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b(family)g(of)h(cones.)43 b(This)27 b(together)456 4492 y(with)32 b(Remark)g(1)g(completes)h(the)g(pro)s(of)e(of)i(the)g (\014rst)g(part)f(of)g(Theorem)h(1.)43 b(T)-8 b(o)456 4608 y(pro)m(v)m(e)33 b(Prop)s(osition)e(2)i(w)m(e)g(need)h(an)e (auxiliary)456 4751 y Fm(Lemma)40 b(1.)j Fl(L)-5 b(et)38 b Fk(p)32 b Fn(:)h Fk(E)38 b Fj(\000)-16 b(!)32 b Fk(M)48 b Fl(b)-5 b(e)37 b(a)g(r)-5 b(e)g(al)38 b(ve)-5 b(ctor)37 b(bund)5 b(le)37 b(of)g(r)-5 b(ank)37 b Fk(n)g Fl(with)h(a)456 4867 y(metric.)j(Assume)26 b(that)h(for)f(any)h(p)-5 b(oint)26 b Fk(x)i Fj(2)g Fk(M)10 b Fl(,)29 b(the)d(\014b)-5 b(er)26 b Fk(E)2682 4882 y Fg(x)2754 4867 y Fn(=)h Fk(p)2906 4831 y Ff(\000)p Fh(1)3000 4867 y Fn(\()p Fk(x)p Fn(\))g Fl(c)-5 b(arries)456 4983 y(the)40 b(fol)5 b(lowing)39 b(structur)-5 b(e:)58 b Fk(E)1583 4947 y Fg(h)1577 5008 y(x)1668 4983 y Fl(and)40 b Fk(E)1941 4947 y Fg(v)1935 5008 y(x)2023 4983 y Fl(ar)-5 b(e)40 b Fk(k)k Fl(and)c Fk(n)27 b Fj(\000)g Fk(k)43 b Fl(line)-5 b(ar)40 b(subsp)-5 b(ac)g(es)456 5099 y(of)42 b(dimensions)f Fk(k)46 b Fl(and)c Fk(n)28 b Fj(\000)h Fk(k)45 b Fl(r)-5 b(esp)g(e)g(ctively)43 b(tr)-5 b(ansverse)42 b(to)h(e)-5 b(ach)42 b(other)h(\(the)456 5216 y(dir)-5 b(e)g(ct)45 b(sum)g(gives)g(us)h(the)f(whole)g Fk(E)1863 5231 y Fg(x)1907 5216 y Fl(\);)51 b Fk(A)2101 5179 y Fg(h)2101 5240 y(x)2191 5216 y Fl(and)45 b Fk(A)2464 5179 y Fg(v)2464 5241 y(h)2554 5216 y Fl(ar)-5 b(e)46 b(close)-5 b(d)44 b(subsets)i(of)p eop %%Page: 3 3 3 2 bop 3406 251 a Fe(3)456 450 y Fk(E)528 465 y Fg(x)612 450 y Fl(c)-5 b(ontaining)39 b Fk(E)1171 414 y Fg(h)1165 475 y(x)1256 450 y Fl(and)h Fk(E)1529 414 y Fg(v)1523 475 y(x)1610 450 y Fl(r)-5 b(esp)g(e)g(ctively.)60 b(L)-5 b(et)41 b Fn(\010)c(:)h Fk(E)44 b Fj(\000)-16 b(!)37 b Fk(E)46 b Fl(b)-5 b(e)40 b(a)g(bund)5 b(le)456 566 y(isomorphism)33 b(with)h(the)h(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties:)456 847 y Fn(\(4\))686 b(\010)p Fk(A)1410 806 y Fg(h)1410 872 y(x)1483 847 y Fj(\032)29 b Fk(A)1662 806 y Fg(h)1662 872 y(f)7 b Fh(\()p Fg(x)p Fh(\))1818 847 y Fk(;)50 b Fn(\010)1965 806 y Ff(\000)p Fh(1)2060 847 y Fk(A)2133 806 y Fg(v)2133 872 y(x)2205 847 y Fj(\032)28 b Fk(A)2383 806 y Fg(v)2383 873 y(f)2424 854 y Fi(\000)p Fd(1)2507 873 y Fh(\()p Fg(x)p Fh(\))2606 847 y Fk(;)456 1176 y Fn(\(5\))435 b Fj(jj)p Fn(\010\()p Fk(u;)17 b(v)t Fn(\))p Fj(jj)26 b(\025)i Fk(\026)1615 1111 y Fi(0)1641 1176 y Fj(jj)p Fn(\()p Fk(u;)17 b(v)t Fn(\))p Fj(jj)g Fk(;)47 b Fn(\()p Fk(u;)17 b(v)t Fn(\))27 b Fj(2)h Fk(A)2492 1135 y Fg(h)2492 1200 y(x)2537 1176 y Fk(;)17 b(x)27 b Fj(2)i Fk(M)5 b(;)456 1505 y Fn(\(6\))436 b Fj(jj)p Fn(\010\()p Fk(u;)17 b(v)t Fn(\))p Fj(jj)26 b(\024)i Fk(\025)1614 1440 y Fi(0)1641 1505 y Fj(jj)p Fn(\()p Fk(u;)17 b(v)t Fn(\))p Fj(jj)g Fk(;)47 b Fn(\()p Fk(u;)17 b(v)t Fn(\))26 b Fj(2)i Fk(A)2491 1464 y Fg(v)2491 1529 y(x)2535 1505 y Fk(;)17 b(x)28 b Fj(2)g Fk(M)5 b(:)555 1669 y Fl(Writing)31 b(an)f(element)g(fr)-5 b(om)30 b Fk(E)36 b Fl(as)30 b(p)-5 b(air)30 b Fn(\()p Fk(u;)17 b(v)t Fn(\))29 b Fl(c)-5 b(orr)g(esp)g(onds)30 b(to)g(the)h(splitting) 456 1790 y Fk(E)528 1805 y Fg(x)599 1790 y Fn(=)d Fk(E)781 1753 y Fg(h)775 1815 y(h)839 1790 y Fj(\010)13 b Fk(E)1007 1753 y Fg(v)1001 1814 y(x)1049 1790 y Fl(.)44 b(Her)-5 b(e)31 b Fk(\025)1409 1730 y Fi(0)1466 1790 y Fl(and)f Fk(\026)1710 1730 y Fi(0)1768 1790 y Fl(ar)-5 b(e)30 b(some)h(p)-5 b(ositive)30 b(numb)-5 b(ers)30 b(with)h Fk(\025)3170 1730 y Fi(0)3224 1790 y Fj(\024)d Fk(\026)3388 1730 y Fi(0)3415 1790 y Fl(.)456 1906 y(Set)47 b Fk(T)706 1870 y Fg(i)692 1931 y(x)786 1906 y Fn(=)i(\010)981 1870 y Fg(i)1010 1906 y Fk(A)1083 1870 y Fg(h)1083 1938 y Fh(\010)1134 1919 y Fi(\000)p Fb(i)1208 1938 y Fg(x)1252 1906 y Fl(,)h Fk(R)1407 1870 y Fg(i)1406 1931 y(x)1500 1906 y Fn(=)g(\010)1696 1870 y Ff(\000)p Fg(i)1780 1906 y Fk(A)1853 1870 y Fg(v)1853 1938 y Fh(\010)1904 1919 y Fb(i)1930 1938 y Fg(x)1974 1906 y Fl(,)g Fk(E)2132 1870 y Fh(+)2126 1931 y Fg(x)2241 1906 y Fn(=)2367 1831 y Fa(T)2450 1858 y Ff(1)2450 1935 y Fg(i)p Fh(=0)2585 1906 y Fk(T)2656 1870 y Fg(i)2642 1931 y(x)2686 1906 y Fl(,)f Fk(E)2843 1870 y Ff(\000)2837 1931 y Fg(x)2953 1906 y Fn(=)3078 1831 y Fa(T)3161 1858 y Ff(1)3161 1935 y Fg(i)p Fh(=0)3296 1906 y Fk(R)3371 1870 y Fg(i)3370 1931 y(x)3415 1906 y Fl(.)456 2028 y(Then)30 b Fk(E)784 1992 y Fh(+)778 2053 y Fg(x)875 2028 y Fl(is)h(a)g Fk(k)k Fl(dimensional)29 b(line)-5 b(ar)31 b(sp)-5 b(ac)g(e)30 b(inside)h Fk(A)2558 1992 y Fg(h)2558 2053 y(x)2634 2028 y Fl(and)g Fk(E)2898 1992 y Ff(\000)2892 2053 y Fg(x)2989 2028 y Fl(is)g(an)g Fk(n)14 b Fj(\000)g Fk(k)456 2145 y Fl(dimensional)33 b(line)-5 b(ar)34 b(sp)-5 b(ac)g(e)34 b(inside)g Fk(A)1885 2108 y Fg(v)1885 2169 y(x)1929 2145 y Fl(.)555 2285 y(Pr)-5 b(o)g(of)43 b Fn(As)g(it)f(follo)m(ws)g(from)f (\(4\),)k Fk(T)1926 2249 y Fg(i)p Fh(+1)1912 2309 y Fg(x)2090 2285 y Fj(\032)g Fk(T)2283 2249 y Fg(i)2269 2309 y(x)2358 2285 y Fj(\032)h Fk(T)2552 2249 y Fh(0)2538 2309 y Fg(x)2636 2285 y Fn(=)f Fk(A)2830 2249 y Fg(h)2830 2309 y(x)2875 2285 y Fk(:)e Fn(This)g(means)456 2401 y(that)35 b Fk(E)748 2365 y Fh(+)742 2426 y Fg(x)842 2401 y Fn(is)g(the)h(in)m(tersection)f (of)g(a)g(nested)h(sequence)i(of)d(closed)h(sets.)52 b(More-)456 2517 y(o)m(v)m(er,)37 b Fk(E)773 2481 y Fh(+)767 2542 y Fg(x)867 2517 y Fn(p)s(ossesses)i(a)c(kind)g(of)g(semi-in)m(v)-5 b(ariance)34 b(prop)s(ert)m(y)i(with)f(resp)s(ect)i(to)456 2634 y(\010,)45 b(namely)c(\010)p Fk(E)1094 2597 y Fh(+)1088 2658 y Fg(x)1198 2634 y Fj(\032)j Fk(E)1397 2592 y Fh(+)1391 2665 y(\010\()p Fg(x)p Fh(\))1541 2634 y Fn(.)73 b(Next)43 b(construction)f(sho)m(ws)i(that)e(the)g(set)h Fk(E)3385 2597 y Fh(+)3379 2658 y Fg(x)456 2772 y Fn(con)m(tains)48 b(a)h(linear)e(space.)92 b(Namely)-8 b(,)52 b(since)d Fk(E)2315 2736 y Fg(h)2309 2797 y(x)2415 2772 y Fj(\032)56 b Fk(A)2621 2736 y Fg(h)2621 2797 y(x)2665 2772 y Fn(,)d(\(4\))48 b(implies)f(that)456 2888 y Fk(S)522 2852 y Fg(j)516 2913 y(x)602 2888 y Fn(=)42 b(\010)790 2852 y Fg(j)827 2888 y Fk(E)905 2852 y Fg(h)899 2920 y Fh(\010)950 2901 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Fg(l)720 4006 y(j)798 3981 y Fj(2)h Fk(S)971 3945 y Fg(j)965 4006 y(x)1051 3981 y Fj(\032)g Fk(T)1240 3945 y Fg(j)1226 4006 y(x)1318 3981 y Fj(\032)g Fk(T)1507 3945 y Fg(i)1493 4006 y(x)1537 3981 y Fn(.)68 b(If)40 b(y)m(ou)h(let)f Fk(j)47 b Fn(go)40 b(to)g(in\014nit)m(y)g(the)h(closeness)h(of)456 4109 y Fk(T)527 4073 y Fg(i)513 4134 y(x)593 4109 y Fn(will)33 b(giv)m(e)j(y)m(ou)h(that)f Fk(e)1432 4073 y Fg(l)1491 4109 y Fj(2)e Fk(T)1662 4073 y Fg(i)1648 4134 y(x)1692 4109 y Fn(.)54 b(Since)36 b(this)g(is)f(true)i(for)e(all)f Fk(i)g Fn(=)f(0)p Fk(;)17 b Fn(1)p Fk(;)g(:::)p Fn(,)36 b(w)m(e)456 4225 y(ha)m(v)m(e)i(that)f Fk(e)946 4189 y Fg(l)1007 4225 y Fj(2)e Fk(E)1186 4189 y Fh(+)1180 4250 y Fg(x)1245 4225 y Fn(.)57 b(And)37 b(this)g(is,)g(in)g(turn,)h (true)f(for)f(all)f Fk(l)r Fn(,)k(and)e(hence)h(the)456 4341 y(inclusion)e Fk(S)928 4356 y Fg(x)1008 4341 y Fj(\032)g Fk(E)1199 4305 y Fh(+)1193 4366 y Fg(x)1296 4341 y Fn(is)h(v)m (eri\014ed.)60 b(So)37 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Fk(;)g(v)2407 4867 y Fi(00)2451 4927 y Fn(\),)31 b(where)h(\()p Fk(u;)17 b(v)3016 4867 y Fi(0)3042 4927 y Fn(\))27 b Fj(2)h Fk(S)3261 4942 y Fg(x)3305 4927 y Fn(,)j(in)456 5043 y(a)h(unique)h(w)m(a)m(y)-8 b(.)45 b(If)32 b(w)m(e)i(let)637 5211 y(\()p Fk(u)731 5226 y Fg(j)767 5211 y Fk(;)17 b(v)858 5226 y Fg(j)894 5211 y Fn(\))28 b(=)f(\010)1133 5170 y Ff(\000)p Fg(j)1225 5211 y Fn(\()p Fk(u;)17 b(v)t Fn(\))g Fk(;)48 b Fn(\()p Fk(u)1638 5147 y Fi(0)1638 5236 y Fg(j)1674 5211 y Fk(v)1725 5147 y Fi(0)1721 5236 y Fg(j)1758 5211 y Fn(\))28 b(=)f(\010)1997 5170 y Ff(\000)p Fg(j)2089 5211 y Fn(\()p Fk(u;)17 b(v)2278 5147 y Fi(0)2303 5211 y Fn(\))g Fk(;)50 b Fn(\()p Fk(u)2529 5147 y Fi(00)2529 5236 y Fg(j)2573 5211 y Fk(v)2624 5147 y Fi(00)2620 5236 y Fg(j)2669 5211 y Fn(\))27 b(=)h(\010)2908 5170 y Ff(\000)p Fg(j)3000 5211 y Fn(\(0)p Fk(;)17 b(v)3182 5147 y Fi(00)3225 5211 y Fn(\))p eop %%Page: 4 4 4 3 bop 456 251 a Fe(4)456 450 y Fn(then)44 b(\()p Fk(u;)17 b(v)t Fn(\))46 b Fj(2)h Fk(E)1153 414 y Fh(+)1147 475 y Fg(x)1256 450 y Fn(implies)42 b(that)i(\()p Fk(u)1916 465 y Fg(j)1951 450 y Fk(;)17 b(v)2042 465 y Fg(j)2079 450 y Fn(\))47 b Fj(2)g Fk(A)2350 414 y Fg(h)2350 480 y Fh(\010)p Ff(\000)p Fg(j)t Fh(\()p Fg(x)p Fh(\))2587 450 y Fn(,)g(and)d(hence)h(b)m(y)i(\(5\))o(,)456 592 y(w)m(e)42 b(ha)m(v)m(e)h Fj(jj)p Fn(\()p Fk(u)992 607 y Fg(j)1027 592 y Fk(;)17 b(v)1118 607 y Fg(j)1155 592 y Fn(\))p Fj(jj)42 b(\024)h Fn(\()p Fk(\026)1508 532 y Fi(0)1534 592 y Fn(\))1572 556 y Ff(\000)p Fg(j)1664 592 y Fj(jj)p Fn(\()p Fk(u;)17 b(v)1909 532 y Fi(0)1933 592 y Fn(\))p Fj(jj)p Fk(:)41 b Fn(By)h(the)g(same)f(tok)m(en)i Fj(jj)p Fn(\()p Fk(u)3112 532 y Fi(0)3112 617 y Fg(j)3147 592 y Fk(v)3198 532 y Fi(0)3194 617 y Fg(j)3231 592 y Fn(\))p Fj(jj)f(\024)456 725 y Fn(\()p Fk(\026)553 665 y Fi(0)579 725 y Fn(\))617 689 y Ff(\000)p Fg(j)708 725 y Fj(jj)p Fn(\()p Fk(u;)17 b(v)953 665 y Fi(0)978 725 y Fn(\))p Fj(jj)p Fk(:)31 b Fn(Similarly)-8 b(,)28 b(b)m(y)34 b(\(6\))o(,)e(w)m(e)h(ha)m(v)m(e)g Fj(jj)p Fk(v)2356 665 y Fi(00)2400 725 y Fj(jj)27 b(\024)h Fn(\()p Fk(\025)2683 665 y Fi(0)2709 725 y Fn(\))2747 689 y Fg(j)2784 725 y Fj(jj)p Fn(\()p Fk(u)2934 665 y Fi(00)2934 750 y Fg(j)2977 725 y Fk(v)3028 665 y Fi(00)3024 750 y Fg(j)3073 725 y Fn(\))p Fj(jj)p Fk(:)j Fn(Alto-)456 841 y(gether:)456 1118 y Fj(jj)p Fk(v)563 1054 y Fi(00)606 1118 y Fj(jj)c(\024)i Fn(\()p Fk(\025)890 1054 y Fi(0)916 1118 y Fn(\))954 1077 y Fg(j)990 1118 y Fj(jj)p Fn(\()p Fk(u)1140 1054 y Fi(00)1140 1143 y Fg(j)1184 1118 y Fk(v)1235 1054 y Fi(00)1231 1143 y Fg(j)1279 1118 y Fn(\))p Fj(jj)e(\024)i Fn(\()p Fk(\025)1601 1054 y Fi(0)1627 1118 y Fn(\))1665 1077 y Fg(j)1701 1118 y Fj(jj)p Fn(\()p Fk(u)1851 1133 y Fg(j)1887 1118 y Fk(;)17 b(v)1978 1133 y Fg(j)2014 1118 y Fn(\))p Fj(jj)p Fn(+)p Fj(jj)p Fn(\()p Fk(u)2334 1054 y Fi(0)2334 1143 y Fg(j)2369 1118 y Fk(v)2420 1054 y Fi(0)2416 1143 y Fg(j)2452 1118 y Fn(\))p Fj(jj)27 b(\024)h Fn(\()2727 1051 y Fk(\025)2784 991 y Fi(0)p 2726 1095 86 4 v 2726 1187 a Fk(\026)2785 1139 y Fi(0)2821 1118 y Fn(\))2859 1077 y Fg(j)2896 1118 y Fn(\()p Fj(jj)p Fn(\()p Fk(u;)17 b(v)t Fn(\))p Fj(jj)p Fn(+)p Fj(jj)p Fn(\()p Fk(u;)g(v)3594 1054 y Fi(0)3616 1118 y Fn(\))p Fj(jj)p Fn(\))p Fk(:)456 1396 y Fn(F)-8 b(or)26 b(all)f Fk(j)34 b Fj(2)28 b Fc(N)15 b Fn(,)28 b(whence)h Fk(v)1437 1337 y Fi(00)1510 1396 y Fn(=)e(0)g(and)h(\()p Fk(u;)17 b(v)t Fn(\))26 b Fj(2)i Fk(S)2281 1411 y Fg(x)2325 1396 y Fn(.)42 b(This)27 b(pro)m(v)m(es)i(the)f(inclusion)456 1513 y Fk(E)534 1477 y Fh(+)528 1537 y Fg(x)620 1513 y Fj(\032)h Fk(S)786 1528 y Fg(x)829 1513 y Fn(,)k(and)g(hence)h(the)f (equalit)m(y)f Fk(E)1967 1477 y Fh(+)1961 1537 y Fg(x)2054 1513 y Fn(=)27 b Fk(S)2217 1528 y Fg(x)2261 1513 y Fn(.)44 b(Q.E.D.)456 1629 y(Note)33 b(that)h(this,)g(for)f(dimension)f (reasons,)j(immediately)c(giv)m(es)j(the)g(impro)m(v)m(e-)456 1745 y(men)m(t)25 b(of)f(semi-in)m(v)-5 b(ariance)23 b(to)i(the)g(true)g(in)m(v)-5 b(ariance)24 b(of)h Fj(f)p Fk(E)2629 1709 y Fh(+)2623 1770 y Fg(x)2688 1745 y Fj(g)2738 1760 y Fg(x)p Ff(2)p Fg(M)2928 1745 y Fn(with)g(resp)s(ect)456 1861 y(to)32 b(\010,)h(namely)f(\010)p Fk(E)1192 1825 y Fh(+)1186 1886 y Fg(x)1279 1861 y Fn(=)27 b Fk(E)1460 1820 y Fh(+)1454 1893 y(\010\()p Fg(x)p Fh(\))1604 1861 y Fn(.)456 1987 y Fl(Pr)-5 b(o)g(of)35 b(of)h(Pr)-5 b(op)g(osition)35 b(2)f Fn(Let)f(the)i(v)m(ector)f(bundle)g Fk(p)c Fn(:)g Fk(E)35 b Fj(\000)-16 b(!)29 b Fk(M)45 b Fn(b)s(e)34 b(the)g(tan-)456 2103 y(gen)m(t)d(bundle)h Fk(T)14 b(M)41 b Fn(of)31 b(our)g(Riemannian)e(manifold)f Fk(M)10 b Fn(.)44 b(F)-8 b(or)30 b(ev)m(ery)j Fk(x)28 b Fj(2)g Fk(M)42 b Fn(let)456 2219 y(the)28 b(subspaces)i Fk(E)1138 2183 y Fg(h)1132 2244 y(x)1211 2219 y Fn(and)e Fk(E)1474 2183 y Fg(v)1468 2244 y(x)1543 2219 y Fn(b)s(e)g(those)h(from)e(the)h (generating)g(splitting)d(for)j(the)456 2336 y(family)23 b Fj(F)829 2299 y Fg(\015)900 2336 y Fn(and)j(set)h Fk(A)1302 2299 y Fg(h)1302 2360 y(x)1375 2336 y Fn(=)g Fk(H)1567 2299 y 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Fg(x)1118 1496 y Fn(=)f Fk(D)s(f)1367 1513 y Fg(f)1408 1494 y Fi(\000)p Fd(1)1491 1513 y Fh(\()p Fg(x)p Fh(\))1590 1496 y Fk(Q)1667 1513 y Fg(f)1708 1494 y Fi(\000)p Fd(1)1791 1513 y Fh(\()p Fg(x)p Fh(\))1890 1496 y Fn(.)68 b(The)42 b(op)s(erator)e Fk(f)2654 1460 y Fg(??)2769 1496 y Fn(is)g(a)h(con)m (traction)456 1613 y(an)33 b(a)h(neigh)m(b)s(ourho)s(o)s(d)f Fk(U)44 b Fn(of)33 b(the)h(unstable)g(plane)g(\014eld)f (\(distribution\))f Fk(E)3287 1576 y Fg(u)3366 1613 y Fn(of)456 1729 y Fk(f)11 b Fn(.)55 b(This)37 b(is)f(easy)h(and)g(follo) m(ws)e(the)i(pattern)g(of)f(Prop)s(osition)f(1.)55 b(Since)37 b Fk(E)3297 1693 y Fg(u)3379 1729 y Fn(is)456 1845 y(a)d(\014xed)i(p)s (oin)m(t)f(of)f(the)h(con)m(traction)g Fk(f)1892 1809 y Fg(??)2001 1845 y Fn(w)m(e)h(ha)m(v)m(e)h(that)d Fk(E)2666 1809 y Fg(u)2743 1845 y Fn(=)e(lim)2986 1860 y Fg(n)p Ff(!1)3191 1845 y Fk(f)3250 1809 y Fg(??n)3367 1845 y Fk(Q)456 1961 y Fn(uniformly)e(with)i(resp)s(ect)i(to)e Fk(Q)c Fj(2)g Fk(U)10 b Fn(.)555 2077 y(In)m(tegrabilit)m(y)36 b(is)g(a)h(lo)s(cal)e(prop)s(ert)m(y)-8 b(,)39 b(so)e(it)f(is)h(enough) g(to)g(sho)m(w)h(that)f Fk(E)3297 2041 y Fg(u)3379 2077 y Fn(is)456 2194 y(in)m(tegrable)43 b(at)g(a)h(p)s(oin)m(t.)77 b(Giv)m(en)43 b(a)h(p)s(oin)m(t)f Fk(x)48 b Fj(2)f Fk(M)55 b Fn(consider)44 b(its)f(successiv)m(e)456 2310 y(preimages)c Fk(x)974 2325 y Fg(l)1042 2310 y Fn(=)h Fk(f)1217 2274 y Ff(\000)p Fg(l)1298 2310 y Fn(\()p Fk(x)p Fn(\))17 b Fk(;)50 b(l)43 b Fn(=)e(0)p Fk(;)17 b Fn(1)p Fk(;)g Fn(2)p Fk(;)g(:::)p Fn(.)65 b(Around)41 b(ev)m(ery)h(p)s(oin)m(t)e Fk(x)3110 2325 y Fg(l)3176 2310 y Fn(tak)m(e)i(a)456 2426 y(c)m(hart)36 b(\()p Fk(U)815 2441 y Fg(l)842 2426 y Fk(;)17 b(\036)944 2441 y Fg(l)969 2426 y Fn(\).)55 b(Consider)37 b(a)f(foliation)d Fk(\037)2032 2441 y Fg(l)2094 2426 y Fn(of)j Fk(U)2275 2441 y Fg(l)2338 2426 y Fn(b)m(y)h(parallel)d Fk(k)s Fn(-planes)i(\(with)456 2542 y(resp)s(ect)f(to)f(the)h(map)f Fk(\036)1359 2557 y Fg(l)1384 2542 y Fn(\))g(suc)m(h)i(that)e(the)h (tangen)m(t)g(space)g(to)f Fk(C)2871 2557 y Fg(l)2931 2542 y Fn(at)g(the)h(p)s(oin)m(t)456 2659 y Fk(x)511 2674 y Fg(l)573 2659 y Fn(coincides)h(with)g Fk(E)1293 2622 y Fg(u)1287 2683 y(x)1338 2659 y Fn(.)55 b(Let)37 b Fk(X)1680 2674 y Fg(l)1742 2659 y Fn(b)s(e)f(a)g(distribution)f (tangen)m(t)h(to)g Fk(\037)3041 2674 y Fg(l)3067 2659 y Fn(.)55 b(If)36 b Fk(U)3316 2674 y Fg(l)3379 2659 y Fn(is)456 2775 y(c)m(hosen)j(small)d(enough,)k(then)e Fk(X)1712 2790 y Fg(l)1776 2775 y Fn(will)e(b)s(e)i(close)g(to)f Fk(E)2545 2739 y Fg(u)2628 2775 y Fn(in)h Fk(U)2814 2790 y Fg(l)2840 2775 y Fn(.)59 b(Using)38 b(parti-)456 2891 y(tion)d(of)i(unit)m(y)g(w)m(e)h(can)f(extend)i(the)e(lo)s(cal)e (distribution)g Fk(X)2712 2906 y Fg(l)2774 2891 y Fn(to)i(a)g(global,)e (not)456 3007 y(necessarily)47 b(in)m(tegrable)e(distribution)f Fk(GX)2125 3022 y Fg(l)2198 3007 y Fn(close)i(to)g Fk(E)2656 2971 y Fg(u)2701 3007 y Fn(,)k(i.e.)85 b Fk(GX)3146 3022 y Fg(l)3223 3007 y Fj(2)52 b Fk(U)10 b Fn(.)456 3124 y(Th)m(us)30 b(w)m(e)f(kno)m(w)h(that)e Fk(f)1355 3087 y Fg(??n)1472 3124 y Fn(\()p Fk(GX)1668 3139 y Fg(l)1694 3124 y Fn(\))g(con)m(v)m(erges)j(to)d Fk(E)2386 3087 y Fg(u)2459 3124 y Fn(when)i Fk(n)e Fn(go)s(es)h(to)f(in\014nit)m(y)-8 b(,)456 3240 y(uniformly)41 b(with)i(resp)s(ect)h(to)f Fk(l)r Fn(.)76 b(In)43 b(particular,)i(lim)2510 3255 y Fg(n)p Ff(!1)2715 3240 y Fk(C)2785 3255 y Fg(n)2877 3240 y Fn(=)h Fk(E)3077 3204 y Fg(u)3122 3240 y Fn(,)g(where)456 3356 y Fk(C)526 3371 y Fg(n)613 3356 y Fn(=)40 b Fk(f)788 3320 y Fg(??n)905 3356 y Fn(\()p Fk(GX)1101 3371 y Fg(n)1148 3356 y Fn(\))g(\(the)g(diagonal)e(tric)m(k\).)66 b(All)38 b(the)i Fk(C)2575 3371 y Fg(n)2622 3356 y Fn('s)h(are)f(in)m(tegrable)f (at)456 3472 y Fk(x)p Fn(,)31 b(namely)e Fk(C)975 3487 y Fg(n)1052 3472 y Fn(in)m(tegrates)h(to)g Fk(f)1673 3436 y Fg(n)1720 3472 y Fn(\()p Fk(\037)1819 3487 y Fg(n)1866 3472 y Fn(\))g(in)f(the)i(neigh)m(b)s(ourho)s(o)s(d)e Fk(f)2934 3436 y Fg(n)2981 3472 y Fn(\()p Fk(U)3085 3487 y Fg(n)3132 3472 y Fn(\))h(of)g(the)456 3588 y(p)s(oin)m(t)40 b Fk(x)p Fn(.)72 b(Because)43 b(in)m(tegrabilit)m(y)c(is)i(a)h(closed)g (condition,)g Fk(E)2853 3552 y Fg(u)2940 3588 y Fn(is)f(also)g(in)m (te-)456 3705 y(grable)31 b(at)h Fk(x)p Fn(.)43 b(Since)33 b Fk(x)f Fn(w)m(as)h(arbitrary)-8 b(,)31 b(the)i(unstable)f (distribution)e Fk(E)3121 3669 y Fg(u)3198 3705 y Fn(of)i(the)456 3821 y(di\013eomorphism)e Fk(f)44 b Fn(is)32 b(in)m(tegrable.)44 b(In)m(tegrabilit)m(y)32 b(of)g(the)h(stable)g(distribution)456 3937 y(can)f(b)s(e)h(obtained)f(b)m(y)i(replacing)d Fk(f)43 b Fn(with)32 b Fk(f)2093 3901 y Ff(\000)p Fh(1)2188 3937 y Fn(.)555 4053 y(Let)40 b(no)m(w)g Fk(f)51 b Fn(b)s(e)39 b(Anoso)m(v)i(in)e(the)h(standard)g(de\014nition,)g(then,)j(as)c(w)m(e) i(ha)m(v)m(e)456 4170 y(already)30 b(men)m(tioned)h Fk(f)1330 4133 y Fg(n)1408 4170 y Fn(is)f(\()p Fk(\025;)17 b(\026)p Fn(\)-Anoso)m(v)31 b(for)f(some)h Fk(n)p Fn(,)h(and)f(hence)i(has)e(in) m(te-)456 4286 y(grable)i(stable)h(and)g(instable)g(distributions,)f (but)i(these)g(coincide)f(with)g(those)456 4402 y(of)e Fk(f)11 b Fn(.)43 b(This)33 b(completes)f(the)h(pro)s(of)f(of)g (Theorem)h(1.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF