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b(jection)25 b Fk(P)35 b Fg(!)1858 4580 y Fm(~)1840 4601 y Fk(P)i Fm(b)n(y)25 b(pr.)g(Let)34 b(~)-50 b Fk(\013)25 b Fm(b)r(e)h(de\014ned) g(as)f(follo)n(ws.)f(F)-7 b(or)854 4679 y(~)830 4700 y Fk(X)30 b Fg(2)23 b Fk(T)1056 4712 y Fh(q)1111 4679 y Fm(~)1092 4700 y Fk(P)40 b Fm(c)n(ho)r(ose)26 b Fk(X)j Fg(2)24 b Fk(T)12 b(P)38 b Fm(suc)n(h)28 b(that)g(pr)2223 4721 y Fe(\003)2261 4700 y Fm(\()p Fk(X)7 b Fm(\))23 b(=)2536 4679 y(~)2512 4700 y Fk(X)6 b Fm(.)28 b(Then)g(let)1737 4886 y(~)-50 b Fk(\013)p Fm(\()1838 4865 y(~)1814 4886 y Fk(X)7 b Fm(\))23 b(=)g(tr)o(\()p Fk(\013)p Fm(\()p Fk(X)7 b Fm(\)\))29 b Fk(:)830 5063 y Fm(Pro)n(v)n(e)24 b(that)j(this)g(is)g(w)n(ell)f(de\014ned)h(and)f(that)35 b(~)-50 b Fk(\013)27 b Fm(is)f(an)h Fk(S)2610 5032 y Fj(1)2647 5063 y Fm({connection)e(on)3235 5042 y(~)3216 5063 y Fk(P)12 b Fm(.)720 5193 y(c\))41 b(Sho)n(w)25 b(that)h(the)g(curv)-5 b(ature)1745 5172 y(~)1735 5193 y(\012)26 b(of)33 b(~)-50 b Fk(\013)26 b Fm(and)f(the)h(curv)-5 b(ature)25 b(\012)g(of)h Fk(\013)g Fm(are)e(related)830 5293 y(b)n(y)1803 5392 y(pr)1882 5358 y Fe(\003)1930 5371 y Fm(~)1920 5392 y(\012)f(=)g(tr\(\012\))28 b Fk(:)p eop %%Page: 2 2 2 1 bop 541 531 a Fm(3.)42 b(Consider)26 b(the)i(em)n(b)r(edding)g Fk(\023)23 b Fm(:)g Fk(S)5 b(O)r Fm(\()p Fk(m)p Fm(\))24 b Fk(,)-14 b Fg(!)24 b Fk(S)5 b(O)r Fm(\()p Fk(m)19 b Fm(+)f(1\))27 b(de\014ned)h(b)n(y)1575 891 y Fk(A)23 b Fg(7\000)-14 b(!)1817 649 y Fc(0)1817 796 y(B)1817 845 y(B)1817 895 y(B)1817 948 y(@)1931 713 y Fm(1)82 b(0)14 b Fk(:)g(:)g(:)g Fm(0)1931 813 y(0)1940 901 y(.)1940 934 y(.)1940 968 y(.)165 b Fk(A)1931 1067 y Fm(0)2305 649 y Fc(1)2305 796 y(C)2305 845 y(C)2305 895 y(C)2305 948 y(A)648 1251 y Fm(Let)26 b(pr)874 1271 y Fj(1)934 1251 y Fm(:)d Fk(S)5 b(O)r Fm(\()p Fk(m)17 b Fm(+)f(1\))23 b Fg(!)g Fk(S)1563 1221 y Fh(m)1652 1251 y Fm(map)k(an)f(orthogonal)e (matrix)i(to)g(its)h(\014rst)f(column.)648 1350 y(F)-7 b(urthermore,)22 b(let)i Fk(f)32 b Fm(:)23 b Fd(so)o Fm(\()p Fk(m)11 b Fm(+)g(1\))22 b Fg(!)h Fd(so)p Fm(\()p Fk(m)p Fm(\))h(b)r(e)g(the)g(map)f(whic)n(h)h(forgets)f(the)h(\014rst) 648 1450 y(ro)n(w)i(and)h(the)h(\014rst)g(column)f(of)h(matrices)f(in)g Fd(so)p Fm(\()p Fk(m)18 b Fm(+)g(1\).)715 1616 y(a\))41 b(Sho)n(w)27 b(that)h(with)g(the)g(righ)n(t)f(action)1395 1799 y Fk(S)5 b(O)r Fm(\()p Fk(m)19 b Fm(+)f(1\))h Fg(\002)f Fk(S)5 b(O)r Fm(\()p Fk(m)p Fm(\))24 b Fg(\000)-15 b(!)23 b Fk(S)5 b(O)r Fm(\()p Fk(m)19 b Fm(+)f(1\))1927 1923 y(\()p Fk(B)t(;)c(A)p Fm(\))24 b Fg(7\000)-15 b(!)23 b Fk(B)g Fg(\001)18 b Fk(\023)p Fm(\()p Fk(A)p Fm(\))29 b Fk(;)830 2106 y Fm(the)f(bundle)g(pr)1324 2126 y Fj(1)1384 2106 y Fm(:)23 b Fk(S)5 b(O)r Fm(\()p Fk(m)19 b Fm(+)f(1\))23 b Fg(!)g Fk(S)2017 2076 y Fh(m)2108 2106 y Fm(is)k(a)g(principal)h Fk(S)5 b(O)r Fm(\()p Fk(m)p Fm(\){bundle.)710 2239 y(b\))42 b(Let)31 b Fk(X)k Fg(2)29 b Fk(T)1219 2251 y Fh(g)1257 2239 y Fk(S)5 b(O)r Fm(\()p Fk(m)21 b Fm(+)g(1\))30 b(and)h Fk(L)1916 2254 y Fh(g)1950 2238 y Fb(\000)p Fa(1)2062 2239 y Fm(is)g(the)h(left)f(m)n(ultiplication)g(with)h Fk(g)3216 2209 y Fe(\000)p Fj(1)830 2338 y Fm(on)c Fk(S)5 b(O)r Fm(\()p Fk(m)18 b Fm(+)h(1\).)27 b(Then)h(let)1672 2521 y Fk(\013)p Fm(\()p Fk(X)7 b Fm(\))24 b(=)e Fk(f)2040 2454 y Fc(\000)2078 2521 y Fk(L)2135 2537 y Fh(g)2169 2520 y Fb(\000)p Fa(1)2246 2537 y Fe(\003)2284 2521 y Fk(X)2360 2454 y Fc(\001)2439 2521 y Fk(:)830 2704 y Fm(Sho)n(w)27 b(that)h Fk(\013)g Fm(is)g(an)f Fk(S)5 b(O)r Fm(\()p Fk(m)p Fm(\){connection)27 b(on)h Fk(S)5 b(O)r Fm(\()p Fk(m)19 b Fm(+)f(1\).)720 2836 y(c\))41 b(Let)25 b Fk(X)p 976 2849 76 4 v 6 w(;)14 b(Y)p 1088 2849 67 4 v 43 w Fm(b)r(e)25 b(the)g(fundamen)n(tal)f(v)n(ector)f (\014elds)i(of)f Fk(X)r(;)14 b(Y)42 b Fg(2)23 b Fd(so)o Fm(\()p Fk(m)12 b Fm(+)g(1\))25 b(of)f(the)830 2936 y(righ)n(t)j (action)g(of)h Fk(S)5 b(O)r Fm(\()p Fk(m)19 b Fm(+)f(1\))27 b(on)g(itself.)i(Sho)n(w)e(that)1390 3119 y(\012\()p Fk(X)p 1482 3132 76 4 v 7 w(;)14 b(Y)p 1595 3132 67 4 v 18 w Fm(\))24 b(=)e([)p Fk(f)9 b Fm(\()p Fk(X)e Fm(\))p Fk(;)14 b(f)9 b Fm(\()p Fk(Y)18 b Fm(\)])h Fg(\000)f Fk(f)9 b Fm(\([)p Fk(X)r(;)14 b(Y)k Fm(]\))28 b Fk(:)541 3318 y Fm(4.)42 b(W)-7 b(e)26 b(use)h(the)g(construction)e(of)i(\015at) f(bundles)h(from)f(exercise)g(4)g(on)g(the)h(2.)f(problem)648 3418 y(sheet)h(and)h(the)g(notation)f(in)n(tro)r(duced)g(there.)h(Let)f Fk(G)h Fm(b)r(e)g(a)f(Lie)h(group.)715 3584 y(a\))41 b(Sho)n(w)32 b(that)g(a)g(principal)f Fk(G)p Fm({bundle)i(is)e(trivial) h(if)g(and)g(only)g(if)g(it)h(admits)f(a)830 3683 y(section.)710 3816 y(b\))42 b(Supp)r(ose)27 b Fk(\032)c Fm(:)g Fk(\031)1314 3828 y Fj(1)1352 3816 y Fm(\()p Fk(B)t Fm(\))h Fg(\000)-14 b(!)23 b Fk(G)k Fm(is)g(a)g(represen)n(tation)e(with)j(\014nite)f (image.)g(Sho)n(w)830 3916 y(that)e(there)g(is)g(a)g(\014nite)g(co)n(v) n(er)p 1786 3849 68 4 v 23 w Fk(B)i Fg(!)c Fk(B)30 b Fm(suc)n(h)24 b(that)h(the)h(pullbac)n(k)e(to)p 3005 3849 V 25 w Fk(B)29 b Fm(of)c(the)830 4015 y(\015at)j(principal)f Fk(G)p Fm({bundle)h Fk(P)35 b Fg(!)23 b Fk(B)32 b Fm(giv)n(en)26 b(b)n(y)i Fk(\032)f Fm(is)h(trivial.)565 4198 y(Please)e(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(20)p Fk(:)p Fm(11)p Fk(:)p Fm(2002.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF