Content-Type: multipart/mixed; boundary="-------------9910270338808" This is a multi-part message in MIME format. ---------------9910270338808 Content-Type: text/plain; name="99-408.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-408.keywords" killing spinors, lorentzian manifolds ---------------9910270338808 Content-Type: application/postscript; name="preprint417.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="preprint417.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.85 Copyright 1999 Radical Eye Software %%Title: article.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips article.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.09.24:1354 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul 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5168 y Fm(0)2209 5154 y Fs(\))2244 5116 y Fm(2)2304 5154 y Fs(+)20 b Fr(\017)2442 5092 y Fs(1)p 2442 5133 46 4 v 2442 5216 a(4)2497 5154 y(\()p Fr(f)2587 5116 y Fl(0)2610 5154 y Fs(\))2645 5116 y Fm(2)2685 5154 y Fs(\()p Fr(t)2753 5168 y Fm(0)2793 5154 y Fs(\))p Fr(:)p eop %%Page: 6 6 6 5 bop 75 46 a Fm(6)1449 b(CHRISTOPH)30 b(BOHLE)75 241 y Fs(In)k(the)g(case)i Fr(f)605 208 y Fl(0)627 241 y Fs(\()p Fr(t)695 255 y Fm(0)735 241 y Fs(\))c(=)g(0)i(nothing)g(is)f (to)i(b)s(e)f(done)g(\(b)s(ecause)h(then)f(a)h(solution)e(of)h(\(I'\))h (corresp)s(onds)e(to)75 349 y(a)38 b(pair)f(of)h(Killing)c(spinors)i (to)j Fp(\006)p Fr(\025)e Fs(and)g(w)m(e)i(ha)m(v)m(e)g Fr(\025)1962 363 y Fq(F)2058 349 y Fp(2)e(f\006)p Fr(\025)p Fp(g)p Fs(\).)64 b(So)38 b(in)f(the)h(follo)m(wing)e(w)m(e)i(supp)s (ose)75 457 y Fr(f)130 424 y Fl(0)153 457 y Fs(\()p Fr(t)221 471 y Fm(0)260 457 y Fs(\))26 b Fp(6)p Fs(=)f(0.)75 618 y Fo(Case)38 b Fs(R)370 582 y Fq(F)453 618 y Fp(6)p Fs(=)25 b(0)31 b(\(or)g(equiv)-5 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Fq(')2155 5098 y Fr(<)g Fs(0.)40 b(This)24 b(is)h(done)h(using)e(the)i(classi\014cation)75 5205 y(results)j(for)h(complete)h(1-connected)h(Riemannian)c(manifolds) h(with)g(real)h(Killing)d(spinors.)p eop %%Page: 11 11 11 10 bop 1036 46 a Fm(KILLING)32 b(SPINORS)f(ON)f(LORENTZIAN)h(MANIF)n (OLDS)890 b(11)75 241 y Fs(5.1.)54 b Fy(Lo)s(cal)47 b(W)-9 b(arp)s(ed)47 b(Pro)s(duct)h(Structure.)d Fs(Let)d(\()p Fr(M)2204 208 y Fq(n;)p Fm(1)2306 241 y Fr(;)15 b(g)s Fs(\))42 b(b)s(e)e(a)h(connected)h(Loren)m(tzian)f(spin)75 349 y(manifold.)c(Let)28 b Fr(')d Fp(2)g Fs(\000\()p Fr(S)5 b Fs(\))27 b(b)s(e)f(a)h(real)f(Killing)e(spinor)g(to)k(Killing) 23 b(n)m(um)m(b)s(er)i Fr(\025)h Fp(2)e Fn(R)s Fp(n)q(f)p Fs(0)p Fp(g)q Fs(.)45 b(The)26 b(asso)s(ciated)75 457 y(v)m(ector)39 b(\014eld)e Fr(V)613 471 y Fq(')701 457 y Fs(of)h(the)g(spinor)e Fr(')i Fp(2)g Fs(\000\()p Fr(S)5 b Fs(\))38 b(is)f(de\014ned)g(to)h(b)s(e)f(the)h(v)m(ector)i(\014eld)c (dual)h(to)i(the)f(1-form)75 567 y Fr(!)132 581 y Fq(')182 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b(de\014ned)f(on)i(the)f(op)s(en)g(sets)h Fp(V)1265 5112 y Fl(\006)1363 5098 y Fs(=)g Fp(f)p Fr(p)g Fp(2)g Fr(M)49 b Fs(:)39 b Fr(v)s Fs(\()p Fr(p)p Fs(\))h Fd(?)f Fs(0)p Fp(g)g Fs(=)g Fp(f)p Fr(p)g Fp(2)g Fr(M)49 b Fs(:)39 b Fr(Q)2959 5112 y Fq(')3048 5098 y Fd(?)g Fr(u)3210 5065 y Fm(2)3250 5098 y Fs(\()p Fr(p)p Fs(\))p Fp(g)p Fs(.)66 b(In)38 b(the)75 5205 y(follo)m(wing,)d(de\014ne)f Fr(\017)10 b Fs(:)32 b Fp(V)917 5219 y Fm(+)999 5205 y Fp([)22 b(V)1138 5219 y Fl(\000)1230 5205 y Fp(\000)-16 b(!)33 b(f\006)p Fs(1)p Fp(g)92 b Fr(p)32 b Fp(7!)h Fs(sgn)o(\()p Fr(v)s Fs(\()p Fr(p)p Fs(\)\))k(to)e(b)s(e)f(the)h(lo)s(cally)f (constan)m(t)i(sign)d(of)i Fr(v)s Fs(.)p eop %%Page: 12 12 12 11 bop 75 46 a Fm(12)1414 b(CHRISTOPH)30 b(BOHLE)75 241 y Fs(The)f(normalized)g(gradien)m(t)g(\014eld)g Fr(\030)k Fs(is)c(ligh)m(tlik)m(e)f(on)i Fp(V)1969 255 y Fl(\000)2057 241 y Fs(and)g(spacelik)m(e)f(on)h Fp(V)2794 255 y Fm(+)2853 241 y Fs(.)40 b(The)29 b(union)f(of)i Fp(V)3515 255 y Fm(+)3604 241 y Fs(and)75 349 y Fp(V)131 363 y Fl(\000)220 349 y Fs(is)f(dense,)i(b)s(ecause)f(its)g(complemen)m(t)g(consists)g (of)h(h)m(yp)s(ersurfaces.)75 507 y(By)g(Corollary)e(18,)i(there)g(are) g(three)f(p)s(ossibilities)c(dep)s(ending)i(on)i(the)h(\014rst)e(in)m (tegral)h Fr(Q)3182 521 y Fq(')3232 507 y Fs(:)200 693 y Fp(\017)42 b Fr(Q)359 707 y Fq(')434 693 y Fr(<)25 b Fs(0)31 b(and)f Fr(M)35 b Fs(=)25 b Fp(V)1058 707 y Fl(\000)1117 693 y Fs(,)200 803 y Fp(\017)42 b Fr(Q)359 817 y Fq(')434 803 y Fs(=)25 b(0)31 b(and)f Fr(M)35 b Fs(=)25 b Fp(V)1058 817 y Fl(\000)1225 792 y Fs(_)1208 803 y Fp([)90 b Fr(v)1406 770 y Fl(\000)p Fm(1)1501 803 y Fs(\(0\))122 b(or)200 914 y Fp(\017)42 b Fr(Q)359 928 y Fq(')434 914 y Fr(>)25 b Fs(0)31 b(and)f Fr(M)35 b Fs(=)25 b Fp(V)1058 928 y Fl(\000)1225 903 y Fs(_)1208 914 y Fp([)90 b Fr(v)1406 881 y Fl(\000)p Fm(1)1501 914 y Fs(\(0\))1725 903 y(_)1708 914 y Fp([)g(V)1915 928 y Fm(+)1974 914 y Fs(.)75 1100 y(Using)30 b(equation)g(\(2\))h(one)g (can)g(see)g(that)g Fr(\030)j Fs(satis\014es)985 1289 y Fr(\030)t Fs(\()p Fr(u)p Fs(\))26 b(=)f Fr(\017)1310 1219 y Fp(p)p 1385 1219 85 4 v 1385 1289 a Fr(\017v)367 b Fp(r)1909 1303 y Fq(X)1976 1289 y Fr(\030)30 b Fs(=)25 b Fp(\000)2276 1227 y Fr(u)p 2223 1268 160 4 v 2223 1286 a Fp(p)p 2298 1286 85 4 v 2298 1351 a Fr(\017v)2407 1289 y Fs(pro)5 b(j)2572 1311 y Fq(\030)2606 1292 y Fg(?)2662 1289 y Fs(\()p Fr(X)i Fs(\))p Fr(:)-2764 b Fs(\(3\))75 1507 y(The)25 b(last)h(equation)f(implies)e(that)j Fr(\030)k Fs(is)25 b(closed)g(\(i.e.)39 b(rot)q(\()p Fr(V)20 b Fs(\))26 b(=)f Fr(g)s Fs(\()p Fp(r)2436 1521 y Fq(:)2460 1507 y Fr(\030)t(;)15 b(:)p Fs(\))c Fp(\000)g Fr(g)s Fs(\()p Fr(:;)k Fp(r)2919 1521 y Fq(:)2945 1507 y Fr(\030)t Fs(\))25 b(=)g(0\))i(and)e(geo)s(desic)75 1615 y(\(i.e.)41 b Fp(r)343 1630 y Fq(\030)380 1615 y Fr(\030)30 b Fs(=)25 b(0\).)41 b(Consequen)m(tly)-8 b(,)30 b(the)h(follo)m(wing)e(lemma)h (applies)e(to)j(the)g(in)m(tegral)f(curv)m(es)h(of)f Fr(\030)t Fs(.)75 1791 y Fy(Lemma)j(19.)46 b Fo(L)-5 b(et)34 b Fs(\()p Fr(M)916 1758 y Fq(n;)p Fm(1)1019 1791 y Fr(;)15 b(g)s Fs(\))36 b Fo(b)-5 b(e)34 b(a)h(L)-5 b(or)g(entzian)36 b(spin)f(manifold)i(and)e(let)g Fr(u)28 b Fs(=)p Fr(<)g(';)15 b(')30 b(>)k Fo(b)-5 b(e)35 b(the)f(length)75 1901 y(function)i(of)f(a)h(r)-5 b(e)g(al)37 b(Kil)5 b(ling)35 b(spinor)i Fr(')31 b Fp(2)f Fs(\000\()p Fr(S)5 b Fs(\))36 b Fo(to)g Fp(\006)1989 1865 y Fm(1)p 1989 1880 36 4 v 1989 1932 a(2)2034 1901 y Fo(.)50 b(L)-5 b(et)36 b Fr(\015)k Fo(b)-5 b(e)36 b(an)g(arbitr)-5 b(ary)37 b(ge)-5 b(o)g(desic)37 b(having)f(the)75 2010 y(length)e Fr(\017)27 b Fs(=)f Fr(g)s Fs(\()p Fr(\015)637 1977 y Fl(0)661 2010 y Fs(\(0\))p Fr(;)15 b(\015)868 1977 y Fl(0)893 2010 y Fs(\(0\)\))29 b Fp(2)d(f\000)p Fs(1)p Fr(;)15 b Fs(0)p Fr(;)g Fs(1)p Fp(g)p Fo(.)47 b(F)-7 b(or)34 b(the)g(length)g(function)g Fr(u)p Fs(\()p Fr(t)p Fs(\))27 b(=)g Fr(u)p Fs(\()p Fr(\015)5 b Fs(\()p Fr(t)p Fs(\)\))34 b Fo(of)g(the)g(spinor)h Fr(')75 2118 y Fo(along)f Fr(\015)5 b Fo(,)32 b(the)h(fol)5 b(lowing)34 b(holds:)1011 2418 y Fr(u)p Fs(\()p Fr(t)p Fs(\))26 b(=)1288 2204 y Fj(8)1288 2286 y(>)1288 2314 y(<)1288 2477 y(>)1288 2504 y(:)1369 2292 y Fr(u)p Fs(\(0\))15 b(cos)r(\()p Fr(t)p Fs(\))21 b(+)f Fr(u)1941 2259 y Fl(0)1964 2292 y Fs(\(0\))15 b(sin\()p Fr(t)p Fs(\))194 b Fr(\017)25 b Fs(=)g(1)1369 2421 y Fr(u)p Fs(\(0\))c(+)f Fr(u)1700 2388 y Fl(0)1723 2421 y Fs(\(0\))p Fr(t)633 b(\017)25 b Fs(=)g(0)1369 2551 y Fr(u)p Fs(\(0\))15 b(cosh)q(\()p Fr(t)p Fs(\))21 b(+)f Fr(u)1991 2518 y Fl(0)2014 2551 y Fs(\(0\))15 b(sinh\()p Fr(t)p Fs(\))93 b Fr(\017)25 b Fs(=)g Fp(\000)p Fs(1)p Fr(:)75 2791 y Fo(Pr)-5 b(o)g(of.)43 b Fs(Equation)30 b(\(2\))h(implies)d(that)j Fr(u)f Fs(along)g Fr(\015)36 b Fs(has)30 b(to)h(satisfy)f(the)g(di\013eren)m(tial)f (equation)211 2956 y Fr(u)263 2918 y Fl(00)305 2956 y Fs(\()p Fr(t)p Fs(\))d(=)f Fr(\015)582 2918 y Fl(0)606 2956 y Fs(\()p Fr(t)p Fs(\))p Fr(g)s Fs(\(grad)q(\()p Fr(u)p Fs(\))p Fr(;)15 b(\015)1182 2918 y Fl(0)1207 2956 y Fs(\()p 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b(we)g(have)g Fr(d)p Fs(\010)2665 3415 y Fq(t)2694 3401 y Fs(\()p Fr(\030)t Fs(\))31 b(=)f Fr(\030)40 b Fo(and)c Fr(d)p Fs(\010)3312 3415 y Fq(t)3342 3401 y Fs(\()p Fr(\030)3421 3368 y Fl(?)3480 3401 y Fs(\))31 b(=)f Fr(\030)3691 3368 y Fl(?)75 3520 y Fo(for)j(any)g Fr(t)g Fo(wher)-5 b(e)34 b(these)f(expr)-5 b(essions)34 b(ar)-5 b(e)33 b(de\014ne)-5 b(d.)43 b(F)-7 b(urthermor)i(e,)35 b(for)f(any)f Fr(X)r(;)15 b(Y)46 b Fp(2)25 b Fr(\030)3139 3487 y Fl(?)3135 3542 y Fq(p)3230 3520 y Fo(we)33 b(have)703 3752 y Fs(\010)769 3714 y Fl(\003)769 3774 y Fq(t)808 3752 y Fr(g)s Fs(\()p Fr(X)r(;)15 b(Y)22 b Fs(\))j(=)g Fr(e)1279 3702 y Fl(\000)p Fm(2)1381 3648 y Ff(R)1428 3669 y Fh(t)1415 3726 y Ft(0)1514 3675 y Fh(u)p 1478 3687 109 3 v 1478 3695 a Fg(p)p 1528 3695 59 3 v 36 x Fh(\017v)1597 3702 y Fm(\(\010)1675 3710 y Fh(s)1709 3702 y Fm(\()p Fq(p)p Fm(\)\))p Fq(ds)1920 3752 y Fp(\001)20 b Fr(g)2008 3766 y Fq(p)2048 3752 y Fs(\()p Fr(X)r(;)15 b(Y)21 b Fs(\))26 b(=)f(\()2483 3690 y Fr(u)2535 3657 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Fr(t)p Fs(\))h(satis\014es)f(the)g(linear)f(di\013eren)m(tial)g (equation)1422 5154 y Fr(a)1470 5117 y Fl(0)1493 5154 y Fs(\()p Fr(t)p Fs(\))d(=)f Fp(\000)1830 5093 y Fs(2)p Fr(u)p 1799 5133 160 4 v 1799 5151 a Fp(p)p 1875 5151 85 4 v 66 x Fr(\017v)1969 5154 y Fs(\(\010)2070 5168 y Fq(t)2099 5154 y Fs(\()p Fr(p)p Fs(\)\))p Fr(a)p Fs(\()p Fr(t)p Fs(\))p eop %%Page: 13 13 13 12 bop 1036 46 a Fm(KILLING)32 b(SPINORS)f(ON)f(LORENTZIAN)h(MANIF)n (OLDS)890 b(13)75 271 y Fs(b)s(eing)24 b(solv)m(ed)h(b)m(y)g(\010)770 238 y Fl(\003)770 294 y Fq(t)809 271 y Fr(g)s Fs(\()p Fr(X)r(;)15 b(Y)21 b Fs(\))26 b(=)f Fr(e)1280 221 y Fl(\000)p Fm(2)1382 168 y Ff(R)1429 188 y Fh(t)1416 246 y Ft(0)1515 194 y Fh(u)p 1479 206 109 3 v 1479 214 a Fg(p)p 1529 214 59 3 v 36 x Fh(\017v)1598 221 y Fm(\(\010)1676 229 y Fh(s)1709 221 y Fm(\()p Fq(p)p Fm(\)\))p Fq(ds)1910 271 y Fp(\001)10 b Fr(g)1988 285 y Fq(p)2028 271 y Fs(\()p Fr(X)r(;)15 b(Y)21 b Fs(\))p Fr(:)26 b Fs(Using)e(the)h(equations)g (\(3\))h(and)f(Lemma)75 379 y(19,)31 b(w)m(e)g(ha)m(v)m(e)h Fr(u)617 346 y Fl(0)665 379 y Fs(=)25 b Fr(\017)798 314 y Fp(p)p 874 314 85 4 v 65 x Fr(\017v)34 b Fs(and)29 b Fr(u)1217 346 y Fl(0)q(0)1285 379 y Fs(=)c Fp(\000)p Fr(\017u)p Fs(.)40 b(So)31 b(w)m(e)f(get)746 581 y(\010)812 544 y Fl(\003)812 604 y Fq(t)851 581 y Fr(g)s Fs(\()p Fr(X)r(;)15 b(Y)22 b Fs(\))j(=)g Fr(e)1322 541 y Fm(2)1370 488 y Ff(R)1417 508 y Fh(t)1403 566 y Ft(0)1466 514 y Fh(u)1503 493 y Fg(00)p 1466 526 78 3 v 1476 571 a Fh(u)1513 557 y Fg(0)1554 541 y Fm(\(\010)1632 549 y Fh(s)1665 541 y Fm(\()p Fq(p)p Fm(\)\))p Fq(ds)1877 581 y Fp(\001)20 b Fr(g)1965 595 y Fq(p)2005 581 y Fs(\()p Fr(X)r(;)15 b(Y)21 b Fs(\))26 b(=)f(\()2439 520 y Fr(u)2491 487 y Fl(0)2515 520 y Fs(\()p Fr(t)p Fs(\))p 2433 560 192 4 v 2433 643 a Fr(u)2485 617 y Fl(0)2508 643 y Fs(\(0\))2634 581 y(\))2669 544 y Fm(2)2709 581 y Fr(g)2752 595 y Fq(p)2792 581 y Fs(\()p Fr(X)r(;)15 b(Y)21 b Fs(\))p Fr(:)p 3684 776 4 62 v 3688 718 55 4 v 3688 776 V 3742 776 4 62 v 75 980 a Fs(By)26 b(de\014nition,)f(the)h(lev)m(el)g(surfaces)g(of)g Fr(u)g Fs(are)g(in)m(tegral)g(manifolds)d(of)j(the)h(geometric)g (distribution)22 b(giv)m(en)75 1088 y(b)m(y)29 b Fr(\030)244 1055 y Fl(?)303 1088 y Fs(.)41 b(The)29 b(preceding)f(lemma)h(implies)e (that)j(\010)1825 1102 y Fq(t)1855 1088 y Fs(,)g(whenev)m(er)f (de\014ned)f(on)i(a)g(connected)g(piece)g(of)f(lev)m(el)75 1196 y(surface,)36 b(maps)f(that)h(piece)e(of)i(lev)m(el)e(surface)h (on)g(another)g(connected)h(piece)f(of)g(lev)m(el)g(surface.)55 b(Th)m(us,)75 1304 y(on)30 b(the)h(dense)f(set)h Fp(V)804 1318 y Fm(+)883 1304 y Fp([)19 b(V)1019 1318 y Fl(\000)1109 1304 y Fs(w)m(e)30 b(ha)m(v)m(e)i(the)f(follo)m(wing)d(lo)s(cal)i(form) g(of)g Fr(M)10 b Fs(.)75 1467 y Fy(Lemma)33 b(21.)46 b Fo(L)-5 b(et)38 b Fs(\()p Fr(M)920 1434 y Fq(n;)p Fm(1)1022 1467 y Fr(;)15 b(g)s Fs(\))39 b Fo(b)-5 b(e)37 b(a)h(c)-5 b(onne)g(cte)g(d)39 b(L)-5 b(or)g(entzian)40 b(spin)e(manifold)h (admitting)g(a)f(non-trivial)75 1575 y(r)-5 b(e)g(al)40 b(Kil)5 b(ling)39 b(spinor)h Fr(')f Fo(and)h(let)f Fr(u)d Fs(=)p Fr(<)g(';)15 b(')38 b(>)p Fo(.)60 b(L)-5 b(et)39 b Fr(p)e Fp(2)f Fr(M)2326 1542 y Fq(n;)p Fm(1)2466 1575 y Fo(such)j(that)i Fr(v)s Fs(\()p Fr(p)p Fs(\))c Fp(6)p Fs(=)f(0)p Fo(.)61 b(Then)39 b(ther)-5 b(e)75 1683 y(is)38 b(a)h(c)-5 b(onne)g(cte)g(d)39 b(op)-5 b(en)40 b(neighb)-5 b(orho)g(o)g(d)41 b Fp(V)1500 1697 y Fq(p)1575 1683 y Fp(\022)35 b(V)1737 1697 y Fq(\017)1808 1683 y Fo(\()p Fr(\017)g Fs(=)g Fr(sg)s(n)p Fs(\()p Fr(v)s Fs(\()p Fr(p)p Fs(\)\))p Fo(\))40 b(isometric)f(to)f(the)h(warp)-5 b(e)g(d)41 b(pr)-5 b(o)g(duct)75 1793 y Fs(\()p Fr(F)181 1760 y Fq(n)p Fl(\000)p Fm(1)346 1793 y Fp(\002)28 b Fs(\()p Fp(\000)p Fr(\016)n(;)15 b(\016)s Fs(\))p Fr(;)g(f)802 1760 y Fm(2)844 1793 y Fs(\()p Fr(t)p Fs(\))p Fr(g)990 1812 y Fl(j)1010 1823 y Fh(F)1093 1793 y Fs(+)27 b Fr(\017dt)1308 1760 y Fm(2)1348 1793 y Fs(\))p Fo(,)46 b(wher)-5 b(e)44 b Fr(F)1795 1760 y Fq(n)p Fl(\000)p Fm(1)1977 1793 y Fs(=)g 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b(geo)s(desic)g(with)f Fr(\015)3108 770 y Fl(0)3131 803 y Fs(\(0\))i(=)e Fr(\030)t Fs(\()p Fr(p)p Fs(\).)62 b(By)75 936 y(equation)30 b(\(6\))i(and)d Fr(u)821 903 y Fl(0)845 936 y Fs(\(0\))d Fr(<)f Fs(0)31 b(w)m(e)g(ha)m(v)m(e) 1522 892 y Fq(u)p Fm(\(0\))p 1511 915 154 4 v 1511 969 a Fq(u)1552 950 y Fg(0)1575 969 y Fm(\(0\))1700 936 y Fs(=)25 b Fp(\007)p Fs(1,)31 b(so)f(the)h(length)f(function)f Fr(u)h Fs(along)g Fr(\015)36 b Fs(satis\014es)1032 1110 y Fr(u)p Fs(\()p Fr(t)p Fs(\))26 b(=)f Fr(u)p Fs(\(0\))15 b(cosh)q(\()p Fr(t)p Fs(\))21 b(+)f Fr(u)1931 1073 y Fl(0)1954 1110 y Fs(\(0\))15 b(sinh\()p Fr(t)p Fs(\))26 b(=)f Fr(u)p Fs(\(0\))p Fr(e)2682 1073 y Fl(\007)p Fq(t)2768 1110 y Fr(:)75 1273 y Fs(Hence,)36 b(for)d(an)m(y)h Fr(p)d Fp(2)g(V)924 1227 y Fc(?)917 1296 y Fl(\000)983 1273 y Fs(,)k(the)f(Geo)s(desic)g Fr(\015)39 b Fs(sta)m(ys)34 b(in)f Fp(V)2079 1227 y Fc(?)2072 1296 y Fl(\000)2172 1273 y Fs(and)g(runs)f(through)h(all)g(lev)m(el)g(surfaces)h(of)g Fr(u)75 1405 y Fs(lying)29 b(in)g Fp(V)469 1359 y Fc(?)462 1428 y Fl(\000)528 1405 y Fs(.)75 1561 y(If)h(w)m(e)h(tak)m(e)h Fr(F)38 b Fs(=)25 b Fr(u)741 1528 y Fl(\000)p Fm(1)835 1561 y Fs(\()p Fp(\006)p Fs(1\),)32 b(the)e(map)1192 1724 y(\011)10 b(:)31 b Fr(F)i Fp(\002)20 b Fn(R)34 b Fp(\000)-16 b(!)25 b(V)1836 1678 y Fc(?)1829 1747 y Fl(\000)2077 1724 y Fs(\()p Fr(x;)15 b(t)p Fs(\))27 b Fp(7!)e Fs(\010)2481 1738 y Fq(t)2510 1724 y Fs(\()p Fr(x)p Fs(\))75 1870 y(de\014ned)39 b(in)g(Lemma)i(21)g(is)e(a)i(di\013eomorphism.)68 b(It)40 b(is)g(injectiv)m(e,)i(b)s(ecause)f(u)e(is)h(strictly)f (decreasing)75 1989 y(along)32 b(geo)s(desic)f(\(equation)h(\(3\)\),)i (and)d(it)g(is)g(surjectiv)m(e,)h(b)s(ecause)f(for)h(an)m(y)g(p)s(oin)m (t)e(in)h Fp(V)3150 1943 y Fc(?)3143 2013 y Fl(\000)3209 1989 y Fs(,)h(the)g(geo)s(desic)75 2121 y Fr(\015)5 b Fs(,)31 b(whic)m(h)e(is)g(the)i(in)m(tegral)f(curv)m(e)g(of)h Fr(\030)j Fs(runs)29 b(through)h(all)f(lev)m(el)h(surfaces)g(of)h Fr(u)f Fs(lying)f(in)g Fp(V)3244 2075 y Fc(?)3237 2145 y Fl(\000)3303 2121 y Fs(.)75 2295 y(Therefore,)c(b)m(y)e(Lemma)h(21)g (the)g(sets)g Fp(V)1442 2249 y Fc(?)1435 2318 y Fl(\000)1524 2295 y Fs(are)g(isomorphic)e(to)i(w)m(arp)s(ed)e(pro)s(ducts,)i(the)g (w)m(arping)e(function)75 2407 y(b)s(eing)29 b Fr(f)10 b Fs(\()p Fr(t)p Fs(\))25 b(=)g Fr(e)641 2374 y Fl(\007)p Fq(t)756 2407 y Fs(b)m(y)30 b(the)h(ab)s(o)m(v)m(e)h(form)m(ula)d(for)h Fr(u)h Fs(along)f(in)m(tegral)g(curv)m(es)h(of)f Fr(\030)t Fs(.)75 2557 y(The)g(spin)e(geometric)k(part)e(of)g(the)h(statemen)m(t) h(follo)m(ws)e(from)g(Theorems)g(6)g(and)g(7.)75 2707 y(Sho)m(wing)g(that)i Fr(F)45 b Fs(is)31 b(complete)h(is)e(a)i(little)f (bit)f(more)i(tec)m(hnical)g(as)f(in)g(the)g(preceding)g(case.)45 b(W)-8 b(e)33 b(pro)m(v)m(e)75 2814 y(that)f(there)f(is)f(an)g Fr(\017)c(>)g Fs(0)32 b(suc)m(h)e(that)i(for)e(an)m(y)i Fr(p)26 b Fp(2)f Fr(F)44 b Fs(and)31 b(an)m(y)g Fr(X)i Fp(2)26 b Fr(T)2534 2828 y Fq(p)2574 2814 y Fr(F)44 b Fs(of)31 b(length)f(1,)i(the)f(geo)s(desic)g(in)75 2922 y Fr(F)41 b Fs(b)s(eing)26 b(tangen)m(t)j(to)g Fr(X)35 b Fs(in)27 b Fr(p)g Fs(exists)h(at)g(least)g(on)g(the)g(in)m(terv)-5 b(al)27 b(\()p Fp(\000)p Fr(\017;)15 b(\017)p Fs(\).)40 b(As)28 b Fr(\017)g Fs(is)f(indep)s(enden)m(t)e(of)j Fr(p)g Fs(and)75 3030 y Fr(X)7 b Fs(,)31 b(ev)m(ery)g(geo)s(desic)g(in) e Fr(F)43 b Fs(can)31 b(b)s(e)f(extended)g(to)h Fn(R)s Fs(.)75 3180 y(Let)25 b Fr(\015)k Fs(b)s(e)24 b(the)g(geo)s(desic)g(in) f Fr(M)35 b Fs(with)23 b Fr(\015)1402 3147 y Fl(0)1425 3180 y Fs(\(0\))j(=)f Fr(X)33 b Fp(2)25 b Fr(T)1909 3194 y Fq(p)1949 3180 y Fr(F)13 b Fs(.)38 b(Because)26 b Fr(\015)j Fs(is)24 b(spacelik)m(e)g(and)f(has)h(length)g(1,)i(b)m(y)75 3308 y(Lemma)k(19)h(w)m(e)g(ha)m(v)m(e)g Fr(u)p Fs(\()p Fr(\015)5 b Fs(\()p Fr(s)p Fs(\)\))27 b(=)e Fp(\006)15 b Fs(cos\()p Fr(s)p Fs(\).)41 b(Th)m(us,)30 b(for)g Fr(s)25 b Fp(2)f Fs(\()p Fp(\000)2323 3272 y Fq(\031)p 2323 3287 43 4 v 2327 3340 a Fm(2)2376 3308 y Fr(;)2427 3272 y Fq(\031)p 2427 3287 V 2431 3340 a Fm(2)2480 3308 y Fs(\),)30 b Fr(\015)36 b Fs(sta)m(ys)31 b(in)e Fp(V)3050 3262 y Fc(?)3043 3332 y Fl(\000)3109 3308 y Fs(.)40 b(There)30 b(w)m(e)h(ha)m(v)m(e)75 3418 y Fr(\015)g Fs(=)26 b(\()p Fr(\014)5 b(;)15 b(\013)p Fs(\))27 b Fp(2)f Fr(F)33 b Fp(\002)20 b Fn(R)s Fs(.)48 b(By)31 b(Lemma)g(24,)h Fr(\014)k Fs(is)29 b(pregeo)s(desic)i(in)e Fr(F)44 b Fs(with)29 b Fr(\014)2651 3385 y Fl(0)2675 3418 y Fs(\(0\))e(=)f Fr(X)7 b Fs(.)42 b(P)m(arameterizing)31 b Fr(\014)75 3533 y Fs(on)g(arc-length)h(yields)d(a)j(geo)s(desic)1336 3509 y(~)1323 3533 y Fr(\014)6 b Fs(.)43 b(What)32 b(w)m(e)g(ha)m(v)m (e)g(to)g(sho)m(w)g(is)e(that)i(the)f(arc)h(length)f(of)g Fr(\014)37 b Fs(in)30 b(b)s(oth)75 3641 y(directions)35 b(from)h(0)g(is)g(at)h(least)f Fr(\017)p Fs(,)i(where)e Fr(\017)g Fs(is)g(indep)s(enden)m(t)e(of)i Fr(p)g Fs(and)g Fr(X)7 b Fs(.)59 b(Then)35 b Fr(F)49 b Fs(is)36 b(geo)s(desically)75 3749 y(complete,)31 b(b)s(ecause)f(ev)m(ery)i(geo)s(desic)e(can)h(b)s (e)f(extended.)75 3899 y(Because)40 b Fr(u)f 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