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b Ff(r)1930 3840 y Fc(Y)1988 3828 y Fg(X)34 b Fi(for)27 b(all)g(v)n(ector)g(\014elds)g Fg(X)r(;)14 b(Y)19 b Fi(.)565 3928 y(b\))37 b(Let)h Ff(r)g Fi(b)r(e)f(an)h(a\016ne) f(connection.)g(De\014ne)h(the)f(co)n(v)-5 b(arian)n(t)36 b(deriv)-5 b(ativ)n(e)37 b(of)g Fg(!)42 b Ff(2)440 4027 y Fi(\012)500 3997 y Fc(k)541 4027 y Fi(\()p Fg(M)9 b Fi(\))28 b(b)n(y)667 4265 y(\()p Ff(r)768 4231 y Fe(\003)768 4286 y Fc(X)831 4265 y Fg(!)s Fi(\)\()p Fg(Y)998 4277 y Fd(1)1036 4265 y Fg(;)14 b(:::;)g(Y)1227 4277 y Fc(k)1268 4265 y Fi(\))23 b(=)g Fg(L)1468 4277 y Fc(X)1530 4265 y Fi(\()p Fg(!)s Fi(\()p Fg(Y)1697 4277 y Fd(1)1735 4265 y Fg(;)14 b(:::;)g(Y)1926 4277 y Fc(k)1967 4265 y Fi(\)\))19 b Ff(\000)2175 4161 y Fc(k)2133 4186 y Fb(X)2139 4363 y Fc(i)p Fd(=1)2267 4265 y Fg(!)s Fi(\()p Fg(Y)2402 4277 y Fd(1)2439 4265 y Fg(;)14 b(:::;)g Ff(r)2651 4277 y Fc(X)2715 4265 y Fg(Y)2763 4277 y Fc(i)2790 4265 y Fg(;)g(:::;)g(Y)2981 4277 y Fc(k)3022 4265 y Fi(\))p Fg(:)440 4504 y Ff(r)509 4474 y Fe(\003)509 4527 y Fc(X)572 4504 y Fg(!)41 b Fi(is)e(again)e(a)h Fg(k)s Fi(-form.)f(Supp)r(ose)h Ff(r)h Fi(is)f(torsion-free.)f(Sho)n(w) g(for)h Fg(!)44 b Ff(2)d Fi(\012)3003 4474 y Fd(2)3040 4504 y Fi(\()p Fg(M)9 b Fi(\):)39 b(if)440 4603 y Ff(r)509 4573 y Fe(\003)547 4603 y Fg(!)26 b Fi(=)d(0)k(then)h Fg(d!)e Fi(=)d(0.)440 4941 y(4.)41 b(Let)g(\()p Fg(M)t(;)14 b(g)s Fi(\))41 b(b)r(e)h(a)e(Riemannian)h(manifold)g(and)g Fg(T)57 b Fi(:)45 b Fa(X)p Fi(\()p Fg(M)9 b Fi(\))27 b Ff(\002)g Fa(X)p Fi(\()p Fg(M)9 b Fi(\))46 b Ff(!)f Fa(X)p Fi(\()p Fg(M)9 b Fi(\))440 5041 y(a)38 b Fg(C)585 5011 y Fe(1)655 5041 y Fi(\()p Fg(M)9 b Fi(\)-bilinear,)37 b(an)n(tisymmetric)h(mapping.)g(Sho)n(w)f(that)h(there)g(exists)g(a)f (unique)440 5141 y(a\016ne)27 b(Riemannian)h(connection)f(on)g Fg(M)36 b Fi(whic)n(h)28 b(has)f(torsion)g Fg(T)12 b Fi(.)440 5397 y(Please)26 b(hand)h(in)h(y)n(our)d(solutions)i(on)g(Th)n (ursda)n(y)-7 b(,)26 b(the)h(15th)g(of)g(Jan)n(uary)-7 b(,)25 b(in)j(the)g(lecture.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF