallel)g(with)e(resp)r(ect)h(to)g(the)g(giv)n(en)g(connec-)521 2796 y(tion.)k(Ov)n(er)22 b Fr(M)29 b Fu(the)21 b(condition)g Fq(r)1400 2806 y Fp(_)-19 b Fs(c)1418 2796 y Fr(s)19 b Fu(=)g(0)i(mak)n(es)g(sense)h(although)f Fr(s)g Fu(is)521 2874 y(not)f(a)g(connection)h(o)n(v)n(er)f(all)h(of)g Fr(M)27 b Fu(b)r(ecause)21 b(the)f(co)n(v)l(arian)n(t)g(deriv)l(ativ)n (e)521 2953 y(is)j(only)e(considered)h(in)g(the)g(direction)g(of)g Fr(c)p Fu(,)h(where)f Fr(s)g Fu(is)g(de\014ned.)589 3032 y(The)17 b(Prop)r(osition)h(follo)n(ws)h(from)e(the)h(existence)h(and)f (uniqueness)g(of)521 3110 y(the)g(solutions)h(of)f(systems)g(of)g (linear)h(ordinary)f(di\013eren)n(tial)g(equations)521 3189 y(with)29 b(giv)n(en)h(initial)g(conditions,)h(together)e(with)g (the)g(linear)h(dep)r(en-)521 3268 y(dence)23 b(of)f(the)f(solutions)h (on)g(the)f(initial)i(v)l(alues.)p 308 3317 338 3 v 376 3362 a Fp(1)400 3384 y Fh(In)15 b(ph)n(ysics)i Fg(g)g Fh(is)f(called)h(a)e(gauge)h(transformation,)h(and)e(these)h(form)n (ulae)g(sho)n(w)g(ho)n(w)g(con-)308 3451 y(nection)f(and)f(curv)m (ature)g(matrices)g(b)r(eha)n(v)n(e)g(under)h(gauge)f(transformations.) 24 b(The)14 b(curv)m(ature)308 3519 y(is)20 b(more)e(in)n(v)m(arian)n (t)j(than)e(the)g(connection.)10 b(.)f(.)p eop %%Page: 7 7 7 6 bop 2305 159 a Fk(7)521 294 y Fv(Corollary)20 b(8.)25 b Fn(Over)19 b(a)g(curve)h(every)g(ve)m(ctor)f(bund)s(le)h(with)f(c)m (onne)m(ction)521 373 y(admits)k(a)g(fr)m(aming)h(by)g(p)m(ar)m(al)s (lel)e(se)m(ctions.)376 469 y Fu(\(31\))27 b(The)14 b(last)h(corollary) g(fails)h(for)e(base)h(spaces)g(whic)n(h)f(are)h(not)f (one-dimensional,)521 547 y(and)31 b(this)h(leads)g(to)g(geometric)g (in)n(terpretations)e(of)j(the)e(curv)l(ature.)521 626 y(It)c(will)h(turn)e(out)h(that)f(the)h(corollary)g(enco)r(des)h(the)f (fact)g(that)g(on)g(a)521 705 y(one-manifold)21 b(there)g(is)g(no)g (curv)l(ature)g(\(as)g(ev)n(ery)h(t)n(w)n(o-form)d(v)l(anishes)521 783 y(iden)n(tically\).)589 862 y(Recall)25 b(that)f(with)g(resp)r(ect) h(to)f(a)h(framing)f Fr(s)1737 872 y Fp(1)1764 862 y Fr(;)11 b(:)g(:)g(:)i(;)e(s)1942 872 y Fs(k)1996 862 y Fu(of)25 b Fr(E)k Fu(a)c(con-)521 941 y(nection)32 b Fq(r)g Fu(is)f(expressed)h(b)n(y)f(the)h(matrix)e(\()p Fr(!)1747 951 y Fs(ij)1787 941 y Fu(\))i(of)f(one-forms.)58 b(If)521 1019 y(w)n(e)23 b(c)n(ho)r(ose)h(a)f(c)n(hart)g(in)g Fr(M)30 b Fu(with)22 b(lo)r(cal)i(co)r(ordinates)f Fr(y)1922 1029 y Fp(1)1949 1019 y Fr(;)11 b(:)g(:)g(:)i(;)e(y)2128 1029 y Fs(n)2161 1019 y Fu(,)23 b(then)521 1098 y(in)30 b(the)f(domain)f(of)h(this)g(c)n(hart)g(ev)n(ery)h(one-form)e(can)i(b)r (e)f(expressed)521 1177 y(uniquely)19 b(as)g(a)g(linear)g(com)n (bination)f(of)h(the)g Fr(dy)1718 1187 y Fs(i)1738 1177 y Fu(.)28 b(In)19 b(particular,)g(there)521 1255 y(are)j(smo)r(oth)e (functions)i Fr(!)1192 1231 y Fs(\013)1190 1272 y(ij)1253 1255 y Fu(on)g(the)f(domain)g(of)h(the)g(c)n(hart)f(suc)n(h)h(that)308 1431 y(\(7\))676 b Fr(!)1110 1441 y Fs(ij)1169 1431 y Fu(=)1274 1347 y Fs(n)1239 1367 y Fi(X)1242 1508 y Fs(\013)p Fp(=1)1348 1431 y Fr(!)1391 1403 y Fs(\013)1389 1448 y(ij)1430 1431 y Fr(dy)1496 1441 y Fs(\013)1552 1431 y Fr(:)521 1613 y Fu(Denoting)22 b(the)g(v)n(ector)g(\014elds)1321 1586 y Fs(@)p 1294 1597 82 3 v 1294 1636 a(@)s(y)1346 1642 y Fm(\013)1404 1613 y Fu(b)n(y)g Fr(@)1531 1623 y Fs(\013)1564 1613 y Fu(,)g(w)n(e)g(\014nd)f(the)h(follo)n(wing:)640 1807 y Fq(r)696 1817 y Fs(@)721 1823 y Fm(\013)754 1807 y Fr(s)785 1817 y Fs(i)823 1807 y Fu(=)d Fq(h)p Fr(@)954 1817 y Fs(\013)987 1807 y Fr(;)11 b Fq(r)p Fr(s)1103 1817 y Fs(i)1123 1807 y Fq(i)19 b Fu(=)1274 1723 y Fs(k)1238 1743 y Fi(X)1245 1885 y Fs(j)s Fp(=1)1336 1807 y Fq(h)p Fr(@)1397 1817 y Fs(\013)1430 1807 y Fr(;)11 b(!)1500 1817 y Fs(ij)1542 1807 y Fq(i)p Fr(s)1599 1817 y Fs(j)1642 1807 y Fu(=)1748 1723 y Fs(k)1713 1743 y Fi(X)1720 1885 y Fs(j)s Fp(=1)1821 1807 y Fr(!)1864 1779 y Fs(\013)1862 1824 y(ij)1903 1807 y Fr(s)1934 1817 y Fs(j)1981 1807 y Fr(:)521 1987 y Fu(More)23 b(generally)-6 b(,)23 b(if)1132 2163 y Fr(s)c Fu(=)1287 2078 y Fs(k)1252 2099 y Fi(X)1262 2241 y Fs(i)p Fp(=1)1360 2163 y Fr(x)1398 2173 y Fs(i)1417 2163 y Fr(s)1448 2173 y Fs(i)1489 2163 y Fr(;)521 2334 y Fu(then)857 2469 y Fq(r)913 2479 y Fs(@)938 2485 y Fm(\013)971 2469 y Fr(s)g Fu(=)1127 2385 y Fs(k)1091 2405 y Fi(X)1098 2547 y Fs(j)s Fp(=1)1188 2469 y Fu(\()1222 2423 y Fr(@)t(x)1299 2433 y Fs(j)p 1221 2453 105 3 v 1221 2515 a Fr(@)t(y)1292 2525 y Fs(\013)1347 2469 y Fu(+)1449 2385 y Fs(k)1413 2405 y Fi(X)1423 2547 y Fs(i)p Fp(=1)1522 2469 y Fr(x)1560 2479 y Fs(i)1578 2469 y Fr(!)1621 2441 y Fs(\013)1619 2486 y(ij)1660 2469 y Fu(\))p Fr(s)1717 2479 y Fs(j)1764 2469 y Fr(:)521 2639 y Fu(W)-6 b(riting)29 b Fr(A)819 2615 y Fs(\013)883 2639 y Fu(for)g(the)g(matrix)e(\()p Fr(!)1402 2615 y Fs(\013)1400 2656 y(ij)1441 2639 y Fu(\))i(of)h (functions)e(w)n(e)i(see)g(that)e(the)521 2718 y(op)r(erator)23 b Fq(r)845 2728 y Fs(@)870 2734 y Fm(\013)903 2718 y Fu(,)h(whic)n(h)g(w)n(e)g(abbreviate)f(to)g Fq(r)1699 2728 y Fs(\013)1733 2718 y Fu(,)i(has)e(the)g(form)g Fq(r)2224 2728 y Fs(\013)2279 2718 y Fu(=)521 2796 y Fr(@)556 2806 y Fs(\013)604 2796 y Fu(+)16 b Fr(A)720 2772 y Fs(\013)754 2796 y Fu(.)589 2875 y(W)-6 b(e)21 b(can)f(no)n(w)g(c)n(haracterize)h(\015atness)f(of)h Fq(r)f Fu(through)f(the)h(condition)521 2954 y(that)h(the)h (directional)g(deriv)l(ativ)n(es)h Fq(r)1489 2964 y Fs(\013)1544 2954 y Fu(comm)n(ute:)521 3050 y Fv(Prop)r(osition)17 b(9.)23 b Fn(The)17 b(c)m(onne)m(ction)g Fq(r)g Fn(is)g(\015at)f(if)i (and)e(only)g(if)i Fu([)p Fq(r)2149 3060 y Fs(\013)2183 3050 y Fr(;)11 b Fq(r)2268 3060 y Fs(\014)2300 3050 y Fu(])19 b(=)521 3128 y(0)24 b Fn(for)g(every)h(lo)m(c)m(al)e(c)m(o)m (or)m(dinate)f(system)j Fr(y)1573 3138 y Fp(1)1599 3128 y Fr(;)11 b(:)g(:)g(:)j(;)d(y)1779 3138 y Fs(n)1835 3128 y Fn(on)24 b(the)g(b)m(ase)g(man-)521 3207 y(ifold)f Fr(M)7 b Fn(.)1056 3371 y Fu(15)22 b(Decem)n(b)r(er)g(2003)376 3519 y(\(32\))27 b(W)-6 b(e)23 b(no)n(w)e(pro)n(v)n(e:)p eop %%Page: 8 8 8 7 bop 308 159 a Fk(8)521 294 y Fv(Theorem)36 b(10.)d Fn(A)g(ve)m(ctor)g(bund)s(le)g Fr(E)40 b Fq(!)c Fr(M)j Fn(with)33 b(c)m(onne)m(ction)g Fq(r)521 373 y Fn(admits)23 b(lo)m(c)m(al)f(fr)m(ames)h(c)m(onsisting)h(of)f(p)m(ar)m(al)s(lel)f (se)m(ctions)i(if)f(and)g(only)g(if)521 451 y Fq(r)h Fn(is)g(\015at,)f(i.)h(e.)31 b Fr(F)1007 427 y Fo(r)1068 451 y Fu(=)19 b(0)p Fn(.)376 550 y Fu(\(33\))27 b(As)c(a)f(consequence) g(of)g(the)g(previous)g(theorem)e(w)n(e)j(obtain:)521 649 y Fv(Corollary)k(11.)i Fn(A)24 b(ve)m(ctor)h(bund)s(le)g Fr(E)g Fq(!)20 b Fr(M)32 b Fn(admits)24 b(a)g(\015at)g(c)m(onne)m(c-) 521 728 y(tion)g(if)g(and)f(only)g(if)h(it)g(admits)f(a)g(system)h(of)f (lo)m(c)m(al)g(trivializations)991 847 y Fr( )1034 857 y Fs(i)1060 847 y Fu(:)g Fr(\031)1140 819 y Fo(\000)p Fp(1)1205 847 y Fu(\()p Fr(U)1276 857 y Fs(i)1294 847 y Fu(\))18 b Fq(!)i Fr(U)1470 857 y Fs(i)1503 847 y Fq(\002)c Ft(R)1618 819 y Fs(k)521 965 y Fn(for)23 b(which)h(al)s(l)f(tr)m (ansition)g(maps)682 1084 y Fr( )725 1094 y Fs(j)765 1084 y Fq(\016)15 b Fr( )859 1056 y Fo(\000)p Fp(1)857 1102 y Fs(i)930 1084 y Fu(:)23 b(\()p Fr(U)1042 1094 y Fs(i)1076 1084 y Fq(\\)15 b Fr(U)1181 1094 y Fs(j)1205 1084 y Fu(\))g Fq(\002)g Ft(R)1360 1056 y Fs(k)1410 1084 y Fq(\000)-11 b(!)19 b Fu(\()p Fr(U)1608 1094 y Fs(i)1641 1084 y Fq(\\)c Fr(U)1746 1094 y Fs(j)1771 1084 y Fu(\))g Fq(\002)g Ft(R)1926 1056 y Fs(k)1238 1187 y Fu(\()p Fr(x;)c(v)r Fu(\))19 b Fq(7\000)-11 b(!)19 b Fu(\()p Fr(x;)11 b(g)1662 1197 y Fs(ij)1702 1187 y Fu(\()p Fr(x)p Fu(\))p Fr(v)r Fu(\))521 1309 y Fn(have)23 b(the)f(pr)m(op)m(erty)f(that)h(the)g (functions)i Fr(g)1597 1319 y Fs(ij)1645 1309 y Fu(:)e Fr(U)1730 1319 y Fs(i)1762 1309 y Fq(\\)12 b Fr(U)1864 1319 y Fs(j)1908 1309 y Fq(!)19 b Fr(GL)2091 1319 y Fs(k)2120 1309 y Fu(\()p Ft(R)p Fu(\))k Fn(ar)m(e)521 1388 y(c)m(onstant.)1056 1567 y Fu(18)f(Decem)n(b)r(er)g(2003)376 1726 y(\(34\))27 b Fv(Metrics)15 b Fu(on)f(v)n(ector)h(bundles)f(are)g(smo)r(othly)f(v)l (arying)i(\014b)r(erwise)f(scalar)521 1805 y(pro)r(ducts.)28 b(Using)20 b(a)g(partition)e(of)i(unit)n(y)f(w)n(e)h(pro)n(v)n(e)g (that)f(ev)n(ery)h(v)n(ector)521 1884 y(bundle)i(admits)e(a)i(metric.) 376 1962 y(\(35\))27 b(A)21 b(connection)g Fq(r)g Fu(on)f(a)h(v)n (ector)f(bundle)g Fr(E)j Fq(!)c Fr(M)27 b Fu(is)21 b(compatible)f(with) 521 2041 y(a)i(metric)g Fq(h)g Fr(;)33 b Fq(i)23 b Fu(if)f(and)f(only)h (if)865 2160 y Fr(d)p Fq(h)p Fr(s)956 2170 y Fp(1)983 2160 y Fr(;)11 b(s)1043 2170 y Fp(2)1070 2160 y Fq(i)19 b Fu(=)g Fq(hr)p Fr(s)1298 2170 y Fp(1)1326 2160 y Fr(;)11 b(s)1386 2170 y Fp(2)1413 2160 y Fq(i)k Fu(+)g Fq(h)p Fr(s)1577 2170 y Fp(1)1604 2160 y Fr(;)c Fq(r)p Fr(s)1720 2170 y Fp(2)1748 2160 y Fq(i)521 2279 y Fu(for)23 b(all)i(pairs)e(of)g (sections)h Fr(s)1235 2289 y Fp(1)1262 2279 y Fr(;)11 b(s)1322 2289 y Fp(2)1371 2279 y Fq(2)21 b Fu(\000\()p Fr(E)t Fu(\).)33 b(Sometimes)22 b(a)i(connection)521 2357 y(compatible)e(with)f(some)g(metric)h(is)g(called)h(a)f(metric)f (connection.)521 2456 y Fv(Lemma)k(12.)k Fn(A)23 b(c)m(onne)m(ction)h Fq(r)h Fn(is)f(c)m(omp)m(atible)f(with)h(a)f(metric)i Fq(h)f Fr(;)35 b Fq(i)521 2535 y Fn(if)30 b(and)f(only)h(if)g(its)f(c)m (onne)m(ction)h(matrix)g Fr(!)i Fn(with)d(r)m(esp)m(e)m(ct)g(to)g(any)h (lo-)521 2613 y(c)m(al)k(fr)m(ame)f(that)g(is)h(orthonormal)e(with)h(r) m(esp)m(e)m(ct)h(to)f Fq(h)h Fr(;)46 b Fq(i)34 b Fn(is)g(skew-)521 2692 y(symmetric,)d(i.)e(e.)45 b Fr(!)1074 2702 y Fs(ij)1143 2692 y Fu(=)29 b Fq(\000)p Fr(!)1316 2702 y Fs(j)s(i)1386 2692 y Fn(for)f(al)s(l)g Fr(i)g Fn(and)g Fr(j)t Fn(.)46 b(In)29 b(this)f(c)m(ase)h(the)521 2771 y(curvatur)m(e)22 b(matrix)g Fu(\012)f Fn(with)g(r)m(esp)m(e)m(ct)g(to)g(a)h(lo)m(c)m(al) e(orthonormal)f(fr)m(ame)j(is)521 2849 y(also)h(skew-symmetric:)31 b Fu(\012)1207 2859 y Fs(ij)1266 2849 y Fu(=)19 b Fq(\000)p Fu(\012)1436 2859 y Fs(j)s(i)1478 2849 y Fn(.)589 2948 y Fu(Finally)-6 b(,)35 b(metric)d(connections)g(alw)n(a)n(ys)g(exist.) 59 b(The)32 b(follo)n(wing)g(is)521 3027 y(pro)n(v)n(ed)23 b(com)n(bining)f(the)h(ab)r(o)n(v)n(e)h(lemma)e(with)g(the)h(pro)r(of)g (of)g(Prop)r(osi-)521 3105 y(tion)f(5.)521 3204 y Fv(Prop)r(osition)f (13.)k Fn(Every)19 b(ve)m(ctor)h(bund)s(le)h Fr(E)i Fn(with)c(a)h (metric)g Fq(h)g Fr(;)31 b Fq(i)20 b Fn(ad-)521 3283 y(mits)27 b(c)m(omp)m(atible)g(c)m(onne)m(ctions.)42 b(The)27 b(sp)m(ac)m(e)g(of)g(al)s(l)g(c)m(omp)m(atible)f(c)m(on-)521 3362 y(ne)m(ctions)k(is)f(an)g(a\016ne)g(sp)m(ac)m(e)g(for)g(the)g(sp)m (ac)m(e)g Fu(\012)1779 3337 y Fp(1)1805 3362 y Fu(\()p Fr(S)t(k)r(ew)r(E)t(nd)p Fu(\()p Fr(E)t Fu(\)\))f Fn(of)521 3440 y Fu(1)p Fn(-forms)19 b(with)g(values)h(in)g(the)g(endomorphisms)e (of)i Fr(E)j Fn(which)d(ar)m(e)f(skew-)521 3519 y(symmetric)24 b(with)f(r)m(esp)m(e)m(ct)g(to)h Fq(h)f Fr(;)36 b Fq(i)p Fn(.)p eop %%Page: 9 9 9 8 bop 2305 159 a Fk(9)1056 294 y Fu(22)22 b(Decem)n(b)r(er)g(2003)376 445 y(\(36\))27 b(W)-6 b(e)28 b(no)n(w)f(consider)h(v)n(ector)g (bundles)f(of)g(small)h(rank)e(equipp)r(ed)h(with)521 523 y(metric)18 b(connections.)29 b(If)18 b(the)h(rank)e(is)i(=)g(1,)h (then)e(the)g(sk)n(ew-symmetry)521 602 y(of)29 b(the)g(connection)g (matrix)f(sho)n(ws)g(that)g(ev)n(ery)h(metric)g(connection)521 680 y(is)34 b(\015at.)63 b(As)34 b(ev)n(ery)f(bundle)g(admits)f(a)i (metric)e(and)h(a)g(compatible)521 759 y(connection,)d(w)n(e)e (conclude)h(that)e(all)i(rank)e(one)h(bundles)g(admit)e(\015at)521 838 y(connections.)52 b(In)29 b(some)g(sense,)j(rank)c(=)k(2)d(is)h (the)f(\014rst)g(in)n(teresting)521 916 y(case.)376 995 y(\(37\))e(Supp)r(ose)14 b Fr(E)22 b Fq(!)d Fr(M)j Fu(is)15 b(a)g(smo)r(oth)e(v)n(ector)h(bundle)g(of)h(rank)f(=)19 b(2)c(equipp)r(ed)521 1074 y(with)g(a)h(metric.)27 b(In)15 b(addition,)i(w)n(e)f(assume)f(that)g Fr(E)k Fu(is)d(orien)n(ted,)i(i.) e(e.)h(w)n(e)521 1152 y(ha)n(v)n(e)h(c)n(hosen)g(a)g(system)f(of)h(lo)r (cal)h(trivialisations)f(for)g(whic)n(h)g(all)g(transi-)521 1231 y(tion)i(maps)e(tak)n(e)i(v)l(alues)h(in)f(the)g(orien)n (tation-preserving)f(linear)h(maps)521 1310 y(only)-6 b(.)35 b(W)-6 b(e)24 b(w)n(ork)f(only)g(with)g(orien)n(ted)h (orthonormal)d(lo)r(cal)j(frames)f Fr(s)2304 1320 y Fp(1)521 1388 y Fu(and)30 b Fr(s)689 1398 y Fp(2)716 1388 y Fu(.)55 b(Due)31 b(to)f(the)g(orien)n(tation,)i(equiv)l(alen)n(tly)g(the)e (ordering)g(of)521 1467 y(the)18 b(orthonormal)e(frames,)j(w)n(e)f(kno) n(w)g(unam)n(biguously)f(what)g(the)h(cur-)521 1546 y(v)l(ature)25 b(form)g(\012)936 1556 y Fp(12)1011 1546 y Fu(is)1055 1519 y Fp(2)1080 1546 y Fu(.)40 b(P)n(assing)26 b(to)f(a)g(di\013eren)n (t)g(orthonormal)f(frame)521 1624 y(b)n(y)e(an)g(orien)n (tation-preserving)f(gauge)h(transformation)e(c)n(hanges)i(the)521 1703 y(curv)l(ature)17 b(matrix)f(\012)h(b)n(y)g(conjugating)g(it)g (with)g(an)g(elemen)n(t)h(of)f Fr(S)t(O)r Fu(\(2\),)521 1782 y(whic)n(h)25 b(turns)f(out)g(not)h(to)g(b)r(e)g(a)g(c)n(hange)g (at)g(all.)40 b(So)24 b(\012)1939 1792 y Fp(12)2015 1782 y Fu(is)h(in)g(fact)h(a)521 1860 y(globally)h(de\014ned)e(2-form,)i (whic)n(h)f(is)g(indep)r(enden)n(t)f(of)h(the)g(c)n(hoice)i(of)521 1939 y(frame.)37 b(Moreo)n(v)n(er,)27 b(it)e(is)g(lo)r(cally)h(exact)e (\012)1626 1949 y Fp(12)1700 1939 y Fu(=)g Fr(d!)1850 1949 y Fp(12)1901 1939 y Fu(,)i(and)e(therefore)521 2017 y(closed.)31 b(Th)n(us)21 b(the)g(follo)n(wing)i(de\014nition)f(mak)n (es)f(sense:)521 2114 y Fv(De\014nition)28 b(14.)h Fu(The)23 b Fv(Euler)k(class)d Fu(of)g(the)f(orien)n(ted)h(bundle)f Fr(E)i Fq(!)521 2193 y Fr(M)42 b Fu(is)35 b(the)g(de)f(Rham)f (cohomology)h(class)i Fr(e)p Fu(\()p Fr(E)t(;)11 b Fq(r)p Fu(\))40 b(=)2043 2166 y Fo(\000)p Fp(1)p 2043 2177 62 3 v 2047 2216 a(2)p Fs(\031)2110 2193 y Fu([\012)2176 2203 y Fp(12)2227 2193 y Fu(])h Fq(2)521 2275 y Fr(H)581 2251 y Fp(2)576 2293 y Fs(dR)639 2275 y Fu(\()p Fr(M)7 b Fu(\).)589 2372 y(Here)26 b Fq(r)f Fu(is)h(an)n(y)f(connection)g (compatible)g(with)g(the)g(giv)n(en)g(metric)521 2450 y Fq(h)d Fr(;)34 b Fq(i)p Fu(.)c(W)-6 b(e)22 b(ha)n(v)n(e)h(the)e (follo)n(wing)i(list)f(of)h(prop)r(erties:)594 2529 y Fq(\017)28 b Fu(If)22 b Fq(r)g Fu(is)g(\015at,)g(then)f Fr(e)p Fu(\()p Fr(E)t(;)11 b Fq(r)p Fu(\))18 b(=)h(0.)594 2608 y Fq(\017)28 b Fu(Denoting)g(b)n(y)1060 2590 y(\026)1045 2608 y Fr(E)k Fu(the)c(v)n(ector)g(bundle)f(obtained)h(from)f(the)h (ori-)656 2686 y(en)n(ted)14 b(bundle)g Fr(E)19 b Fu(b)n(y)14 b(c)n(hanging)g(the)h(orien)n(tation,)h(w)n(e)f(ha)n(v)n(e)g Fr(e)p Fu(\()2217 2669 y(\026)2203 2686 y Fr(E)s(;)c Fq(r)p Fu(\))19 b(=)656 2765 y Fq(\000)p Fr(e)p Fu(\()p Fr(E)t(;)11 b Fq(r)p Fu(\).)594 2843 y Fq(\017)28 b Fu(If)19 b(w)n(e)h(c)n(hange)f(the)g(connection)h Fq(r)f Fu(to)g(a)h(di\013eren) n(t)e(connection)i Fq(r)2315 2819 y Fo(0)656 2922 y Fu(compatible)25 b(with)f(the)i(same)f(metric,)h(then)f Fq(r)1875 2898 y Fo(0)1916 2922 y Fu(has)h Fr(!)2079 2932 y Fp(12)2147 2922 y Fu(+)17 b Fr(a)26 b Fu(in)656 3001 y(its)e(connection)g(matrix)f (where)g Fq(r)i Fu(had)e Fr(!)1743 3011 y Fp(12)1794 3001 y Fu(,)i(with)e Fr(a)h Fu(a)g(globally)656 3079 y(de\014ned)c(1-form)h(on)g Fr(M)7 b Fu(.)30 b(The)21 b(Euler)g(class)h(with)f(resp)r(ect)h(to)f(this)656 3158 y(new)e(connection)g(is)h Fr(e)p Fu(\()p Fr(E)t(;)11 b Fq(r)1375 3134 y Fo(0)1390 3158 y Fu(\))19 b(=)g([\012)1571 3168 y Fp(12)1631 3158 y Fu(+)10 b Fr(da)p Fu(])19 b(=)g([\012)1934 3168 y Fp(12)1985 3158 y Fu(])g(=)g Fr(e)p Fu(\()p Fr(E)t(;)11 b Fq(r)p Fu(\).)521 3237 y(Th)n(us,)17 b(w)n(e)f(see)h(that)e(the)h (Euler)g(class,)j(as)d(a)g(cohomology)g(class,)i(is)f(inde-)521 3315 y(p)r(enden)n(t)g(of)i(the)f(connection)g(w)n(e)h(c)n(ho)r(ose,)g (as)g(long)f(as)g(it)h(is)f(compatible)521 3394 y(with)25 b(the)h(metric.)40 b(Therefore,)27 b(w)n(e)f(write)f Fr(e)p Fu(\()p Fr(E)t(;)11 b Fq(h)26 b Fr(;)37 b Fq(i)p Fu(\))26 b(for)f(the)h(Euler)p 308 3452 338 3 v 376 3497 a Fp(2)400 3519 y Fh(Changing)19 b(the)g(order)g(of)g(the)g(frame)g(c)n (hanges)g(the)g(sign)h(due)f(to)g(the)g(sk)n(ew-symmetry)-5 b(.)p eop %%Page: 10 10 10 9 bop 308 159 a Fk(10)521 294 y Fu(class.)49 b(W)-6 b(e)29 b(will)g(see)g(later,)h(that)d(it)h(is)h(in)f(fact)g(indep)r (enden)n(t)g(of)g(the)521 373 y(metric,)d(and)f(is)h(therefore)g(a)g (top)r(ological)g(in)n(v)l(arian)n(t)g(of)f(the)h(orien)n(ted)521 451 y(bundle)d Fr(E)t Fu(.)376 530 y(\(38\))27 b(W)-6 b(e)27 b(consider)f Fr(M)32 b Fu(=)26 b Fr(S)1118 506 y Fp(2)1145 530 y Fu(,)h(co)n(v)n(ered)f(b)n(y)g(t)n(w)n(o)g(c)n(harts) f(di\013eomorphic)g(to)521 609 y Ft(R)568 584 y Fp(2)597 609 y Fu(.)k(Construct)20 b(an)h(orien)n(table)h(v)n(ector)g(bundle)f Fr(E)k Fu(of)d(rank)f(2)h(o)n(v)n(er)g Fr(S)2305 584 y Fp(2)521 687 y Fu(b)n(y)k(making)g(an)g(iden)n(ti\014cation)h(of)f (\()p Fr(x;)11 b(v)r Fu(\))26 b Fq(2)h Fu(\()p Ft(R)1757 663 y Fp(2)1803 687 y Fq(n)18 b(f)p Fu(0)p Fq(g)p Fu(\))f Fq(\002)i Ft(R)2117 663 y Fp(2)2171 687 y Fu(=)27 b Fr(X)2304 697 y Fp(1)521 766 y Fu(with)c(\()p Fq(\000)783 740 y Fs(x)p 758 751 77 3 v 758 789 a Fo(j)p Fs(x)p Fo(j)811 777 y Fl(2)842 766 y Fr(;)11 b(g)r Fu(\()p Fr(x)p Fu(\))16 b Fq(\001)g Fr(v)r Fu(\))22 b Fq(2)g Fu(\()p Ft(R)1268 742 y Fp(2)1312 766 y Fq(n)16 b(f)p Fu(0)p Fq(g)p Fu(\))f Fq(\002)i Ft(R)1620 742 y Fp(2)1671 766 y Fu(=)22 b Fr(X)1799 776 y Fp(2)1849 766 y Fu(for)i(some)f(smo)r(oth)521 860 y(map)17 b Fr(g)10 b Fu(:)22 b Ft(R)793 835 y Fp(2)829 860 y Fq(n)7 b(f)p Fu(0)p Fq(g)18 b(!)h Fr(S)t(O)r Fu(\(2\).)27 b(On)18 b(these)h(t)n(w)n(o)e(trivializations)i(w)n(e)g(ha)n(v)n(e)521 938 y(metrics)25 b(de\014ned)h(b)n(y)f(the)h(standard)e(scalar)j(pro)r (duct)d(on)i Ft(R)2066 914 y Fp(2)2094 938 y Fu(,)h(and)f(as)521 1017 y Fr(g)e Fu(is)e(assumed)e(to)h(tak)n(e)h(v)l(alues)g(in)g(the)f (orthogonal)g(group,)g(w)n(e)g(obtain)521 1096 y(an)33 b(induced)f(metric)g(on)g Fr(E)t Fu(.)62 b(W)-6 b(e)33 b(tak)n(e)g(the)f(t)n(w)n(o)h(\015at)f(connections)521 1174 y(induced)21 b(b)n(y)g(the)g(t)n(w)n(o)f(trivializations)i(and)f (glue)g(them)f(together)g(with)521 1253 y(an)j(explicit)i(partition)e (of)g(unit)n(y)g(to)h(obtain)f(a)g(metric)g(connection)h(on)521 1331 y Fr(E)t Fu(.)589 1410 y(In)e(this)g(case)h(an)g(orien)n(tation)f (of)g Fr(S)1490 1386 y Fp(2)1539 1410 y Fu(giv)n(es)h(us)g(an)f(iden)n (ti\014cation)h(of)521 1489 y Fr(H)581 1464 y Fp(2)576 1506 y Fs(dR)639 1489 y Fu(\()p Fr(S)710 1464 y Fp(2)736 1489 y Fu(\))i(with)e Ft(R)p Fu(,)28 b(so)c(that)g(the)g(Euler)g(class) i(is)f(simply)e(the)i(n)n(um)n(b)r(er)521 1567 y(obtained)g(b)n(y)g(in) n(tegration)f(of)1315 1541 y Fo(\000)p Fp(1)p 1315 1552 62 3 v 1319 1591 a(2)p Fs(\031)1382 1567 y Fu(\012)1430 1577 y Fp(12)1506 1567 y Fu(o)n(v)n(er)h Fr(S)1695 1543 y Fp(2)1747 1567 y Fu(\(with)f(resp)r(ect)h(to)g(the)521 1650 y(c)n(hosen)19 b(orien)n(tation)f(of)g Fr(S)1177 1625 y Fp(2)1204 1650 y Fu(\).)28 b(Iden)n(tifying)18 b Ft(R)1653 1625 y Fp(2)1689 1650 y Fq(n)8 b(f)p Fu(0)p Fq(g)17 b Fu(with)h Fr(S)2041 1625 y Fp(1)2075 1650 y Fq(\002)8 b Ft(R)20 b Fu(with)521 1728 y(co)r(ordinates)j(\()p Fr(\022)r(;)11 b(t)p Fu(\),)23 b(w)n(e)h(sp)r(ecify)f Fr(g)r Fu(\()p Fr(\022)r(;)11 b(t)p Fu(\))23 b(to)g(b)r(e)g(rotation)f (b)n(y)h(angle)h Fr(n\022)521 1807 y Fu(\(indep)r(enden)n(tly)c(of)h Fr(t)p Fu(,)g(for)g(some)g(in)n(teger)g Fr(n)p Fu(\).)29 b(Then)20 b(w)n(e)h(obtain)g(Euler)521 1886 y(class)i(=)d Fq(\006)p Fr(n)j Fu(for)f(our)g(bundle,)g(dep)r(ending)g(on)g(ho)n(w)f (w)n(e)i(orien)n(t)f Fr(E)k Fu(and)521 1964 y Fr(S)566 1940 y Fp(2)592 1964 y Fu(.)42 b(\(When)25 b Fr(n)h Fu(=)g(0,)h(the)f (bundle)f(is)h(trivial)g(and)g(the)f(connection)h(is)521 2043 y(\015at.\))1099 2228 y(8)c(Jan)n(uary)f(2004)376 2414 y(\(39\))27 b(An)17 b Fv(a\016ne)i(connection)e Fu(on)f(a)h(smo)r(oth)e(manifold)h Fr(M)24 b Fu(is)17 b(a)g(connection)521 2492 y Fq(r)23 b Fu(on)f(its)g(tangen)n(t)g (bundle)g Fr(T)9 b(M)26 b Fq(!)20 b Fr(M)7 b Fu(.)31 b(F)-6 b(or)23 b(a\016ne)f(connections)h(the)521 2571 y(v)l(ariables)31 b Fr(X)36 b Fu(and)30 b Fr(s)h Fu(in)f Fq(r)1234 2581 y Fs(X)1280 2571 y Fr(s)h Fu(are)f(on)g(equal)h(fo)r (oting,)i(as)e(they)f(are)521 2650 y(b)r(oth)h(sections)i(of)f(the)g (tangen)n(t)g(bundle.)60 b(This)32 b(leads)g(to)g(p)r(ossible)521 2728 y(symmetries)19 b(whic)n(h)i(mak)n(e)f(no)g(sense)h(in)g(the)f (more)g(general)h(setting)g(of)521 2807 y(arbitrary)g(v)n(ector)h (bundles.)376 2886 y(\(40\))27 b(The)22 b Fv(torsion)g Fu(of)g(an)g(a\016ne)g(connection)g Fq(r)g Fu(is)h(de\014ned)e(b)n(y) 830 3018 y Fr(T)9 b Fu(\()p Fr(X)r(;)i(Y)j Fu(\))19 b(=)g Fq(r)1213 3028 y Fs(X)1258 3018 y Fr(Y)30 b Fq(\000)16 b(r)1450 3028 y Fs(Y)1491 3018 y Fr(X)k Fq(\000)c Fu([)p Fr(X)r(;)11 b(Y)k Fu(])521 3150 y(for)22 b(all)h Fr(X)r(;)11 b(Y)33 b Fq(2)19 b(X)10 b Fu(\()p Fr(M)d Fu(\).)521 3255 y Fv(Lemma)25 b(15.)j Fn(The)c(torsion)f(de\014nes)h(a)f (skew-symmetric)h(map)887 3387 y Fr(T)16 b Fu(:)23 b Fq(X)10 b Fu(\()p Fr(M)d Fu(\))14 b Fq(\002)i(X)10 b Fu(\()p Fr(M)d Fu(\))18 b Fq(\000)-11 b(!)19 b(X)10 b Fu(\()p Fr(M)d Fu(\))521 3519 y Fn(that)23 b(is)h(biline)m(ar)g(over)f Fr(C)1161 3494 y Fo(1)1212 3519 y Fu(\()p Fr(M)7 b Fu(\))p Fn(.)p eop %%Page: 11 11 11 10 bop 2279 159 a Fk(11)589 294 y Fu(An)23 b(a\016ne)h(connection)f Fq(r)h Fu(is)g(called)g Fv(symmetric)g Fu(if)g(it)f(is)h(torsion-)521 373 y(free,)f(i.)g(e.)f Fr(T)31 b Fu(v)l(anishes)22 b(iden)n(tically) 1419 347 y Fp(3)1444 373 y Fu(.)376 451 y(\(41\))27 b(T)-6 b(o)30 b(explain)h(wh)n(y)e(torsion-freeness)h(is)g(indeed)g(a)g (symmetry)e(condi-)521 530 y(tion,)33 b(w)n(e)d(consider)h(the)f (expression)g(of)h(an)f(a\016ne)h(connection)f(in)h(a)521 609 y(lo)r(cal)23 b(co)r(ordinate)f(system)g(\()p Fr(y)1280 619 y Fp(1)1306 609 y Fr(;)11 b(:)g(:)g(:)j(;)d(y)1486 619 y Fs(n)1518 609 y Fu(\))22 b(on)g Fr(M)7 b Fu(.)31 b(W)-6 b(e)23 b(write)f Fr(@)2096 619 y Fs(i)2138 609 y Fu(for)g(the)521 689 y(co)r(ordinate)g(v)n(ector)g(\014elds)1241 662 y Fs(@)p 1221 673 70 3 v 1221 712 a(@)s(y)1273 719 y Fm(i)1297 689 y Fu(,)g(and)g(use)h(the)f(lo)r(cal)h(frame)e Fr(@)2071 699 y Fp(1)2098 689 y Fr(;)11 b(:)g(:)g(:)i(;)e(@)2280 699 y Fs(n)2312 689 y Fu(.)521 773 y(Then)1034 890 y Fq(r)p Fr(@)1125 900 y Fs(i)1162 890 y Fu(=)1267 806 y Fs(n)1232 826 y Fi(X)1240 968 y Fs(j)s Fp(=1)1341 890 y Fr(!)1382 900 y Fs(ij)1438 890 y Fq(\012)16 b Fr(@)1541 900 y Fs(j)1587 890 y Fr(;)521 1054 y Fu(and)21 b(using)i(\(7\))e(w)n (e)i(obtain)1042 1215 y Fq(r)1098 1225 y Fs(@)1123 1232 y Fm(i)1144 1215 y Fr(@)1179 1225 y Fs(j)1222 1215 y Fu(=)1327 1131 y Fs(n)1292 1151 y Fi(X)1298 1295 y Fs(k)q Fp(=1)1401 1215 y Fr(!)1444 1187 y Fs(i)1442 1232 y(j)s(k)1493 1215 y Fr(@)1528 1225 y Fs(k)1579 1215 y Fr(;)521 1382 y Fu(whic)n(h)f(is)g(usually)g(written)f(as)1067 1539 y Fq(r)1123 1549 y Fs(@)1148 1556 y Fm(i)1169 1539 y Fr(@)1204 1549 y Fs(j)1247 1539 y Fu(=)1352 1455 y Fs(n)1317 1475 y Fi(X)1323 1619 y Fs(k)q Fp(=1)1426 1539 y Fu(\000)1467 1511 y Fs(k)1467 1556 y(ij)1508 1539 y Fr(@)1543 1549 y Fs(k)521 1706 y Fu(in)36 b(classical)h(notation.)69 b(Therefore,)39 b(w)n(e)c(de\014ne)g(the)h Fv(Christo\013el)521 1785 y(sym)n(b)r(ols)c Fu(of)g(the)f(a\016ne)g(connection)h Fq(r)g Fu(with)f(resp)r(ect)g(to)g(the)g(co-)521 1863 y(ordinate)22 b(system)e(\()p Fr(y)1058 1873 y Fp(1)1085 1863 y Fr(;)11 b(:)g(:)g(:)i(;)e(y)1264 1873 y Fs(n)1297 1863 y Fu(\))21 b(to)h(b)r(e)g(\000)1556 1839 y Fs(k)1556 1880 y(ij)1616 1863 y Fu(=)d Fr(!)1729 1839 y Fs(i)1727 1881 y(j)s(k)1778 1863 y Fu(.)589 1942 y(Returning)h(to)i(the)g (de\014nition)f(of)h(torsion,)g(w)n(e)g(see)h(that)623 2099 y Fr(T)9 b Fu(\()p Fr(@)732 2109 y Fs(i)751 2099 y Fr(;)i(@)815 2109 y Fs(j)840 2099 y Fu(\))18 b(=)989 2015 y Fs(n)954 2035 y Fi(X)960 2179 y Fs(k)q Fp(=1)1052 2099 y Fu(\()p Fr(!)1121 2071 y Fs(i)1119 2116 y(j)s(k)1184 2099 y Fq(\000)e Fr(!)1295 2067 y Fs(j)1293 2118 y(ik)1338 2099 y Fu(\))p Fr(@)1399 2109 y Fs(k)1446 2099 y Fu(=)1550 2015 y Fs(n)1516 2035 y Fi(X)1521 2179 y Fs(k)q Fp(=1)1613 2099 y Fu(\(\000)1680 2071 y Fs(k)1680 2116 y(ij)1736 2099 y Fq(\000)g Fu(\000)1845 2071 y Fs(k)1845 2116 y(j)s(i)1886 2099 y Fu(\))p Fr(@)1947 2109 y Fs(k)1997 2099 y Fr(:)521 2266 y Fu(As)28 b(the)f(torsion)g(is)h(linear)g(o)n(v)n(er)g(the)f(smo) r(oth)f(functions,)j(w)n(e)e(obtain)521 2345 y(the)22 b(follo)n(wing:)521 2437 y Fv(Lemma)h(16.)j Fn(A)n(n)d(a\016ne)f(c)m (onne)m(ction)g Fq(r)g Fn(is)g(torsion-fr)m(e)m(e)f(if)i(and)e(only)521 2516 y(if)j Fu(\000)626 2492 y Fs(k)626 2533 y(ij)686 2516 y Fu(=)19 b(\000)797 2492 y Fs(k)797 2533 y(j)s(i)862 2516 y Fn(for)k(any)h(lo)m(c)m(al)e(c)m(o)m(or)m(dinate)g(system.)589 2609 y Fu(Th)n(us)17 b(symmetry)f(of)i(the)g(connection)g(really)h (refers)f(to)g(a)g(symmetry)521 2687 y(of)30 b(the)f(Christo\013el)g (sym)n(b)r(ols)f(expressing)i(this)f(connection)g(in)h(lo)r(cal)521 2766 y(co)r(ordinates.)376 2845 y(\(42\))d(Consider)21 b(a)g Fv(Riemannian)j(manifold)d Fr(M)7 b Fu(,)22 b(that)e(is)i(a)f (smo)r(oth)e(man-)521 2923 y(ifold)24 b(with)e(a)i(metric)e(on)h(its)g (tangen)n(t)g(bundle.)32 b(In)23 b(this)g(case)h(it)f(turns)521 3002 y(out)g(that)f(there)g(is)i(a)f(unique)f(symmetric)g(connection)h (that)f(is)h(at)g(the)521 3081 y(same)f(time)f(compatible)g(with)h(the) f(giv)n(en)i(metric:)521 3173 y Fv(Prop)r(osition)k(17)f Fu(\(F)-6 b(undamen)n(tal)21 b(Lemma)g(of)i(Riemannian)f(Geom-)521 3252 y(etry\))p Fv(.)28 b Fn(The)d(tangent)g(bund)s(le)h(of)e(a)g(R)n (iemannian)i(manifold)e(admits)g(a)521 3331 y(unique)h(torsion-fr)m(e)m (e)e(c)m(onne)m(ction)h(c)m(omp)m(atible)f(with)g(the)g(metric.)p 308 3385 338 3 v 376 3430 a Fp(3)400 3451 y Fh(Note)e(that)g(requiring) h(the)f(naiv)n(e)h(symmetry)e Ff(r)1463 3459 y Fe(X)1505 3451 y Fg(Y)32 b Fh(=)19 b Ff(r)1680 3459 y Fe(Y)1719 3451 y Fg(X)25 b Fh(for)c(all)h Fg(X)j Fh(and)d Fg(Y)33 b Fh(leads)308 3519 y(to)19 b(a)g(con)n(tradiction.)p eop %%Page: 12 12 12 11 bop 308 159 a Fk(12)521 294 y Fn(Pr)m(o)m(of.)27 b Fu(Giv)n(en)k(v)n(ector)h(\014elds)f Fr(X)5 b Fu(,)34 b Fr(Y)46 b Fu(and)31 b Fr(Z)5 b Fu(,)34 b(w)n(e)d(can)h(use)f(the)g(t) n(w)n(o)521 373 y(requiremen)n(ts,)25 b(compatibilit)n(y)g(with)f(the)h (metric)f Fq(h)i Fr(;)36 b Fq(i)26 b Fu(and)e(torsion-)521 451 y(freeness,)f(to)e(conclude)h(that)f(the)h(only)f(p)r(ossible)h(v)l (alue)g(for)g Fq(hr)2137 461 y Fs(X)2183 451 y Fr(Y)t(;)11 b(Z)5 b Fq(i)521 530 y Fu(is)308 676 y(\(8\))56 b Fq(hr)531 686 y Fs(X)577 676 y Fr(Y)t(;)11 b(Z)5 b Fq(i)19 b Fu(=)820 631 y(1)p 820 661 34 3 v 820 722 a(2)860 676 y(\()p Fq(h)p Fu([)p Fr(X)r(;)11 b(Y)k Fu(])p Fr(;)c(Z)5 b Fq(i)16 b Fu(+)f Fq(h)p Fu([)p Fr(Z)q(;)c(X)5 b Fu(])p Fr(;)11 b(Y)18 b Fq(i)d Fu(+)g Fq(h)p Fu([)p Fr(Z)q(;)c(Y)17 b Fu(])p Fr(;)11 b(X)5 b Fq(i)p Fu(+)1231 798 y Fr(L)1276 808 y Fs(X)1322 798 y Fq(h)p Fr(Y)t(;)11 b(Z)5 b Fq(i)16 b Fu(+)f Fr(L)1622 808 y Fs(Y)1663 798 y Fq(h)p Fr(X)r(;)c(Z)5 b Fq(i)15 b(\000)h Fr(L)1979 808 y Fs(Z)2017 798 y Fq(h)p Fr(X)r(;)11 b(Y)k Fq(i)p Fu(\))22 b Fr(:)521 913 y Fu(This)c(pro)n(v)n (es)g(uniqueness.)28 b(T)-6 b(o)18 b(see)h(existence,)h(w)n(e)e(use)g (\(8\))g(as)g(a)g(de\014ni-)521 991 y(tion.)27 b(As)16 b Fq(h)f Fr(;)26 b Fq(i)15 b Fu(is)h(non-degenerate,)g(requiring)e (that)g(the)h(equation)f(hold)521 1070 y(for)24 b(all)g Fr(Z)k Fu(uniquely)c(de\014nes)f Fq(r)1334 1080 y Fs(X)1380 1070 y Fr(Y)15 b Fu(.)34 b(W)-6 b(e)24 b(then)f(c)n(hec)n(k)i(that)d (this)i Fq(r)f Fu(is)521 1149 y(indeed)c(a)f(connection,)h(is)g (metric-compatible,)f(and)g(torsion-free.)66 b Fj(\003)1083 1330 y Fu(12)22 b(Jan)n(uary)e(2004)376 1472 y(\(43\))27 b(If)h Fr(E)k Fq(!)d Fr(M)34 b Fu(is)28 b(a)g(v)n(ector)g(bundle)f (with)g(a)h(connection)f Fq(r)p Fu(,)j(then)d(the)521 1550 y(dual)20 b(bundle)f Fr(E)933 1526 y Fo(\003)978 1550 y Fq(!)h Fr(M)26 b Fu(carries)21 b(a)f(w)n(ell-de\014ned)g(dual)g (connection)g Fq(r)2304 1526 y Fo(\003)521 1629 y Fu(c)n(haracterized)j (b)n(y)e(the)h(iden)n(tit)n(y)915 1739 y Fr(d)p Fq(h)p Fr(s;)11 b(\013)p Fq(i)20 b Fu(=)f Fq(hr)p Fr(s;)11 b(\013)p Fq(i)17 b Fu(+)e Fq(h)p Fr(s;)c Fq(r)1628 1712 y Fo(\003)1656 1739 y Fr(\013)p Fq(i)521 1850 y Fu(for)28 b(all)i Fr(s)g Fq(2)f Fu(\000\()p Fr(E)t Fu(\))f(and)g Fr(\013)i Fq(2)g Fu(\000\()p Fr(E)1438 1825 y Fo(\003)1464 1850 y Fu(\).)49 b(\(The)28 b(brac)n(k)n(ets)g(here)h(denote)521 1929 y(the)24 b(natural)e(pairing)i(b)r(et)n(w)n(een)f(a)h(bundle)f(and)g (its)h(dual)f(bundle,)h(not)521 2007 y(a)e(metric.\))589 2086 y(In)c(the)g(case)h(of)g(an)f(a\016ne)h(connection)g Fq(r)p Fu(,)h(the)e(dual)g(connection)h Fq(r)2304 2061 y Fo(\003)521 2164 y Fu(giv)n(es)k(us)f(the)f(follo)n(wing)i(c)n (haracterization)f(of)g(torsion-freeness:)521 2259 y Fv(Prop)r(osition)i(18.)j Fn(A)n(n)c(a\016ne)g(c)m(onne)m(ction)f Fq(r)h Fn(on)f Fr(M)29 b Fn(is)23 b(torsion-fr)m(e)m(e)521 2338 y(if)i(and)f(only)h(if)g(the)f(exterior)h(di\013er)m(ential)g(on)g (one-forms)f(is)g(given)i(by)521 2416 y(the)e(c)m(omp)m(osition)645 2543 y Fu(\000\()p Fr(T)760 2515 y Fo(\003)787 2543 y Fr(M)7 b Fu(\))924 2505 y Fo(r)964 2489 y Fd(\003)902 2543 y Fq(\000)-11 b(!)19 b Fu(\000\()p Fr(T)1144 2515 y Fo(\003)1171 2543 y Fr(M)j Fq(\012)15 b Fr(T)1372 2515 y Fo(\003)1399 2543 y Fr(M)7 b Fu(\))1552 2505 y Fo(^)1514 2543 y Fq(\000)-11 b(!)19 b Fu(\000\(\003)1754 2515 y Fp(2)1781 2543 y Fr(T)1829 2515 y Fo(\003)1855 2543 y Fr(M)7 b Fu(\))23 b Fr(:)376 2654 y Fu(\(44\))k(Next)e(w)n(e)g(discuss) g(a)f(concrete)h(example)f(where)h(w)n(e)g(can)f(write)h(do)n(wn)521 2732 y(the)j(Levi-Civita)g(connection)g(of)g(a)f(Riemannian)g(metric)g (explicitly)-6 b(.)521 2811 y(W)g(e)32 b(shall)f(use)g(this)g(to)f (compute)g(the)g(Euler)h(class)h(of)f(the)g(tangen)n(t)521 2890 y(bundle)22 b(of)g(the)f(2-sphere.)589 2968 y(Consider)28 b(the)i(unit)e(sphere)i Fr(S)1398 2944 y Fp(2)1455 2968 y Fq(\032)i Ft(R)1586 2944 y Fp(3)1614 2968 y Fu(.)53 b(W)-6 b(e)30 b(tak)n(e)f(the)g(standard)521 3047 y(scalar)d(pro)r (duct)f(on)g Ft(R)1106 3022 y Fp(3)1134 3047 y Fu(.)41 b(By)26 b(restriction)f(to)g(the)h(tangen)n(t)f(spaces)h(to)521 3126 y(the)16 b(sphere)f(w)n(e)h(obtain)f(a)h(Riemannian)e(metric)h(on) h Fr(S)1858 3101 y Fp(2)1884 3126 y Fu(.)28 b(Next)16 b(c)n(ho)r(osing)521 3204 y(an)31 b(explicit)i(parametrization)c(of)j (an)f(op)r(en)g(set)h(on)f(the)g(sphere)g(b)n(y)521 3283 y(an)d(op)r(en)f(set)h(in)f Ft(R)1022 3258 y Fp(2)1079 3283 y Fu(w)n(e)g(construct)g(an)h(explicit)g(lo)r(cal)h(orthonormal) 521 3362 y(frame)22 b Fr(s)738 3372 y Fp(1)765 3362 y Fr(;)11 b(s)825 3372 y Fp(2)875 3362 y Fu(for)23 b(the)f(tangen)n(t)g (bundle.)32 b(Let)23 b Fq(r)1752 3372 y Fp(0)1802 3362 y Fu(b)r(e)g(the)f(Levi-Civita)521 3440 y(connection)f(of)f Ft(R)970 3416 y Fp(3)1019 3440 y Fu(considered)g(as)h(a)f(Riemannian)f (manifold.)28 b(This)20 b(is)521 3519 y(the)26 b(\015at)f(connection)h (for)g(whic)n(h)g(the)f(standard)g(global)h(frame)f(of)h Ft(R)2302 3494 y Fp(3)p eop %%Page: 13 13 13 12 bop 2279 159 a Fk(13)521 294 y Fu(is)26 b(parallel.)41 b(W)-6 b(e)26 b(de\014ne)f(an)g(a\016ne)g(connection)h Fq(r)f Fu(on)g Fr(S)2002 270 y Fp(2)2054 294 y Fu(b)n(y)g(taking)521 373 y(co)n(v)l(arian)n(t)18 b(deriv)l(ativ)n(es)f(in)h Ft(R)1244 348 y Fp(3)1289 373 y Fu(using)f Fq(r)1513 383 y Fp(0)1558 373 y Fu(and)f(then)h(pro)t(jecting)g(to)g Fr(T)9 b(S)2305 348 y Fp(2)521 451 y Fu(along)26 b(the)f(orthogonal)g (complemen)n(t)g(of)h Fr(T)9 b(S)1697 427 y Fp(2)1748 451 y Fq(\032)26 b Fr(T)9 b Ft(R)1921 427 y Fp(3)1949 451 y Fu(.)41 b(Computing)521 530 y(the)23 b(connection)g(matrix)f (with)h(resp)r(ect)g(to)g(our)g(c)n(hosen)g(orthonormal)521 609 y(frame,)c(w)n(e)f(see)h(that)e(it)h(is)h(sk)n(ew-symmetric,)f(and) f(so)h Fq(r)h Fu(is)f(compatible)521 687 y(with)32 b(the)g(metric)g(of) g Fr(S)1152 663 y Fp(2)1179 687 y Fu(.)61 b(\(In)31 b(fact,)36 b Fq(r)c Fu(is)h(also)g(torsion-free,)i(and)521 766 y(therefore)29 b(is)h(the)f(Levi-Civita)g(connection,)j(but)c(w)n(e)h(will)h(not)f (need)521 845 y(this)22 b(here.\))589 923 y(W)-6 b(e)31 b(de\014ne)f(the)h(orien)n(tation)f(of)h Fr(S)1506 899 y Fp(2)1563 923 y Fu(so)g(that)e Fr(s)1835 933 y Fp(1)1862 923 y Fr(;)11 b(s)1922 933 y Fp(2)1980 923 y Fu(is)31 b(p)r(ositiv)n(ely)521 1002 y(orien)n(ted.)42 b(Our)26 b(calculation)g(of)h(the)e(connection)h(matrix)f(of)h Fq(r)g Fu(with)521 1081 y(resp)r(ect)21 b(to)f Fr(s)856 1091 y Fp(1)883 1081 y Fr(;)11 b(s)943 1091 y Fp(2)991 1081 y Fu(yields)21 b(\012)1223 1091 y Fp(12)1292 1081 y Fu(=)e Fr(s)1393 1056 y Fo(\003)1393 1097 y Fp(2)1432 1081 y Fq(^)12 b Fr(s)1520 1056 y Fo(\003)1520 1097 y Fp(1)1547 1081 y Fu(.)29 b(Therefore,)21 b(the)f(Euler)h(class)521 1159 y Fr(e)p Fu(\()p Fr(T)9 b(S)671 1135 y Fp(2)696 1159 y Fr(;)i Fq(h)17 b Fr(;)27 b Fq(i)p Fu(\),)17 b(iden)n(ti\014ed)f (with)f(a)g(n)n(um)n(b)r(er)f(b)n(y)h(using)h(the)f(isomorphism)521 1238 y Fr(H)581 1214 y Fp(2)576 1256 y Fs(dR)639 1238 y Fu(\()p Fr(S)710 1214 y Fp(2)736 1238 y Fu(\))k(=)g Ft(R)k Fu(giv)n(en)g(b)n(y)f(the)f(orien)n(tation,)h(turns)f(out)g(to)h (b)r(e:)481 1400 y Fr(e)p Fu(\()p Fr(T)9 b(S)631 1372 y Fp(2)656 1400 y Fr(;)i Fq(h)23 b Fr(;)34 b Fq(i)p Fu(\))18 b(=)933 1355 y Fq(\000)p Fu(1)p 933 1385 86 3 v 939 1446 a(2)p Fr(\031)1037 1308 y Fi(Z)1074 1461 y Fs(S)1106 1448 y Fl(2)1143 1400 y Fr(s)1174 1372 y Fo(\003)1174 1417 y Fp(2)1216 1400 y Fq(^)d Fr(s)1307 1372 y Fo(\003)1307 1417 y Fp(1)1352 1400 y Fu(=)1449 1355 y(1)p 1429 1385 73 3 v 1429 1446 a(2)p Fr(\031)1520 1308 y Fi(Z)1557 1461 y Fs(S)1589 1448 y Fl(2)1626 1400 y Fr(dv)r(ol)21 b Fu(=)1863 1355 y(1)p 1843 1385 V 1843 1446 a(2)p Fr(\031)1923 1400 y Fu(4)p Fr(\031)h Fu(=)d(2)j Fr(:)521 1561 y Fu(Note)c(that)f (this)g(is)h(indep)r(enden)n(t)f(of)h(the)f(orien)n(tation)g(w)n(e)h (ha)n(v)n(e)g(c)n(hosen,)521 1640 y(b)r(ecause)k(if)f(w)n(e)h(c)n (hange)f(the)g(orien)n(tation,)g(then)g(b)r(oth)f(the)h(Euler)g(class) 521 1718 y(and)k(the)g(isomorphism)e Fr(H)1219 1694 y Fp(2)1214 1736 y Fs(dR)1277 1718 y Fu(\()p Fr(S)1348 1694 y Fp(2)1374 1718 y Fu(\))h(=)h Ft(R)i Fu(c)n(hange)e(b)n(y)g(a)h (sign,)g(and)f(the)521 1797 y(t)n(w)n(o)d(signs)g(cancel.)1083 1970 y(15)g(Jan)n(uary)e(2004)376 2144 y(\(45\))27 b(Let)e Fr(M)32 b Fu(b)r(e)24 b(an)h(orien)n(ted)f(smo)r(oth)f(manifold)h(of)h (dimension)f(2,)i(i.)f(e.)h(a)521 2222 y(surface.)38 b(F)-6 b(or)25 b(a)f(Riemannian)g(metric)g Fq(h)h Fr(;)36 b Fq(i)25 b Fu(on)f Fr(M)32 b Fu(w)n(e)24 b(consider)h(the)521 2301 y(unique)d(function)f Fq(K)j Fu(de\014ned)d(b)n(y)h(the)f (equation)1093 2427 y Fq(\000)p Fu(\012)1193 2437 y Fp(12)1262 2427 y Fu(=)e Fq(K)q Fr(dv)r(ol)24 b(;)521 2553 y Fu(where)19 b(\012)757 2563 y Fp(12)825 2553 y Fu(is)g(the)g(curv)l(ature)f(2-form) f(of)i(the)f(Levi-Civita)i(connection)521 2632 y(with)25 b(resp)r(ect)h(to)g(a)g(lo)r(cal)g(orien)n(ted)g(orthonormal)e(frame)h Fr(s)2069 2642 y Fp(1)2096 2632 y Fu(,)i Fr(s)2172 2642 y Fp(2)2224 2632 y Fu(and)521 2710 y Fr(dv)r(ol)17 b Fu(is)f(the)f(Riemannian)f(v)n(olume)h(form)f Fr(s)1587 2686 y Fo(\003)1587 2727 y Fp(1)1615 2710 y Fq(^)q Fr(s)1692 2686 y Fo(\003)1692 2727 y Fp(2)1735 2710 y Fu(de\014ned)g(b)n(y)h(the) g(metric)521 2789 y(and)h(the)h(orien)n(tation.)27 b(The)16 b(function)h Fq(K)h Fu(is)f(smo)r(oth,)f(and)g(is)h(called)h(the)521 2868 y Fv(Gaussian)29 b(curv)l(ature)c Fu(of)h Fq(h)g Fr(;)37 b Fq(i)25 b Fu(on)h Fr(M)7 b Fu(.)40 b(The)25 b(follo)n(wing)h(Theorem)521 2946 y(asserts)k(that)g(on)g(a)h(closed)g (surface)f(the)h(in)n(tegral)f(of)h(the)f(Gaussian)521 3025 y(curv)l(ature)g(is)i(indep)r(enden)n(t)e(of)h(the)g(metric)g (\(and)f(the)g(orien)n(tation\))521 3104 y(that)21 b(w)n(e)h(ha)n(v)n (e)h(c)n(hosen:)521 3206 y Fv(Theorem)k(19)h Fu(\(Gauss{Bonnet)22 b(Theorem\))p Fv(.)28 b Fn(L)m(et)d Fr(M)33 b Fn(b)m(e)26 b(a)f(c)m(omp)m(act)521 3285 y(oriente)m(d)20 b(surfac)m(e)g(without)g (b)m(oundary.)28 b(F)-5 b(or)20 b(any)f(R)n(iemannian)i(metric)521 3363 y(we)j(have)1018 3435 y Fu(1)p 998 3465 V 998 3527 a(2)p Fr(\031)1089 3389 y Fi(Z)1126 3541 y Fs(M)1191 3480 y Fq(K)q Fr(dv)r(ol)c Fu(=)f Fr(e)p Fu(\()p Fr(M)7 b Fu(\))23 b Fr(;)p eop %%Page: 14 14 14 13 bop 308 159 a Fk(14)521 294 y Fn(wher)m(e)21 b Fr(e)p Fu(\()p Fr(M)7 b Fu(\))p Fn(,)22 b(the)f(Euler)h(numb)m(er)g(of) g Fr(M)7 b Fn(,)22 b(is)g(de\014ne)m(d)f(by)h(mapping)g(the)521 373 y(Euler)k(class)f Fr(e)p Fu(\()p Fr(T)9 b(M)e Fu(\))20 b Fq(2)i Fr(H)1211 348 y Fp(2)1206 390 y Fs(dR)1269 373 y Fu(\()p Fr(M)7 b Fu(\))25 b Fn(to)g Ft(R)i Fn(via)e(the)g (isomorphism)f(given)521 451 y(by)g(the)g(orientation)f(of)g Fr(M)7 b Fn(.)521 582 y(Pr)m(o)m(of.)27 b Fu(Our)h(de\014nitions)g(are) h(suc)n(h)g(that)e(the)i(theorem)e(is)i(tautolog-)521 661 y(ical.)58 b(W)-6 b(e)31 b(de\014ned)f(the)h(Euler)g(class)h(of)f Fr(T)9 b(M)37 b Fu(to)31 b(b)r(e)g(the)f(cohomol-)521 740 y(ogy)21 b(class)i(represen)n(ted)e(b)n(y)g(the)g(closed)h(form) 1708 713 y Fo(\000)p Fp(1)p 1708 724 62 3 v 1712 763 a(2)p Fs(\031)1776 740 y Fu(\012)1824 750 y Fp(12)1874 740 y Fu(,)g(and)f(w)n(e)g(pro)n(v)n(ed)521 818 y(that)g(this)h(is)h (indep)r(enden)n(t)e(of)i(the)f(metric.)29 b(Therefore,)23 b(the)f(image)g(of)521 897 y Fr(e)p Fu(\()p Fr(T)9 b(M)e Fu(\))19 b Fq(2)h Fr(H)867 873 y Fp(2)862 915 y Fs(dR)926 897 y Fu(\()p Fr(M)7 b Fu(\))22 b(in)h Ft(R)h Fu(dep)r(ends)e(only)h (on)f(the)h(orien)n(tation)f(of)h Fr(M)7 b Fu(;)521 976 y(but)24 b(if)h(w)n(e)g(c)n(hange)f(the)h(orien)n(tation,)g(then)f(the) g(Euler)h(class)g(and)f(the)521 1054 y(isomorphism)c Fr(H)967 1030 y Fp(2)962 1072 y Fs(dR)1026 1054 y Fu(\()p Fr(M)7 b Fu(\))18 b(=)i Ft(R)k Fu(b)r(oth)d(c)n(hange)h(b)n(y)g(a)h (sign,)f(and)g(the)g(t)n(w)n(o)521 1133 y(signs)g(cancel.)1402 b Fj(\003)589 1264 y Fu(Recall)19 b(that)e(w)n(e)h(calculated)h(a)g(sp) r(eci\014c)f(example)g(for)g Fr(M)26 b Fu(=)19 b Fr(S)2158 1239 y Fp(2)2202 1264 y Fu(with)521 1343 y(the)25 b(metric)g(induced)g (from)f Ft(R)1302 1318 y Fp(3)1330 1343 y Fu(.)40 b(In)25 b(this)g(case)h(the)f(Gaussian)g(curv)l(a-)521 1421 y(ture)i(is)g (constan)n(t)g(and)g(=)h(1.)45 b(The)27 b(Gauss{Bonnet)f(theorem)g(sho) n(ws)521 1500 y(that)16 b(for)h(an)n(y)g(other)g(metric)f(on)h Fr(S)1375 1475 y Fp(2)1419 1500 y Fu(the)f(a)n(v)n(erage)i(of)f(the)g (Gaussian)g(cur-)521 1579 y(v)l(ature)22 b(is)h(also)g(p)r(ositiv)n(e.) 32 b(F)-6 b(or)23 b(another)f(example)g(consider)h Fr(M)k Fu(=)20 b Fr(T)2286 1554 y Fp(2)2312 1579 y Fu(.)521 1657 y(In)29 b(this)h(case)g(w)n(e)g(can)g(c)n(ho)r(ose)g(a)g(global)g (trivialization)g(of)g(the)f(tan-)521 1736 y(gen)n(t)21 b(bundle)g(giv)n(en)h(b)n(y)f(t)n(w)n(o)h(non-v)l(anishing)e(v)n(ector) i(\014elds)f(whic)n(h)h(are)521 1814 y(ev)n(erywhere)j(linearly)g (indep)r(enden)n(t,)g(for)f(example)h(the)f(v)n(ector)h(\014elds)521 1893 y(tangen)n(t)h(to)g(the)g(factors)g(in)h(a)f(decomp)r(osition)g Fr(T)1817 1869 y Fp(2)1870 1893 y Fu(=)g Fr(S)1992 1869 y Fp(1)2036 1893 y Fq(\002)19 b Fr(S)2152 1869 y Fp(1)2178 1893 y Fu(.)43 b(W)-6 b(e)521 1972 y(can)21 b(de\014ne)g(a)g(metric)f (b)n(y)g(declaring)i(the)e(t)n(w)n(o)h(c)n(hosen)g(v)n(ector)g (\014elds)g(to)521 2050 y(b)r(e)d(orthonormal.)26 b(Then,)19 b(b)n(y)f(de\014nition,)g(this)g(metric)f(is)i(\015at,)g(and)e(the)521 2129 y(Euler)25 b(n)n(um)n(b)r(er)d(of)j(the)f(torus)g(v)l(anishes.)38 b(Therefore,)25 b(the)f(a)n(v)n(erage)i(of)521 2208 y(the)d(Gaussian)h (curv)l(ature)f(of)g(an)n(y)h(metric)f(on)g Fr(T)1770 2183 y Fp(2)1820 2208 y Fu(v)l(anishes,)i(ev)n(en)f(for)521 2286 y(non-\015at)d(metrics.)376 2365 y(\(46\))27 b(The)16 b(Gauss{Bonnet)f(theorem)g(w)n(as)h(tautological,)i(but)d(w)n(e)i(can)f (endo)n(w)521 2444 y(it)h(with)f(non-trivial)h(con)n(ten)n(t)g(b)n(y)f (iden)n(tifying)i(the)e(Euler)h(n)n(um)n(b)r(er)e(with)521 2522 y(other)20 b(quan)n(tities,)g(de\014ned)g(in)g(a)h(di\013eren)n(t) e(w)n(a)n(y)-6 b(.)30 b(W)-6 b(e)21 b(shall)f(do)g(this)g(b)n(y)521 2601 y(generalizing)j(the)f(ab)r(o)n(v)n(e)g(discussion)g(for)g(the)g (torus.)589 2680 y(Let)32 b Fr(X)37 b Fu(b)r(e)31 b(a)h(v)n(ector)g (\014eld)g(with)g(isolated)g(zeros)g(on)g(a)g(compact)521 2758 y(orien)n(ted)f(surface)f Fr(M)38 b Fu(without)29 b(b)r(oundary)-6 b(.)54 b(Let)30 b Fr(p)1873 2768 y Fp(1)1900 2758 y Fr(;)11 b(:)g(:)g(:)j(;)d(p)2081 2768 y Fs(k)2141 2758 y Fu(b)r(e)30 b(the)521 2837 y(zeros,)24 b(and)e Fr(p)868 2847 y Fs(i)907 2837 y Fq(2)e Fr(B)1022 2847 y Fs(i)1061 2837 y Fq(\032)g Fr(M)30 b Fu(disjoin)n(t)22 b(neigh)n(b)r(ourho)r(o)r(ds)f(di\013eomorphic)521 2916 y(to)h(op)r(en)f(balls.)30 b(Set)1053 3099 y Fr(N)c Fu(=)19 b Fr(M)j Fq(n)14 b Fu(\()1388 3015 y Fs(k)1363 3035 y Fi([)1362 3177 y Fs(i)p Fp(=1)1451 3099 y Fr(B)1501 3109 y Fs(i)1520 3099 y Fu(\))21 b Fr(:)521 3283 y Fu(Cho)r(ose)29 b(a)h(metric)f Fq(h)h Fr(;)41 b Fq(i)30 b Fu(on)f Fr(T)9 b(M)36 b Fu(whose)30 b(restriction)f(to)g(the)h Fr(B)2238 3293 y Fs(i)2286 3283 y Fu(is)521 3362 y(\015at.)f(On)20 b Fr(N)28 b Fu(w)n(e)21 b(de\014ne)f(a)g(frame)g(b)n(y)h(setting)f Fr(X)1736 3372 y Fp(1)1781 3362 y Fu(=)f Fr(X=)p Fq(j)p Fr(X)5 b Fq(j)p Fu(,)20 b(and)g(then)521 3440 y(c)n(ho)r(osing)j(for)g Fr(X)947 3450 y Fp(2)997 3440 y Fu(the)g(unique)g(v)n(ector)g(\014eld)g (whic)n(h)g(completes)g Fr(X)2222 3450 y Fp(1)2272 3440 y Fu(to)521 3519 y(a)28 b(p)r(ositiv)n(ely)g(orien)n(ted)g(orthonormal) e(frame.)47 b(Let)28 b Fr(!)1924 3529 y Fp(12)2002 3519 y Fu(and)g(\012)2185 3529 y Fp(12)2263 3519 y Fu(b)r(e)p eop %%Page: 15 15 15 14 bop 2279 159 a Fk(15)521 294 y Fu(connection)22 b(and)g(curv)l(ature)f(forms)g(with)h(resp)r(ect)g(to)g(this)g(frame.) 29 b(As)521 373 y(\012)569 383 y Fp(12)641 373 y Fu(v)l(anishes)23 b(on)e(the)h Fr(B)1156 383 y Fs(i)1197 373 y Fu(w)n(e)g(ha)n(v)n(e)596 518 y Fr(e)p Fu(\()p Fr(M)7 b Fu(\))18 b(=)865 472 y(1)p 845 502 73 3 v 845 564 a(2)p Fr(\031)936 426 y Fi(Z)973 579 y Fs(M)1038 518 y Fq(K)q Fr(dv)r(ol)i Fu(=)1307 472 y Fq(\000)p Fu(1)p 1307 502 86 3 v 1313 564 a(2)p Fr(\031)1411 426 y Fi(Z)1448 579 y Fs(M)1513 518 y Fu(\012)1561 528 y Fp(12)1630 518 y Fu(=)1706 472 y Fq(\000)p Fu(1)p 1706 502 V 1712 564 a(2)p Fr(\031)1810 426 y Fi(Z)1847 579 y Fs(N)1904 518 y Fu(\012)1952 528 y Fp(12)2024 518 y Fr(:)521 668 y Fu(By)i(Stok)n(es's)g(Theorem)f(this)g(in)n(tegral)h (equals)884 801 y Fq(\000)p Fu(1)p 884 831 V 890 893 a(2)p Fr(\031)988 755 y Fi(Z)1025 908 y Fs(@)s(N)1110 847 y Fr(!)1151 857 y Fp(12)1220 847 y Fu(=)1326 763 y Fs(k)1290 783 y Fi(X)1300 925 y Fs(i)p Fp(=1)1425 801 y Fu(1)p 1405 831 73 3 v 1405 893 a(2)p Fr(\031)1496 755 y Fi(Z)1534 908 y Fs(@)s(B)1598 915 y Fm(i)1629 847 y Fr(!)1670 857 y Fp(12)1743 847 y Fr(;)521 1021 y Fu(with)16 b(the)f(c)n(hange)i(in)f(sign)g(due)g(to)g(the)f(c)n(hange)i(in)f (orien)n(tation)g(dep)r(end-)521 1100 y(ing)i(on)f(whether)g(a)g(giv)n (en)h(circle)h(is)f(view)n(ed)g(as)g(b)r(eing)g(in)f(the)h(b)r(oundary) 521 1179 y(of)k Fr(N)29 b Fu(or)22 b(of)g Fr(B)884 1189 y Fs(i)903 1179 y Fu(.)589 1257 y(Consider)f(the)h(v)n(ector)g(\014eld) h Fr(X)1375 1267 y Fp(1)1423 1257 y Fu(as)g(a)f(map)f Fr(X)1762 1267 y Fp(1)1796 1257 y Fu(:)i Fr(@)t(B)1926 1267 y Fs(i)1964 1257 y Fq(!)c Fr(S)2095 1233 y Fp(1)2121 1257 y Fu(,)k(where)521 1336 y(the)31 b(target)f(is)i(the)e(unit)h (circle)h(in)f Ft(R)1508 1312 y Fp(2)1568 1336 y Fu(considered)g(as)g (the)g(tangen)n(t)521 1415 y(space)23 b(to)g(the)f(t)n(w)n (o-dimensional)g(disk.)32 b(Then)22 b(w)n(e)h(calculate)g(that)f(the) 521 1493 y(pullbac)n(k)31 b(of)f(the)h(standard)e(v)n(olume)g(form)h (on)g Fr(S)1829 1469 y Fp(1)1886 1493 y Fu(is)h(precisely)g(the)521 1572 y(connection)22 b(form)f(with)g(resp)r(ect)h(to)g Fr(X)1519 1582 y Fp(1)1545 1572 y Fu(,)h Fr(X)1641 1582 y Fp(2)1667 1572 y Fu(:)1132 1688 y Fr(X)1192 1660 y Fo(\003)1187 1705 y Fp(1)1219 1688 y Fr(d\022)e Fu(=)e Fr(!)1416 1698 y Fp(12)1488 1688 y Fr(:)521 1804 y Fu(Th)n(us,)369 1907 y(1)p 349 1937 V 349 1998 a(2)p Fr(\031)440 1860 y Fi(Z)477 2013 y Fs(@)s(B)541 2020 y Fm(i)573 1952 y Fr(!)614 1962 y Fp(12)683 1952 y Fu(=)780 1907 y(1)p 760 1937 V 760 1998 a(2)p Fr(\031)851 1860 y Fi(Z)889 2013 y Fs(@)s(B)953 2020 y Fm(i)984 1952 y Fr(X)1044 1924 y Fo(\003)1039 1969 y Fp(1)1071 1952 y Fr(d\022)i Fu(=)1253 1907 y(1)p 1233 1937 V 1233 1998 a(2)p Fr(\031)1313 1952 y(deg)r Fu(\()p Fr(X)1493 1962 y Fp(1)1519 1952 y Fq(j)1538 1962 y Fs(@)s(B)1602 1969 y Fm(i)1622 1952 y Fu(\))1659 1860 y Fi(Z)1697 2013 y Fs(S)1729 2000 y Fl(1)1766 1952 y Fr(d\022)f Fu(=)f Fr(deg)r Fu(\()p Fr(X)2101 1962 y Fp(1)2127 1952 y Fq(j)2146 1962 y Fs(@)s(B)2210 1969 y Fm(i)2231 1952 y Fu(\))i Fr(:)521 2110 y Fu(Th)n(us)g(w)n(e)i (ha)n(v)n(e)f(pro)n(v)n(ed)f(the)h(follo)n(wing:)521 2207 y Fv(Theorem)40 b(20)h Fu(\(P)n(oincar)n(\023)-31 b(e{Hopf)36 b(Theorem\))p Fv(.)e Fn(L)m(et)i Fr(M)43 b Fn(b)m(e)37 b(a)f(c)m(om-)521 2286 y(p)m(act)27 b(oriente)m(d)h (surfac)m(e)g(without)f(b)m(oundary.)43 b(If)28 b Fr(X)33 b Fn(is)28 b(a)g(ve)m(ctor)g(\014eld)521 2364 y(with)36 b(isolate)m(d)f(zer)m(os)h Fr(p)1139 2374 y Fp(1)1165 2364 y Fr(;)11 b(:)g(:)g(:)j(;)d(p)1346 2374 y Fs(k)1411 2364 y Fn(on)36 b Fr(M)7 b Fn(,)40 b(then)c(ther)m(e)g(ar)m(e)f(inte)m (gers)521 2443 y Fr(I)5 b(nd)p Fu(\()p Fr(X)r(;)11 b(p)773 2453 y Fs(i)793 2443 y Fu(\))23 b Fn(dep)m(ending)h(on)f(the)h(ve)m (ctor)g(\014eld)f Fr(X)29 b Fn(such)24 b(that)770 2580 y Fu(1)p 750 2610 V 750 2671 a(2)p Fr(\031)841 2533 y Fi(Z)878 2686 y Fs(M)943 2625 y Fq(K)q Fr(dv)r(ol)d Fu(=)e Fr(e)p Fu(\()p Fr(M)7 b Fu(\))17 b(=)1483 2541 y Fs(k)1447 2561 y Fi(X)1457 2703 y Fs(i)p Fp(=1)1556 2625 y Fr(I)5 b(nd)p Fu(\()p Fr(X)r(;)11 b(p)1808 2635 y Fs(i)1828 2625 y Fu(\))23 b Fr(:)521 2800 y Fn(In)i(p)m(articular,)e(the)i(index) f(sum)h(on)f(the)g(right-hand-side)f(is)h(indep)m(en-)521 2878 y(dent)g(of)f(the)h(ve)m(c)m(ctor)f(\014eld)h Fr(X)5 b Fn(.)589 2976 y Fu(The)30 b(indices)h Fr(I)5 b(nd)p Fu(\()p Fr(X)r(;)11 b(p)1210 2986 y Fs(i)1230 2976 y Fu(\))30 b(of)h(the)f(zero)r(es)h(are)g(de\014ned)e(to)i(b)r(e)f(the) 521 3054 y(winding)22 b(n)n(um)n(b)r(ers)e(giv)n(en)i(b)n(y)g(the)f (degrees)i Fr(deg)r Fu(\()p Fr(X)1826 3064 y Fp(1)1852 3054 y Fq(j)1871 3064 y Fs(@)s(B)1935 3071 y Fm(i)1955 3054 y Fu(\).)1083 3208 y(19)f(Jan)n(uary)e(2004)376 3362 y(\(47\))27 b(W)-6 b(e)27 b(discuss)f(examples)g(of)h(indices)g (of)f(zeros)h(of)f(v)n(ector)g(\014elds)h(in)f(the)521 3440 y(plane)15 b(and)f(on)g(compact)g(manifolds.)26 b(As)16 b(a)e(corollary)h(of)g(the)f(P)n(oincar)n(\023)-31 b(e{)521 3519 y(Hopf)29 b(Theorem)f(20)h(w)n(e)g(\014nd)f(that)g(the)g (Euler)h(n)n(um)n(b)r(er)e(of)i(a)g(closed)p eop %%Page: 16 16 16 15 bop 308 159 a Fk(16)521 294 y Fu(orien)n(ted)19 b(surface)f(if)h(gen)n(us)g Fr(g)h Fu(is)f(2)8 b Fq(\000)g Fu(2)p Fr(g)r Fu(.)29 b(This)18 b(c)n(hec)n(ks)i(with)e(the)g(calcu-) 521 373 y(lations)k(w)n(e)f(ha)n(v)n(e)h(done)f(for)g(the)g(sphere)g (and)g(the)g(torus,)g(where)g Fr(g)g Fu(=)e(0)521 451 y(and)i Fr(g)g Fu(=)e(1)j(resp)r(ectiv)n(ely)-6 b(.)376 530 y(\(48\))27 b(W)-6 b(e)28 b(calculate)h(the)e(Gaussian)g(curv)l (ature)g(explicitly)i(for)e(the)g(metric)521 609 y(induced)17 b(on)g Fr(T)897 584 y Fp(2)941 609 y Fu(b)n(y)g(an)g(em)n(b)r(edding)e (in)j Ft(R)1563 584 y Fp(3)1608 609 y Fu(as)g(a)f(surface)g(of)h(rev)n (olution.)1083 756 y(22)k(Jan)n(uary)e(2004)376 903 y(\(49\))27 b(Our)f(calculations)g(for)g Fr(S)1174 878 y Fp(2)1225 903 y Fq(\032)g Ft(R)1350 878 y Fp(3)1404 903 y Fu(and)f(for)h Fr(T)1689 878 y Fp(2)1741 903 y Fq(\032)f Ft(R)1865 878 y Fp(3)1920 903 y Fu(in)g(\(44\))h(and)f(in)521 981 y(\(48\))c(ab)r(o)n (v)n(e)i(in)n(v)n(olv)n(e)f(sp)r(ecial)h(cases)g(of)f(the)g(follo)n (wing:)521 1077 y Fv(Lemma)c(21.)23 b Fn(L)m(et)18 b Fu(\()p Fr(N)s(;)11 b Fq(h)18 b Fr(;)30 b Fq(i)p Fu(\))17 b Fn(b)m(e)i(a)e(R)n(iemannian)i(manifold,)f(and)g Fr(M)25 b Fq(\032)521 1156 y Fr(N)i Fn(a)19 b(submanifold.)30 b(The)20 b(L)m(evi-Civita)h(c)m(onne)m(ction)e Fq(r)1893 1131 y Fs(M)1967 1156 y Fn(of)g(the)h(metric)521 1234 y(on)33 b Fr(M)40 b Fn(obtaine)m(d)33 b(by)h(r)m(estricting)f Fq(h)h Fr(;)45 b Fq(i)33 b Fn(to)g(the)g(subbund)s(le)i Fr(T)9 b(M)43 b Fq(\032)521 1313 y Fr(T)9 b(N)e Fq(j)648 1323 y Fs(M)725 1313 y Fn(is)23 b(given)i(by)1076 1431 y Fq(r)1132 1403 y Fs(M)1132 1448 y(X)1185 1431 y Fr(Y)34 b Fu(=)19 b Fr(\031)r Fq(r)1422 1403 y Fs(N)1436 1444 y Fp(~)1422 1456 y Fs(X)1479 1414 y Fu(~)1469 1431 y Fr(Y)38 b(;)521 1557 y Fn(wher)m(e)724 1540 y Fu(~)704 1557 y Fr(X)26 b Fn(and)920 1540 y Fu(~)910 1557 y Fr(Y)36 b Fn(ar)m(e)20 b(lo)m(c)m(al)g(extensions)h(of)g Fr(X)r(;)11 b(Y)34 b Fq(2)18 b(X)10 b Fu(\()p Fr(M)d Fu(\))20 b Fn(to)h Fr(N)7 b Fn(,)21 b Fq(r)2285 1533 y Fs(N)521 1636 y Fn(is)e(the)f(L)m (evi-Civita)h(c)m(onne)m(ction)g(of)f Fu(\()p Fr(N)s(;)11 b Fq(h)19 b Fr(;)30 b Fq(i)p Fu(\))p Fn(,)19 b(and)f Fr(\031)10 b Fu(:)23 b Fr(T)9 b(N)e Fq(j)2054 1646 y Fs(M)2126 1636 y Fq(!)19 b Fr(T)9 b(M)521 1715 y Fn(is)33 b(the)f(pr)m(oje)m(ction)g(along)g(the)g(ortho)m(gonal)e(c)m(omplement) i(of)h Fr(T)9 b(M)41 b Fq(\032)521 1793 y Fr(T)9 b(N)e Fq(j)648 1803 y Fs(M)725 1793 y Fn(with)23 b(r)m(esp)m(e)m(ct)g(to)g Fq(h)h Fr(;)35 b Fq(i)p Fn(.)376 1889 y Fu(\(50\))27 b(Supp)r(ose)j(no)n(w)g(that)g Fr(M)37 b Fu(has)30 b(co)r(dimension)g (one)h(in)g Fr(N)7 b Fu(.)55 b(If)31 b Fr(M)37 b Fu(and)521 1968 y Fr(N)h Fu(are)32 b(orien)n(ted,)i(then)c(the)h(normal)f(bundle)h (inherits)g(an)g(induced)521 2046 y(orien)n(tation.)h(W)-6 b(e)23 b(c)n(ho)r(ose)g(a)g(p)r(ositiv)n(ely)g(orien)n(ted)f(lo)r(cal)i (orthonormal)521 2125 y(frame)15 b Fr(s)731 2135 y Fp(1)757 2125 y Fr(;)c(:)g(:)g(:)j(;)d(s)936 2135 y Fs(n)984 2125 y Fu(for)k Fr(T)9 b(N)22 b Fu(at)15 b Fr(p)k Fq(2)f Fr(M)23 b Fu(in)15 b(suc)n(h)g(a)h(w)n(a)n(y)f(that)f Fr(s)2032 2135 y Fp(1)2059 2125 y Fr(;)d(:)g(:)g(:)i(;)e(s)2237 2135 y Fs(n)p Fo(\000)p Fp(1)521 2203 y Fu(is)23 b(a)f(p)r(ositiv)n (ely)g(orien)n(ted)g(orthonormal)e(frame)h(for)h Fr(T)9 b(M)e Fu(,)22 b(and)f Fr(s)2177 2213 y Fs(n)2231 2203 y Fu(is)i(a)521 2282 y(p)r(ositiv)n(ely)g(orien)n(ted)e(frame)h(for)f (the)h(normal)f(space.)521 2378 y Fv(De\014nition)k(22.)j Fu(The)21 b Fv(W)-6 b(eingarten)24 b(map)d Fu(at)g Fr(p)e Fq(2)g Fr(M)28 b Fu(is)21 b(the)g(linear)521 2456 y(map)1049 2569 y Fr(L)7 b Fu(:)24 b Fr(T)1182 2579 y Fs(p)1208 2569 y Fr(M)i Fq(\000)-11 b(!)19 b Fr(T)1464 2579 y Fs(p)1491 2569 y Fr(M)1245 2679 y(v)i Fq(7\000)-11 b(!)19 b(r)1481 2652 y Fs(N)1481 2696 y(v)1527 2679 y Fr(s)1558 2689 y Fs(n)589 2796 y Fu(This)g(has)g(the)g(follo)n(wing)h(easily)g(pro)n (v)n(ed)f(prop)r(ert)n(y)-6 b(,)19 b(see)h([1],)h(Section)521 2875 y(10.3.)521 2971 y Fv(Lemma)g(23.)k Fn(The)c(Weingarten)g(map)f (is)h(symmetric)g(with)f(r)m(esp)m(e)m(ct)g(to)521 3049 y(the)k(metric,)g(i.)g(e.)31 b Fq(h)p Fr(L)p Fu(\()p Fr(X)5 b Fu(\))p Fr(;)11 b(Y)k Fq(i)k Fu(=)g Fq(h)p Fr(X)r(;)11 b(L)p Fu(\()p Fr(Y)k Fu(\))p Fq(i)23 b Fn(for)h(al)s(l)f Fr(X)r(;)11 b(Y)34 b Fq(2)18 b Fr(T)2155 3059 y Fs(p)2182 3049 y Fr(M)7 b Fn(.)376 3154 y Fu(\(51\))27 b(F)-6 b(or)25 b Fr(X)r(;)11 b(Y)t(;)g(Z)29 b Fq(2)23 b Fr(T)980 3164 y Fs(p)1006 3154 y Fr(M)31 b Fu(w)n(e)25 b(can)g(relate)g(the)f(v)l (alues)h Fr(F)1878 3130 y Fo(r)1918 3114 y Fm(N)1960 3154 y Fu(\()p Fr(X)r(;)11 b(Y)j Fu(\))p Fr(Z)29 b Fu(and)521 3242 y(of)17 b Fr(F)643 3217 y Fo(r)683 3202 y Fm(M)732 3242 y Fu(\()p Fr(X)r(;)11 b(Y)j Fu(\))p Fr(Z)22 b Fu(of)17 b(the)g(curv)l(atures)g(of)g Fq(r)1611 3217 y Fs(N)1674 3242 y Fu(and)f(of)i Fq(r)1924 3217 y Fs(M)1994 3242 y Fu(through)e(the)521 3321 y(W)-6 b(eingarten)22 b(map.)30 b(In)21 b(the)h(sp)r(ecial)h(case)g(that)f Fq(r)1798 3296 y Fs(N)1866 3321 y Fu(is)g(\015at)g(w)n(e)g(obtain)521 3399 y(the)g(follo)n(wing)h(classical)g(iden)n(tities:)564 3519 y Fr(F)616 3491 y Fo(r)656 3475 y Fm(M)704 3519 y Fu(\()p Fr(X)r(;)11 b(Y)k Fu(\))p Fr(Z)23 b Fu(=)c Fq(h)p Fr(L)p Fu(\()p Fr(Y)c Fu(\))p Fr(;)c(Z)5 b Fq(i)22 b Fr(L)p Fu(\()p Fr(X)5 b Fu(\))15 b Fq(\000)g(h)p Fr(L)p Fu(\()p Fr(X)5 b Fu(\))p Fr(;)11 b(Z)5 b Fq(i)22 b Fr(L)p Fu(\()p Fr(Y)15 b Fu(\))22 b Fr(;)p eop %%Page: 17 17 17 16 bop 2279 159 a Fk(17)815 297 y Fr(L)p Fu(\([)p Fr(X)r(;)11 b(Y)k Fu(]\))k(=)g Fq(r)1232 269 y Fs(M)1232 313 y(X)1286 297 y Fr(L)p Fu(\()p Fr(Y)14 b Fu(\))h Fq(\000)g(r)1573 269 y Fs(M)1573 313 y(Y)1627 297 y Fr(L)p Fu(\()p Fr(X)5 b Fu(\))21 b Fr(:)521 390 y Fu(The)29 b(\014rst)g(of)g(these)h(is)g (the)f(so-called)h Fv(Gauss)j(equation)p Fu(,)f(and)d(the)521 469 y(second)22 b(is)h(the)e Fv(Co)r(dazzi{Mainardi)k(equation)1827 443 y Fp(4)1851 469 y Fu(.)1083 604 y(26)d(Jan)n(uary)e(2004)376 740 y(\(52\))27 b(Let)g Fr(M)34 b Fq(\032)28 b Ft(R)869 716 y Fs(n)930 740 y Fu(b)r(e)e(an)h(orien)n(ted)g(smo)r(oth)e(h)n(yp)r (ersurface.)44 b(W)-6 b(e)28 b(orien)n(t)521 819 y Ft(R)568 794 y Fs(n)624 819 y Fu(through)20 b(the)i(standard)e(basis.)30 b(The)22 b Fv(Gauss)i(map)1092 926 y Fr(G)7 b Fu(:)24 b Fr(M)h Fq(\000)-11 b(!)20 b Fr(S)1455 898 y Fs(n)p Fo(\000)p Fp(1)1230 1028 y Fr(p)f Fq(7\000)-11 b(!)20 b Fr(G)p Fu(\()p Fr(p)p Fu(\))521 1136 y(is)33 b(de\014ned)f(b)n(y)g (assigning)h(to)f(eac)n(h)i(p)r(oin)n(t)d(of)i Fr(M)40 b Fu(the)32 b(unit)g(normal)521 1214 y(v)n(ector)d Fr(G)p Fu(\()p Fr(p)p Fu(\))g(to)f Fr(M)36 b Fu(for)29 b(whic)n(h)f Fr(s)1413 1224 y Fp(1)1440 1214 y Fu(\()p Fr(p)p Fu(\))p Fr(;)11 b(:)g(:)g(:)i(;)e(s)1703 1224 y Fs(n)p Fo(\000)p Fp(1)1796 1214 y Fu(\()p Fr(p)p Fu(\))p Fr(;)g(G)p Fu(\()p Fr(p)p Fu(\))29 b(is)g(a)g(p)r(os-)521 1293 y(itiv)n(ely)g(orien)n(ted) e(frame)g(of)h Ft(R)1306 1269 y Fs(n)1367 1293 y Fu(for)g(an)n(y)f(p)r (ositiv)n(ely)h(orien)n(ted)g(frame)521 1372 y Fr(s)552 1382 y Fp(1)579 1372 y Fu(\()p Fr(p)p Fu(\))p Fr(;)11 b(:)g(:)g(:)i(;)e(s)842 1382 y Fs(n)p Fo(\000)p Fp(1)935 1372 y Fu(\()p Fr(p)p Fu(\))22 b(of)g Fr(T)1156 1382 y Fs(p)1182 1372 y Fr(M)7 b Fu(.)521 1465 y Fv(Lemma)16 b(24.)23 b Fn(Ther)m(e)17 b(is)g(a)f(natur)m(al)g(identi\014c)m(ation)h (of)g Fr(T)1919 1475 y Fs(p)1946 1465 y Fr(M)23 b Fn(with)17 b Fr(T)2209 1475 y Fs(G)p Fp(\()p Fs(p)p Fp(\))2310 1465 y Fr(S)2355 1440 y Fs(n)p Fo(\000)p Fp(1)2447 1465 y Fn(.)521 1543 y(Under)25 b(this)f(identi\014c)m(ation,)h(the)g (derivative)g(of)f(the)h(Gauss)g(map)e(at)i Fr(p)521 1622 y Fn(c)m(oincides)f(with)g(the)f(Weingarten)h(map.)521 1736 y(Pr)m(o)m(of.)j Fu(Both)c Fr(T)928 1746 y Fs(p)954 1736 y Fr(M)31 b Fu(and)23 b Fr(T)1218 1746 y Fs(G)p Fp(\()p Fs(p)p Fp(\))1318 1736 y Fr(S)1363 1711 y Fs(n)p Fo(\000)p Fp(1)1479 1736 y Fu(can)h(b)r(e)f(iden)n(ti\014ed)h(with)f (the)g(or-)521 1814 y(thogonal)33 b(complemen)n(t)f(of)i Fr(G)p Fu(\()p Fr(p)p Fu(\))e(in)i Ft(R)1576 1790 y Fs(n)1609 1814 y Fu(.)64 b(The)33 b(deriv)l(ativ)n(e)h(of)g(the)521 1893 y(Gauss)15 b(map)f(can)i(b)r(e)f(calculated)i(from)d(the)h (de\014nition,)i(see)f([1],)i(Section)521 1972 y(10.3.)1622 b Fj(\003)376 2085 y Fu(\(53\))27 b(Assume)k(no)n(w)g(that)g Fr(n)k Fu(=)h(3.)58 b(In)31 b(this)g(case)h(w)n(e)g(can)g(compute)e (the)521 2164 y(curv)l(ature)15 b(of)h(the)f(surface)h Fr(M)23 b Fu(from)15 b(the)g(Gauss)g(map)g(in)g(a)h(particularly)521 2243 y(straigh)n(tforw)n(ard)k(manner:)521 2336 y Fv(Theorem)25 b(25.)j Fn(The)c(curvatur)m(e)h(form)f Fu(\012)1605 2311 y Fs(M)1605 2352 y Fp(12)1682 2336 y Fn(of)f(the)h(L)m(evi-Civita)h(c)m (on-)521 2414 y(ne)m(ction)33 b(of)g(the)g(metric)g(induc)m(e)m(d)g(on) g Fr(M)39 b Fn(fr)m(om)32 b(the)h(sc)m(alar)f(pr)m(o)m(duct)521 2493 y(on)25 b Ft(R)664 2468 y Fp(3)718 2493 y Fn(is)g(given)i(by)e Fq(\000)p Fr(G)1155 2468 y Fo(\003)1183 2493 y Fr(dv)r(ol)1303 2504 y Fs(S)1335 2491 y Fl(2)1360 2493 y Fn(,)h(wher)m(e)f Fr(dv)r(ol)1714 2504 y Fs(S)1746 2491 y Fl(2)1797 2493 y Fn(is)h(the)f(R)n(iemannian)521 2571 y(volume)f(form)g(of)f(the)h (unit)g(spher)m(e.)521 2685 y(Pr)m(o)m(of.)j Fu(Cho)r(ose)k(an)g(orien) n(ted)h(lo)r(cal)g(orthonormal)d(frame)i Fr(s)2106 2695 y Fp(1)2133 2685 y Fr(;)11 b(s)2193 2695 y Fp(2)2252 2685 y Fu(for)521 2764 y Fr(T)e(M)e Fu(,)35 b(and)d(complete)g(it)g (with)f Fr(s)1388 2774 y Fp(3)1415 2764 y Fu(\()p Fr(p)p Fu(\))36 b(=)g Fr(G)p Fu(\()p Fr(p)p Fu(\))c(to)g(an)g(orien)n(ted)g (or-)521 2842 y(thonormal)25 b(frame)h(for)h Ft(R)1188 2818 y Fp(3)1216 2842 y Fu(.)44 b(Let)27 b Fr(!)1442 2852 y Fs(ij)1510 2842 y Fu(b)r(e)g(the)f(connection)h(matrix)f(of)521 2921 y(the)31 b(Levi-Civita)g(connection)g(of)g Ft(R)1468 2897 y Fp(3)1528 2921 y Fu(with)f(resp)r(ect)h(to)g(this)f(frame.)521 3000 y(Then,)24 b(according)g(to)g(Lemma)e(21)i(the)g(upp)r(er)e (left-hand)i(t)n(w)n(o-b)n(y-t)n(w)n(o)521 3078 y(submatrix)29 b(of)i Fr(!)969 3088 y Fs(ij)1041 3078 y Fu(is)g(the)f(connection)h (matrix)f(of)h(the)f(Levi-Civita)521 3157 y(connection)22 b(of)g(the)g(induced)g(metric)f(on)h Fr(M)7 b Fu(.)589 3236 y(F)-6 b(rom)21 b(the)h(de\014nition)f(of)h(the)g(Gauss)f(map)g(w) n(e)h(calculate)801 3343 y Fr(G)853 3315 y Fo(\003)880 3343 y Fr(dv)r(ol)1000 3354 y Fs(S)1032 3341 y Fl(2)1058 3343 y Fu(\()p Fr(s)1115 3353 y Fp(1)1141 3343 y Fr(;)11 b(s)1201 3353 y Fp(2)1228 3343 y Fu(\))19 b(=)g Fr(!)1384 3353 y Fp(13)1450 3343 y Fq(^)14 b Fr(!)1550 3353 y Fp(23)1601 3343 y Fu(\()p Fr(s)1658 3353 y Fp(1)1685 3343 y Fr(;)d(s)1745 3353 y Fp(2)1772 3343 y Fu(\))22 b Fr(:)p 308 3385 338 3 v 376 3430 a Fp(4)400 3451 y Fh(See)k([1)o(],)h(Section)g(10.3,)g (but)g(b)r(ew)n(are)e(of)i(the)f(sign)h(error)f(in)g(the)g(Co)r (dazzi{Mainardi)308 3519 y(equation.)p eop %%Page: 18 18 18 17 bop 308 159 a Fk(18)521 294 y Fu(Th)n(us)22 b Fr(G)741 270 y Fo(\003)768 294 y Fr(dv)r(ol)888 305 y Fs(S)920 293 y Fl(2)965 294 y Fu(=)e Fr(!)1077 304 y Fp(13)1144 294 y Fq(^)15 b Fr(!)1245 304 y Fp(23)1296 294 y Fu(.)31 b(On)23 b(the)f(other)g(hand,)g(the)g(\015atness)h(of)521 375 y Ft(R)568 350 y Fp(3)618 375 y Fu(giv)n(es)f Fr(d!)854 385 y Fp(12)920 375 y Fq(\000)14 b Fr(!)1027 385 y Fp(13)1092 375 y Fq(^)g Fr(!)1192 385 y Fp(32)1262 375 y Fu(=)19 b(0,)j(as)g(\012)1534 350 y Fc(R)1570 335 y Fl(3)1534 392 y Fp(12)1611 375 y Fu(=)d(0.)29 b(Com)n(bining)20 b(the)i(t)n(w)n(o,)521 454 y(w)n(e)g(obtain:)569 578 y(\012)617 551 y Fs(M)617 595 y Fp(12)689 578 y Fu(=)d Fr(d!)834 588 y Fp(12)904 578 y Fu(=)g Fr(!)1015 588 y Fp(13)1081 578 y Fq(^)c Fr(!)1182 588 y Fp(32)1251 578 y Fu(=)k Fq(\000)p Fr(!)1414 588 y Fp(13)1480 578 y Fq(^)c Fr(!)1581 588 y Fp(23)1650 578 y Fu(=)k Fq(\000)p Fr(G)1824 551 y Fo(\003)1852 578 y Fr(dv)r(ol)1972 590 y Fs(S)2004 577 y Fl(2)2051 578 y Fr(:)2278 703 y Fj(\003)589 838 y Fu(The)h(Gauss)g(curv)l(ature)g Fq(K)i Fu(w)n(as)f(de\014ned)f(b) n(y)g(the)h(equation)f Fq(\000)p Fu(\012)2207 814 y Fs(M)2207 855 y Fp(12)2279 838 y Fu(=)521 917 y Fq(K)q Fr(dv)r(ol)693 927 y Fs(M)747 917 y Fu(.)30 b(Th)n(us,)21 b(w)n(e)h(obtain)1094 1074 y Fq(K)e Fu(=)1242 1029 y Fr(G)1294 1004 y Fo(\003)1320 1029 y Fr(dv)r(ol)1440 1040 y Fs(S)1472 1027 y Fl(2)p 1242 1059 257 3 v 1283 1121 a Fr(dv)r(ol)1403 1131 y Fs(M)1527 1074 y Fr(;)521 1229 y Fu(sho)n(wing)29 b(that)g(the)g(Gauss) g(curv)l(ature)f(measures)h(the)g(distortion)g(of)521 1308 y(signed)22 b(area)g(under)f(the)h(Gauss)f(map.)29 b(In)21 b(particular,)521 1410 y Fv(Corollary)29 b(26.)h Fn(F)-5 b(or)26 b Fr(p)d Fq(2)h Fr(M)30 b Fq(\032)24 b Ft(R)1457 1385 y Fp(3)1512 1410 y Fn(the)i(Gauss)h(curvatur)m(e)f Fq(K)q Fu(\()p Fr(p)p Fu(\))h Fn(is)521 1488 y(non-zer)m(o)20 b(if)i(and)e(only)h(if)g(the)g(Gauss)g(map)g(is)g(a)g(lo)m(c)m(al)f (di\013e)m(omorphism)521 1567 y(at)28 b Fr(p)p Fn(.)45 b(In)29 b(this)f(c)m(ase)g(the)h(sign)g(of)f Fq(K)q Fu(\()p Fr(p)p Fu(\))g Fn(is)h(p)m(ositive)f(or)g(ne)m(gative)h(de-)521 1646 y(p)m(ending)24 b(on)g(whether)f Fr(G)h Fn(is)g(orientation-pr)m (eserving)g(or)f(r)m(eversing)i(at)521 1724 y Fr(p)p Fn(.)589 1826 y Fu(The)f(sign)g(of)h(the)f(curv)l(ature)g(can)g(b)r(e)h (seen)g(b)n(y)f(c)n(hec)n(king)h(whether,)521 1905 y(lo)r(cally)f(near) e Fr(p)p Fu(,)h Fr(M)30 b Fu(lies)23 b(en)n(tirely)g(on)f(one)h(side)g (of)g(the)f(tangen)n(t)g(plane)521 1983 y Fr(T)560 1993 y Fs(p)587 1983 y Fr(M)29 b Fu(as)22 b(an)f(a\016ne)h(subspace)g(of)g Ft(R)1426 1959 y Fp(3)1477 1983 y Fu(through)e(the)i(p)r(oin)n(t)f Fr(p)p Fu(.)589 2062 y(In)n(tegrating)g(the)g(equalit)n(y)h(in)g (Theorem)f(25)h(w)n(e)g(obtain:)521 2164 y Fv(Corollary)31 b(27.)g Fn(L)m(et)d Fr(M)33 b Fq(\032)27 b Ft(R)1333 2139 y Fp(3)1389 2164 y Fn(b)m(e)h(a)g(close)m(d)f(oriente)m(d)h (surfac)m(e.)43 b(Its)521 2242 y(Euler)24 b(numb)m(er)g Fr(e)p Fu(\()p Fr(M)7 b Fu(\))22 b Fn(is)i(twic)m(e)g(the)g(de)m(gr)m (e)m(e)f(of)g(the)h(Gauss)g(map.)521 2378 y(Pr)m(o)m(of.)j Fu(By)21 b(de\014nition)h(and)f(b)n(y)h(Theorem)f(25)h(w)n(e)g(ha)n(v)n (e:)604 2534 y Fr(e)p Fu(\()p Fr(M)7 b Fu(\))18 b(=)853 2489 y Fq(\000)p Fu(1)p 853 2519 86 3 v 859 2581 a(2)p Fr(\031)956 2443 y Fi(Z)994 2595 y Fs(M)1058 2534 y Fu(\012)1106 2507 y Fs(M)1106 2551 y Fp(12)1178 2534 y Fu(=)1275 2489 y(1)p 1255 2519 73 3 v 1255 2581 a(2)p Fr(\031)1346 2443 y Fi(Z)1383 2595 y Fs(M)1448 2534 y Fr(G)1500 2507 y Fo(\003)1527 2534 y Fr(dv)r(ol)1647 2546 y Fs(S)1679 2533 y Fl(2)805 2757 y Fu(=)902 2712 y(1)p 882 2742 V 882 2804 a(2)p Fr(\031)961 2757 y(deg)r Fu(\()p Fr(G)p Fu(\))1175 2666 y Fi(Z)1212 2818 y Fs(S)1244 2806 y Fl(2)1281 2757 y Fr(dv)r(ol)1401 2769 y Fs(S)1433 2756 y Fl(2)1478 2757 y Fu(=)1575 2712 y(1)p 1555 2742 V 1555 2804 a(2)p Fr(\031)1634 2757 y(deg)r Fu(\()p Fr(G)p Fu(\))k(4)p Fr(\031)f Fu(=)e(2)p Fr(deg)r Fu(\()p Fr(G)p Fu(\))j Fr(:)2278 2902 y Fj(\003)376 3037 y Fu(\(54\))27 b(Here)f(is)f(y)n(et)g (another)f(w)n(a)n(y)h(to)g(calculate)h(the)f(Gauss)f(curv)l(ature)h (of)g(a)521 3116 y(surface:)521 3217 y Fv(Lemma)41 b(28.)35 b Fn(L)m(et)h Fr(s)1095 3227 y Fp(1)1122 3217 y Fr(;)11 b(s)1182 3227 y Fp(2)1246 3217 y Fn(b)m(e)36 b(a)g(p)m(ositively)h (oriente)m(d)f(orthonormal)521 3296 y(fr)m(ame)29 b(on)f(an)h(oriente)m (d)f(surfac)m(e)h Fr(M)36 b Fn(endowe)m(d)28 b(with)g(a)h(R)n (iemannian)521 3375 y(metric)24 b Fq(h)g Fr(;)35 b Fq(i)24 b Fn(with)f(L)m(evi-Civita)i(c)m(onne)m(ction)f Fq(r)p Fn(.)30 b(Then)984 3502 y Fq(K)20 b Fu(=)f Fq(h)p Fr(F)1203 3474 y Fo(r)1246 3502 y Fu(\()p Fr(s)1303 3512 y Fp(1)1329 3502 y Fr(;)11 b(s)1389 3512 y Fp(2)1416 3502 y Fu(\))p Fr(s)1473 3512 y Fp(2)1500 3502 y Fr(;)g(s)1560 3512 y Fp(1)1587 3502 y Fq(i)24 b Fr(:)p eop %%Page: 19 19 19 18 bop 2279 159 a Fk(19)521 294 y Fn(Pr)m(o)m(of.)27 b Fu(F)-6 b(or)26 b(a)g(surface)h(in)f Ft(R)1259 270 y Fp(3)1313 294 y Fu(w)n(e)g(can)g(argue)g(as)g(follo)n(ws.)43 b(W)-6 b(e)26 b(kno)n(w)521 373 y(from)c(Theorem)g(25)h(and)g(Lemma)e (24)i(that)f Fq(K)j Fu(is)e(the)g(determinan)n(t)f(of)521 451 y(the)14 b(W)-6 b(eingarten)15 b(map,)g(and)f(this)h(is)g(easily)g (seen)g(to)g(equal)g Fq(h)p Fr(F)2062 427 y Fo(r)2104 451 y Fu(\()p Fr(s)2161 461 y Fp(1)2188 451 y Fr(;)c(s)2248 461 y Fp(2)2275 451 y Fu(\))p Fr(s)2332 461 y Fp(2)2359 451 y Fr(;)g(s)2419 461 y Fp(1)2446 451 y Fq(i)p Fu(.)589 530 y(In)21 b(general,)i(for)f(an)f(abstract)g(surface,)i(w)n(e)f (calculate)h(using)608 636 y Fr(F)660 608 y Fo(r)702 636 y Fu(\()p Fr(s)759 646 y Fp(1)786 636 y Fr(;)11 b(s)846 646 y Fp(2)873 636 y Fu(\))p Fr(s)930 646 y Fp(2)975 636 y Fu(=)19 b Fq(r)1101 646 y Fs(s)1123 652 y Fl(1)1150 636 y Fq(r)1206 646 y Fs(s)1228 652 y Fl(2)1254 636 y Fr(s)1285 646 y Fp(2)1327 636 y Fq(\000)c(r)1450 646 y Fs(s)1472 652 y Fl(2)1499 636 y Fq(r)1555 646 y Fs(s)1577 652 y Fl(1)1603 636 y Fr(s)1634 646 y Fp(2)1676 636 y Fq(\000)g(r)1799 647 y Fp([)p Fs(s)1834 653 y Fl(1)1858 647 y Fs(;s)1893 653 y Fl(2)1917 647 y Fp(])1933 636 y Fr(s)1964 646 y Fp(2)2013 636 y Fr(:)521 746 y Fu(This)23 b(leads)h(to)f Fq(h)p Fr(F)1000 722 y Fo(r)1043 746 y Fu(\()p Fr(s)1100 756 y Fp(1)1126 746 y Fr(;)11 b(s)1186 756 y Fp(2)1213 746 y Fu(\))p Fr(s)1270 756 y Fp(2)1297 746 y Fr(;)g(s)1357 756 y Fp(1)1384 746 y Fq(i)21 b Fu(=)g(\()p Fq(\000)p Fr(d!)1656 756 y Fp(12)1708 746 y Fu(\)\()p Fr(s)1791 756 y Fp(1)1817 746 y Fr(;)11 b(s)1877 756 y Fp(2)1904 746 y Fu(\).)33 b(As)24 b Fq(\000)p Fr(d!)2207 756 y Fp(12)2279 746 y Fu(=)521 825 y Fq(K)q Fr(s)604 801 y Fo(\003)604 842 y Fp(1)646 825 y Fq(^)15 b Fr(s)737 801 y Fo(\003)737 842 y Fp(2)764 825 y Fu(,)22 b(the)g(claim)g(follo)n (ws.)972 b Fj(\003)1123 1047 y Fb(References)336 1154 y Fh(1.)28 b(L.)16 b(Conlon,)g Fa(Di\013eren)n(tiable)i(Manifolds)h(|)d (A)f(First)h(Course)p Fh(,)g(Birkh\177)-28 b(auser)17 b(V)-5 b(erlag)17 b(1993.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF