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(bundle)i(is)g(called)f Fb(\015at)g Fk(if)i(it)f(has)f(v)-5 b(anishing)27 b(curv)-5 b(ature.)565 3579 y(a\))20 b(Sho)n(w)f(that)i (ev)n(ery)d(trivial)i(v)n(ector)f(bundle)h(o)n(v)n(er)f(a)g(di\013eren) n(tiable)h(manifold)g(admits)440 3678 y(a)27 b(\015at)h(connection.)565 3778 y(b\))h(Find)g(a)g(v)n(ector)e(bundle)i(o)n(v)n(er)e Fi(T)1704 3748 y Ff(2)1766 3778 y Fk(=)d Fi(S)1911 3748 y Ff(1)1967 3778 y Fh(\002)19 b Fi(S)2107 3748 y Ff(1)2173 3778 y Fk(whic)n(h)29 b(is)f(non-trivial)g(but)h(admits)440 3878 y(a)e(\015at)h(connection.)440 4113 y(4.)h(Let)g Fi(E)h Fh(!)c Fi(M)37 b Fk(b)r(e)30 b(a)e(v)n(ector)g(bundle)h(with)h (connection)e Fh(r)h Fk(and)g(curv)-5 b(ature)28 b Fi(F)3017 4083 y Fe(r)3077 4113 y Fk(.)h(Con-)440 4213 y(sider)e(the)h(v)n(ector) e(bundle)i Fi(E)5 b(nd)p Fk(\()p Fi(E)g Fk(\))24 b Fh(!)f Fi(M)9 b Fk(.)565 4313 y(a\))27 b(Sho)n(w)g(that)1265 4462 y(\()p 1297 4395 70 4 v Fh(r)1367 4474 y Fg(X)1430 4462 y Fi(L)p Fk(\)\()p Fi(s)p Fk(\))c(:=)g Fh(r)1825 4474 y Fg(X)1888 4462 y Fk(\()p Fi(Ls)p Fk(\))c Fh(\000)f Fi(L)p Fk(\()p Fh(r)2308 4474 y Fg(X)2371 4462 y Fi(s)p Fk(\))p Fi(;)440 4623 y Fk(for)27 b Fi(L)c Fh(2)g Fk(\000\()p Fi(E)5 b(nd)p Fk(\()p Fi(E)g Fk(\)\))p Fi(;)29 b(s)23 b Fh(2)g Fk(\000\()p Fi(E)5 b Fk(\))p Fi(;)29 b(X)g Fh(2)23 b Fa(X)p Fk(\()p Fi(M)9 b Fk(\),)28 b(de\014nes)g(a)f(connection)p 2754 4556 V 27 w Fh(r)h Fk(on)f Fi(E)5 b(nd)p Fk(\()p Fi(E)g Fk(\).)565 4736 y(b\))28 b(Pro)n(v)n(e)d(the)j(follo)n(wing)e (form)n(ula)h(for)g(the)h(curv)-5 b(ature)27 b Fi(F)p 2410 4658 60 3 v 2410 4706 a Fe(r)2497 4736 y Fk(of)h(the)g(connection) p 3150 4669 70 4 v 27 w Fh(r)p Fk(:)1344 4903 y Fi(F)p 1409 4820 60 3 v 1409 4868 a Fe(r)1468 4903 y Fk(\()p Fi(X)r(;)14 b(Y)19 b Fk(\))p Fi(L)k Fk(=)f([)p Fi(F)1962 4868 y Fe(r)2022 4903 y Fk(\()p Fi(X)r(;)14 b(Y)19 b Fk(\))p Fi(;)14 b(L)p Fk(])p Fi(:)440 5052 y Fk(\(The)30 b(term)f(on)h(the)f(righ)n(t)g(hand)h(side)f(is)h(the)f(comm)n(utator)g (of)g(t)n(w)n(o)g(endomorphisms)f(of)440 5152 y Fi(E)5 b Fk(.\))440 5397 y(Please)20 b(hand)i(in)g(y)n(our)e(solutions)h(on)g (Th)n(ursda)n(y)-7 b(,)20 b(the)i(18th)f(of)g(Decem)n(b)r(er,)h(in)g (the)g(lecture.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF