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Fm(.)28 b(Sho)n(w)g(that)h Fk(d\013)1572 1125 y Fd(\014)1572 1175 y(\014)1600 1229 y Fi(W)1671 1212 y Fc(?)1753 1196 y Fm(is)f(an)g(orien)n(tation)f(of)h Fk(W)2561 1165 y Fe(?)2617 1196 y Fm(.)h(Conclude)f(that)g Fk(W)830 1295 y Fm(is)g(orien)n(table.)720 1428 y(e\))41 b(Sho)n(w)27 b(that)h(in)g(the)g(decomp)r(osition)f(of)h Fk(T)12 b(M)35 b Fm(in)n(to)27 b(the)h(direct)g(sum)1685 1653 y Fk(W)i Fj(\010)1896 1597 y Fk(D)p 1886 1634 90 4 v 1886 1710 a(W)2005 1653 y Fj(\010)2100 1597 y Fk(E)p 2098 1634 72 4 v 2098 1710 a(D)2197 1653 y Fj(\010)2290 1597 y Fk(T)12 b(M)p 2290 1634 150 4 v 2332 1710 a(E)830 1859 y Fm(of)29 b(four)g(line)g(bundles)g(eac)n(h)f(of)h(these)g(line)g (bundles)g(is)g(orien)n(table)f(and)g(the-)830 1959 y(refore)f (trivial.)g(Conclude)g(that)h Fk(M)36 b Fm(is)28 b(parallelizable.)648 2125 y(Summary)34 b(:)h(An)h(orien)n(table)e(4{manifold)g(whic)n(h)h (admits)g(an)f(orien)n(table)g(Engel)648 2224 y(distribution)27 b(has)g(trivial)g(tangen)n(t)g(bundle.)541 2390 y(4.)42 b(Let)27 b Fj(r)h Fm(b)r(e)g(a)f(connection)g(on)h(a)f(distribution)h Fk(D)d Fj(\032)d Fk(T)12 b(M)d Fm(.)715 2556 y(a\))41 b(Sho)n(w)27 b(that)h(the)g(torsion)1255 2739 y Fk(T)34 b Fm(:)23 b Fb(X)p Fm(\()p Fk(D)r Fm(\))c Fj(\002)f Fb(X)p Fm(\()p Fk(D)r Fm(\))23 b Fj(\000)-14 b(!)23 b Fb(X)p Fm(\()p Fk(M)9 b Fm(\))1637 2863 y(\()p Fk(X)r(;)14 b(Y)19 b Fm(\))k Fj(7\000)-14 b(!)23 b(r)2125 2875 y Fi(X)2188 2863 y Fk(Y)37 b Fj(\000)18 b(r)2425 2875 y Fi(Y)2483 2863 y Fk(X)25 b Fj(\000)18 b Fm([)p Fk(X)r(;)c(Y)k Fm(])830 3046 y(is)28 b(a)f(tensor.)710 3179 y(b\))42 b(Supp)r(ose)30 b Fk(S)h Fm(:)c Fb(X)p Fm(\()p Fk(D)r Fm(\))20 b Fj(\002)g Fk(:)14 b(:)g(:)19 b Fj(\002)h Fb(X)p Fm(\()p Fk(D)r Fm(\))27 b Fj(\000)-15 b(!)27 b Fb(X)p Fm(\()p Fk(D)r Fm(\))j(is)g(a)f(tensor)g(of)g(t)n(yp)r(e)h(\()p Fk(k)s(;)14 b Fm(1\).)830 3279 y(Sho)n(w)27 b(that)830 3461 y Fj(r)p Fk(S)h Fm(:)23 b Fb(X)p Fm(\()p Fk(M)9 b Fm(\))19 b Fj(\002)f Fb(X)p Fm(\()p Fk(D)r Fm(\))g Fj(\002)g Fk(:)c(:)g(:)19 b Fj(\002)f Fb(X)p Fm(\()p Fk(D)r Fm(\))23 b Fj(\000)-14 b(!)23 b Fb(X)p Fm(\()p Fk(D)r Fm(\))1409 3586 y(\()p Fk(X)1510 3598 y Fg(0)1547 3586 y Fk(;)14 b(X)1653 3598 y Fg(1)1690 3586 y Fk(;)g(:)g(:)g(:)27 b(;)14 b(X)1957 3598 y Fi(k)1998 3586 y Fm(\))23 b Fj(7\000)-14 b(!)23 b(r)2279 3598 y Fi(X)2333 3606 y Fa(0)2384 3586 y Fm(\()p Fk(S)5 b Fm(\()p Fk(X)2573 3598 y Fg(1)2610 3586 y Fk(;)14 b(:)g(:)g(:)28 b(;)14 b(X)2878 3598 y Fi(k)2918 3586 y Fm(\)\))2215 3801 y Fj(\000)2339 3697 y Fi(k)2298 3722 y Fd(X)2304 3899 y Fi(i)p Fg(=1)2431 3801 y Fk(S)19 b Fm(\()p Fk(X)2602 3813 y Fg(1)2639 3801 y Fk(;)14 b(:)g(:)g(:)28 b(;)14 b Fj(r)2907 3813 y Fi(X)2961 3821 y Fa(0)2998 3801 y Fk(X)3067 3813 y Fi(i)3094 3801 y Fk(;)g(:)g(:)g(:)28 b(;)14 b(X)3362 3813 y Fi(k)3402 3801 y Fm(\))830 4054 y(is)28 b(a)f(tensor.)720 4187 y(c\))41 b(Sho)n(w)27 b(that)892 4370 y Fk(R)c Fm(:)g Fb(X)p Fm(\()p Fk(M)9 b Fm(\))19 b Fj(\002)f Fb(X)p Fm(\()p Fk(M)9 b Fm(\))18 b Fj(\002)g Fb(X)p Fm(\()p Fk(D)r Fm(\))23 b Fj(\000)-14 b(!)23 b Fb(X)p Fm(\()p Fk(D)r Fm(\))1526 4494 y(\()p Fk(X)r(;)14 b(Y)5 b(;)14 b(Z)6 b Fm(\))22 b Fj(7\000)-14 b(!)23 b(\000r)2164 4506 y Fi(X)2227 4494 y Fj(r)2296 4506 y Fi(Y)2354 4494 y Fk(Z)h Fm(+)18 b Fj(r)2587 4506 y Fi(Y)2645 4494 y Fj(r)2714 4506 y Fi(X)2777 4494 y Fk(Z)24 b Fm(+)18 b Fj(r)3010 4509 y Fg([)p Fi(X)q(;Y)c Fg(])3180 4494 y Fk(Z)830 4677 y Fm(is)28 b(a)f(tensor.)565 4860 y(Please)f(hand)h(in)h(y)n(our)e(solutions)h(during)g(the)h (lecture)f(on)g(W)-7 b(ednesda)n(y)g(,)27 b(27)p Fk(:)p Fm(11)p Fk(:)p Fm(2002.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF