Appendix. The Theorem of Lustemik andSchnirelmann
One owes to Lusternik and Schnirelmann a general axiomatic theory of the calculus of variations; see [LS 2], [Ly]. The greatest achievements of this theory, as applied to closed geodesics, are the so-called Lusternik-Schnirelmann Theorem that there always exist three non-self-intersecting closed geodesics on a surface of genus O (see [LS 1]) and the Lyusternik-Fet Theorem on the existence of at least one closed geodesic on every compact Riemannian manifold (see [LF]). In this appendix we prove these theorems using the Lusternik-Schnirelmann theory. We have made this appendix completely independent of the main part of these Lectures, where we developed the extended Morse theory of the Rie• mannian Hilbert manifold AM of closed H1-curves with its functional E. It will become evident at once-and this is true in particular for the proof of the Lyusternik-Fet Theorem-that the Lusternik-Schnirelmann approach to the theory of closed geodesics is much more elementary and less complicated than Morse's approach in its extended version dealing with (AM,E). But then, the Morse theory of (AM, E) is much more sensitive to the finer aspects of the closed geodesic problem - witness the existence theorem of infinitely many closed geodesics in Chapter 4. The source of the power of this approach is the Morse complex (see (2.5)). So, upon comparison, the extension of Morse's ideas must be considered better adapted to the theory of closed geodesics than the Lusternik-Schnirelmann theory, with one noteworthy exception, namely the Lusternik-Schnirelmann Theorem on the existence of non-self-intersecting closed geodesics. Morse's theory does not seem to be suita bie for finding such special prime closed geo• desics. Section 1 of this appendix gives a brief, elementary introduction to the Lusternik-Schnirelmann theory as relevant to the closed geodesics problem. We obtain a short proof of the Lyusternik-Fet Theorem. In the second section, we consider the length-decreasing (or rather, energy• decreasing) deformations in the class of non-self-intersecting closed curves on the 2-sphere. Our deformations are considerably more elementary than the one used in [LS 2] and [Ly]. In Section 3, finaIly, we give a complete elementary proof of the Lusternik• Schnîrelmann Theorem. The topological prerequisites are kept to a minimum. In particular, we have avoided any reference to Pontrjagin's Theorem of the "removal of cycles", which plays an essential role in the earlier proofs. We condude this introduction by drawing attention to the fact that the Lusternik-Schnirelmann theory can be applied immediately to Finsler manifolds, 204 Appendix. The Theorem of Lustemik and Schnirelmann even with a non-symmetric Finsler structure. It is doubtful that the Morse theory of (AM,E) can also be extended fully from Riemannian manifolds to Finsler manifolds. For a partial extension, we refer to Mercuri [Mer]. In the case of a non-symmetric Finsler structure on S2 the methods of Luster• nik and Schnirelmann yield the existence of only two non-self-intersecting closed geodesics; this reflects the fact that the space of oriented unparameterized great circles on S2 is isomorphic to S2 and therefore its Z2-homology is 2-dimensional, whereas the space of non-oriented unparameterized great circles on S2 is isomor• phic to the projective plane p 2 and thus, its Z2-homology is 3-dimensional. An example due to Katok [Kat] shows that there actually exist non-symmetric Finsler structures on S2 with only two closed geodesics (not counting the multiple coverings). See Matthias [Mat] for an elementary presentation of this example. Using the methods of Lusternik-Schnirelmann, as presented in this appendix, Thorbergsson [Thr] has investigated the existence of closed geodesics on non• compact Riemannian manifolds. In preparing this appendix the assistance of W. Ballmann was very valuable. He has applied the Lusternik-Schnirelmann method to construct non-self-inter• secting closed geodesics on surfaces of arbitrary genus; see [Ba].
A.l The Space PM and the Theorem of Lyusternik and Fet
Let M be a compact Riemannian manifold. The set CO(S,M) of continuous maps c: S= [O, 1]/{0, 1 } ..... M is in a natural way a metric space (Banach manifold) by taking as distance
d",(c, c') = sUP d(c(t), c'(t)}. IeS
Denote by P M the subspace of CO (S, M) formed by the piecewise differentiable closed curves. For cEPM we have the functions
L(c) = J Icldt, and E(c)=tj Icl2dt. s s
They are related by
L(c).:;;V2E(c) with equality if and only if c is parameterized proportionally to arc length, i.e. if L(cl[O,tJ)=tL(c). Since M is compact, there exists an '1 > O such that any two points p and q on M with d(p,q)':;;2'1 can be joined by an uniquely determined geodesic cpq =cpq(t), 0.:;;t.:;;1, L(cpq )=V2E(cpq )=d(p,q). Here, as always, we assume that geodesics are parameterized proportionally to arc length. A.1 The Space PM and the Theorem of Lyusternik and Fet 205
A.I.I Proposition. Let {Cn} be a sequence of piecewise differentiable paths cn:[O,l]--+M. Put cn(O)=Pn, cn(1)=qn and assume that d(Pn,qn)
{cpn(k)rn(k): [O,tn(k)]-+M} {crn(k)qn(k): [tn(k),l]-+M} of unique minimizing geodesics converge to cpr:[O,to]--+M and crq :[to,l]--+M, respectively. From our hypothesis we have
E(cpr ) + E(crq ) = Iim {E(cpn(k)rn(k») + E(crn(k)qn(k)} Since cpq : [O, 1]--+ M is the unique geodesic segment of minimal E-value from P to q it follows that the curve cprucrq from P to q coincides with CPq , In particular, the value r for the parameter t=to must be equal to cpq(to)' This completes the proof of (A.1.1). D Let tI> Obe as above. FixK> Oand choose an even integer k> Owith 4Klk Here, as always, t is to be taken modulo 1. We now proceed to define a continuous E-decreasing deformation from PKM into itself. Letj be an even integer, O f1fl aC(t)=C(t), for tE [O,jjk] or tE[a,1], f1fl .. clU/k, a] = cCWk)c( .. )IU/k, a]. Here, as always, cpq denotes the minimizing geodesic from P to q with the indi• cated parameter interval. Note that d(c(jjk), c(a)) PiJ.,.c(t)=c(t), for tE[1/k,0'+1)/k] or tE[0'-1 +l/k, 1 +l/k], PiJ .,.cl [0' + 1)/k, O' -1 + 1/k] = cC((j+l )/k)c(.,.-l +l/k) I[0' + 1)/k, O' -1 + 1/k]. We now define PiJ(O',c) for O'E[O,2], cEP"M, to be the subsequent application of the mappings PiJ2/k , ••• , PiJ21 /k , PiJ., where 21 is the even integer determined by 21~ kO'< 2'+ 2. A.t.2 Proposition. The mapping PiJ: [0,2] x P"M ...... P"M is continuous. Moreover, E(PiJ(2,c))~E(c), with equality if and only if c is either a constant map or a closed geodesic, i.e. c:8...... M is an immersion satisfying Vc=O, c#O. Proo! The continuity of PiJ, withP"Mbeing considered as subspace of CO (8, M), is obvious from the definition. Let p, q be points with d(p, q) ~ 1'/. If c is a piecewise differentiable path from p to q then with equality if and only if cpq = c. This implies the last statement of (A. 1.2). O A.1.3 Lemma. Choose K > O and denote by PiJ the deformation PiJ (2, ) introduced above. Let {cn} be a sequence in P"M such that {E(cn)} and {E(PiJcn )} are both convergent with the same /imit Ko > O. Then{ cn} possesses a convergent subsequence whose /imit is a closed geodesic co, E (co) = Ko . Proo! To define PiJ we chose an even k>O such that 4K/k~1'/2. Therefore every PiJcn consists ofthe k/2 geodesic segments ~cnl[0'+1)/k, 0'+3)/k],j=0,2, ... ,k-2. Here, tE[1,1+1/k] is to be read as t-1. From (A.1.1), we obtain the existence of a convergent subsequence of {PiJcn } which we again denote by {PiJcn }. The limit Co of this sequence also consists of k/2 geodesic segments. More• over, (A.1.1) implies that Co is also the limit ofthe sequence {Cn}' Since E(PiJ(co)) =E(Co)=Ko>O, we obtain from (A.1.2) that Co is a closed geodesic. O A.t.4 Lemma. Let Ko > O and let r1It be an open neighborhood of the set C of closed geodesics c with E (c) = Ko. In the case C =0, one may choose r1It = (). Let K> Ko and consider the deformation PiJ = PiJ (2, ) on P" M, introduced above. Then there exists a e=e(r1It) such that Proo! Since PiJ is continuous and PiJI C = id there exists an open neighborhood r1It' of C, r1It' c r1It with PiJr1It' c r1It. If there were no e> O with the desired property this would imply the existence of a sequence {cn}, cn~r1It', with A.l The Space PM and the Theorem of Lyustemik and Fet 207 From (A.1.3) we obtain that {Cn} has a convergent subsequence with the limit point c being a c10sed geodesic. Since E(c) = Ko and crţou', this is impossible. In particular, ifthere exists no c10sed geodesic c with E(C)=Ko, then (A.1.3) implies that there must exist a A> O such that there are no c10sed geodesics with E-value in [Ko-A,Ko+A]. O A.l.S Theorem (Lyusternik and Fet [LF]). On every compact Riemannian manifold M, there exists a closed geodesic. Proof We first consider the case 1tl M#O. That is to say, there exists cePM which is not homotopic to a constant map of the circ1e into M. Denote by P' M the space of alI c'ePM, freely homotopic to c. Put inf E P'M=K'. Then K'>O. Indeed, otherwise there exists c'eP'M with L(c')~ 2E(c') <1), 1»0 as above. But a c10sed curve of length < 1) can be retracted onto its initial equal end point. We c1aim that the set C' of c10sed geodesics c' with E(C')=K' is not empty. This follows from (A.I.4), since otherwise there exists e > O such that 5?J (P<' +.M) cPK'-eM. Since 5?J carries P'KM into itself, we obtain a contradiction to the definition of K'. Now assume 1tl M=O. Then there exists a smallest integer k, 1 be a homotopically non-trivial differentiable mapping. J induces a continuous map F=F(j):(Dk,iJDk)-+(PM,pO M)c (CO(S,M),pO M) as follows (cf. (2.1)). First identify the k-disc D k with the half-equator {xo~O, Xl =O} on the unit sphere Sk+l of ]Rk+2 with coordinates (xo, ... , xk+d. Asso• ciate to every peDkcSk+l the parameterized circ1e ap(t), teS, which starts from p orthogonally to the hyperplane {Xl =O} into the half-sphere {Xl~O}. Note that for pe8Dk, ap is the trivial (=constant) circ1e ap(t)=p. Now define F(P) (t) to beJoap(t). Consider a homotopy of F, i.e. a continuous map tP: [0,1] x (Dk, 8Dk)-+(PM, pOM) with tPl{O} x Dk=F. Put tPl{a} x Dk=P.We c1aim that the homotopy Fa =Fa(f) of F(f)=pO (f) determines a homotopyr ofJ=JD such that F(fj=Fa (f). Indeed, since every qeSk possesses the representation q=ap(t), peD\ simply put There exists K> O such that EIF(Dk ) < K. Choose an even integer k> O satis• fying 4K/k~I)2 and consider the deformation 5?J=5?J(2,) defined on P Put KO= Iim max EI!!d"F(D k ). "-+00 We c1aim that Ko > O. Otherwise we get a contradiction as follows. Ko = O implies EI!!d"F(D k ) < '12/2, for all sufficient1y large n, with '1 > O as above, i.e. for every peD\ the c10sed curve !!d"F(P) has length ~'1 and can therefore be retracted into its initial equal end point. That is to say, Fis homotopic to a map F* with F* (D k ) c::. PJ M = M, which implies that F is homotopic to a constant map, the image of which is a single point in PJ M = M. But then also, f is homo• topic to a constant map, which is a contradiction. Hence Ko > O. To complete the proof of (A.1.5), we demonstrate the existence of a c10sed geodesic Co with E(Co)=Ko. If there is no such geodesic, then, according to (A.IA), there exists an 8>0 and a deformation!!d such that !!dp"0+eMc::.p"o-eM. But this is contrary to the detinition of KO. O A.2 Closed Curves without Self-intersections on the 2-sphere From now on we restrict our attention to the 2-sphere M = (82 ,g) with an arbitrary Riemannian metric g. What matters in the subsequent theory is that M is an orientable c10sed surface. We consider the following subset PM of PM. PM contains all non-self-inter• secting c10sed curves and those which might have weak self-intersections, in the following sense. The curve can be approximated by non-self-intersecting c10sed curves and, moreover, if cl[to,t~] and Cl[tl,tn have the same image set, these segments are (trivial or non-trivial) geodesics. It follows that in particular alI point curves also belong to PM. Fix K > O. Put pIC M nPM = pIC M. Choose an even integer k > O such that 4K/k ~ '12, with '1 > Oso small that the 2'1-ball around every peM is strongly convex. In (A. 1) we detined the homotopy !!d on PICM. When restricted to P"M, the image, in general, will no longer be in P"M. We are therefore going to define a modification ~ of !!d which carries pIC M into itself and still possesses properties which allow the proof of results analogous to (A. 1.3) and (A. 1.6). Again, we detine ~ as a composition of deformations ~ fI where (f runs through intervals ofthe form UJk, U+2)/k] or [1 +j/k,l +U+2)/k],j being an even integer satisfying 0~j~k-2. For cepoM=pOM, we put ~fIC=C. Now assume thatE(c) >0, ceP"M. Let c(O)=p, c(2/k)=q. Then O~d(p,q)~'1. RecaB that for (fe[O,2/k], CPC(fI)(t), O~t~(f, !!d fIC(t) = j C(l), (f~t~1. A.2 Clased Curves withaut Self-intersectians an the 2-sphere 209 As (J increases from O to 2/k, it may happen that the segment cpc(,,) starts to inter• sect properly the remaining curve ca=cl[(J, 1]. As soon as this happens, we begin to modify c" by replacing those small arcs of c" (there might be more than one which come to lie on the "wrong" side of cpc(a» by the geodesic segments on cpc(,,) which go from the first point of proper intersection to the next point of proper intersection on C". Clearly, this is a well-defined operation. We denote by :?fi aC the curve obtained in this manner. :?fi "ci [O, (J] is a geodesic segment of length ~fj and :?fi"cl[(J, 1] consists of a modification of cl[(J, 1] which, when the modifi• cation actually takes places, will properly decrease the E-value of cl[(J, 1]. This is a purely local and continuous procedure. To see this more clearly we introduce normal coordinates at p = c(O). Since L(cj[0,2/ k]) ~ fj, the geodesic segments :?fi"cl[O,(J], 0~(J~2/k, belong entirely to the convex neighborhood of diameter 2fj around p. Thus, they are represented by straight segments of length ~fj, starting at the origin. Only those points of c which have distance ~fj from p=c(O) are affected by the deformation :?fi", 0~(J~2/k. :?fi"cEPKM, for all (J. :?fi2 /k cl[0,2/k] is a geodesic segment oflength ~fj. Moreover, with equality if and only if cl[0,2/k] is a geodesic segment parameterized pro• portional to arc length. Note. The previous definition of the deformation :?fi" in the class PKM is considerably less complicated and more direct than the one proposed by Lyusternik [Ly]. Actually, Lyusternik defines his deformation only on locally ilat surfaces. For treatment ofthe general case he refers to the approximation principle between Riemannian and Euclidean geometry. To make this work one needs some ad• ditional estimates for the energy integral which make this approach rather cumbersome. Ballmann [Ba] has filled in all the necessary details. Having defined :?fi" for (JE[0,2/k], we proceed to define :?fi" for (JE[2/k, 4/k] in the same manner. We start by replacing cl[2/k,(J] with the geodesic segment from c(2Ik) to c«(J); at the same time we modify, if necessary, cl[(J,1 + 2/k] so as to stay in the class PM. :?fi4 /k C [2fk,4fk] is a geodesic segment, starting at c(2fk) and ending at a point of distance ~ fj from c(2Ik). We continue in this manner, just as we did for the deformations ~'" untiI we reach the (J-parameter interval [2-2/k,2]. With this we define by letting ~«(J,c) be the subsequentapplication of :?fi2 /k , ••• ,:?fi2 1/k ,:?fi" where 21 is the even integer determined by 21 ~k(J < 21 +2. We now show that :?fi has all the essential properties which were established in (A. 1) for ~. A.2.1 Proposition.The mapping ~ is continuous. Moreover, E(~(2,c»)~E(c) with equality if and only if c is either a constant or a closed geodesic without self• intersections. 210 Appendix. The Theorem of Lusternik and Schnirelmann Proo! The continuity of {JJ; and its E-decreasing property follow from the definitions. It only remains to prove that E(c)=E({JJ;(2,c))>O implies that c is a closed geodesic. This follows in exactly the same way as (A. 1.2), using (A. 1.1). O A.2.2 Lemma. Let K> O and consider a deformat ion {JJ; = {JJ; (2, ) defined on PKM. Let {c.} be a sequence on PKM such that {E(c.)} and {E({JJ;c.)} are both convergent with the same !imit KO > O. Then {c.} possesses a convergent subsequence whose !imit is a closed geodesic Co without selj-intersections, E(co) = Ko ' Prao! As in the proof of (A. 1.3), one shows the existence of a convergent subsequence of {c.} (which we denote again by {c.}) such that Iim c. exists and is a closed geodesic Co , E (co) = Ko . It remains to show that co has no self-intersections, i.e. co: S-+M is an embed• ding. Now, if Co were a multiply covered closed geodesic then it could not be approximated by elements of PM in which the non-self-intersecting closed curves are dense. Nor could Co have isolated points of self-intersections (which is the only other possibility for a closed geodesic with multiple points). Indeed, such a curve could not be approximated by non-self-intersecting closed curves either. O A.2.3 Lemma. Let o/i be an open neighborhoad of the set C of non-selj-inter• secting closed geodesics of E-value Ko. If C = 0, then one may choase o/i = 0. Let K> Ko and consider a deformation {JJ; = {JJ; (2, ) on pK M. Then there exists e > O such tha! Proa! The proof is exactly the same as the proof of (A. 1.4), using (A.2.3) instead of (A. 1.3). O A.3 The Theorem of Lusternik and Schnirelmann We continue to consider the 2-sphere M=(S2,g) with an arbitrary Riemannian metric g. By S2 we denote the unit sphere in lR3 with coordinates (xO,Xi ,X2)' S2 shall be endowed with the induced metric. A distinguished family of non-self-intersecting curves on S2 are the par• ameterized circles. Such a circle is either a constant map (trivial circle) or an embedding c: S= [O,11/{O,1 }~S2, parameterized proportional to arc length, with its image being the intersection of S2 with a plane of distance < 1 from the origin of lR 3 . We denote by AS2 the space of circles, considered as subset of CO(S, S2). AOS2 is the set of point circles, isomorphic to S2. The space of great circles is denoted by BS2 . BS2 is put into 1 : 1 correspondence with the unit tangent bundle Ti S2 of S2 by associating to agreat circle its initial tangent vector. Ti S2, in turn, is isomorphic to the real projective space p 3 . A.3 The Theorem of Lusternik and Schnirelmann 211 Let us consider the space P S2 of all piecewise differentiable curves on S2. As a point set, PS2 coincides with P M. The metrics on P S2 and P M will generally be different. But the derived topologies are the same in both cases, i.e. the compact open topology induced from CO(S, M). We consider the canonical SO(2)- and 0(2)-action on PM, induced from the standard action of these groups on the circ1e SO(2) x PM--+PM, (z,C)=(e2"i',C(t)) .... z. c=(c(t+r)), i.e. z=e2"ir changes the initial point of c from °to r. The reflection on the x-mus: ZI--+Z operates on P M as a reversal of the orienta• tion 8:PM--+PM; c(t) .... c(1-t). By EMwe denote the quotient space PMjO(2). n:PM--+EM is the quotient map. The elements of EM are called unparameterized (non• oriented) c10sed curves. Since E(z. c)=E(c) and E(8c) = E(c), E can also be viewed as a function on EM. We define fM by nPM, fKM by npKM, etc. In particular, EOM=foM =poM=poM~M. The 0(2)-orbit of a parameterized circ1e is simply called a circ1e. Thus, a circ1e is the intersection of S2 with a plane having distance ~ 1 from the origin ofJR3 . The space nAS2 of circ1es and ofthe space nBS2 of great circ1es are denoted by rs2 and .1S2, respectively . .1S2 is isomorphic to the real projective plane p2. Consider the mapping y:rS2--+.1S2 by which we associate to a circ1e the great circ1e parallel to it. In the case of a point circ1e, we take the great circ1e parallel to the tangent plane of this point circ1e. The fibre y-l(Sl) over agreat circ1e Si may be identified with the l-disc D 1 =[-1,1] by taking an oriented line through the origin ofJR3 , orthogonal to the plane carrying si, and identifying the circ1es parallel to Si with their mid• points on that line. This interpretation of rs2 as the total space of a D 1-bundle over .1S2~p2 allows the following description of the Z2-homology of rs2 mod rOs2, rOs2 = space of point circ1es. The notation for these cyc1es corresponds to the notation in (2.3). First we choose basic cyc1es for .1 S2 ~ p2. As O-dimensional cyc1e [O, O] we take the great circ1e in the (xo ,xl)-plane. As l-dimensional Zrcyc1e of .1S2 we take the mapping [0,1]: [0,1]--+.1S2 212 Appcndix. The Theorem of Lusternik and Schnirelmann which associates to T the great circ1e through the xo-axis forming the positive angle T. 180° with the (xo,xl)-plane. As 2-dimensional Zz-cyc1e of LlS 2 we take the mapping [1,1]: [0,1] x [0,1]-+LlS1 by associating with (T, T') the great circ1e which is obtained from the great circ1e [0,1] (T) by the positive rotation of T' . 1800 around the x2-axis. We now define the corresponding Zl-cyc1es v(O,O), v(0,1), v(l,l) of r S2 mod rO S2 to be the counter images under the map yof the ba se cyc1es [O, O], [0,1], [1,1], respectively. Each of the cyc1es v(O,O), v(O,l), v(l,l) can be covered by chains u(O,O), u(O,l), u(l,l) of AS1 , i.e. we can write v(i,j)=nou(i,j). We write these chains explicitly; our notation corresponds to that employed in (5.1). pl-+(ait)=(cos pn cos 2nt, cos pn sin 2nt, sin pn )). 2 2 2 Let iJ;;.2 be the positive rotation by T . 1800 around the xo-axis. Then u(O, 1): (D l ,aDI) x [O, 1]-+(AS2 ,Aos2 ); (P;T)I-+iJ;;·2o u(0,0) (P). Here we use iJ; ~.2 also to denote the induced action of iJ; ;.2 on the circ1es of S2. Let iJ;~,'1 be the positive rotation by T' . 180 0 around the xz-axis. Define u(1,1): (DI,aDI) x [0,lf-+(AS2 ,AOS2 ); (P;T,T')I-+iJ;Ş·loU(O,l) (P;T). We also view the chains u(i,}) and the associated cyc1es v(i,j)=nou(i,j) as being chains and Zl-cyc1es of PM mod pOM and fM mod fO M, respectively, by considering AS2 and rs2 as subsets of PM and fM, respectively. We want to consider homotopies of these chains and cyc1es so as to preserve these properties. To make this precise we first define the class V(O,O) of 1-dimensional cycles ofthe form Here v is obtained from v (O, O) by a finite sequence of homotopies of v(O, O) of the following type. For each VE V(O, O), there exists a A.3 The Theorem of Lusternik and Schnirelmann 213 with V= nou. Moreover, if VE V(O, O) with its "covering" u having already been defined, we consider continuous mappings h: [0,1] x (DI ,aDI )-+(PM,pOM) I a with the following properties. Put h J {O'} X D = u . Assume that UO = u; hence nouo=v, and image ua c::.PM. Then we also take VI =noul as an element of V(O,O). Next we define the family V(O, 1) of2-dimensional Zz-cycles of fM mod fO M; an element VEV(O, 1) shall be obtained from v(O, 1)EV(0, 1) by a finite sequence of homotopies of the following type. Define - :DI -+DI bypf-+-p. Notethat u(0,1)jD I x {1}=Ou(0,1)J-D I X {O} with O being the orientation reversing map. As a consequence, we have v(0,1)JD I x {1}= -v(0,1)JDI x {O}. Now assume that we have already defined VE V(O, 1). This implies the existence ofamap satisfying u(P;l)=Ou( -p;O) and v=nou. For such a pair u,v, we consider homotopies. By this we mean a continuous map h: [O, 1] x (DI ,aDI ) x [O, l]->(PM,p°M) with the following properties. Put h J{ O'} x D I X [0,1] = u"', nou'" = v"', O:E;O':E; 1. Then U"'JDI x {1} = -Ou"'JD I x {O}; image u"'c::.PM, and (UO,VO)=(u,v). Then VI is also considered as an element of V(O,I). In an analogous manneI we define the class V(I, 1) of 3-dimensional Zz-cycles of fM mod fO M. First, v(1, 1)EV(1, 1). Note that u(1,1) (p;1,r')=Ou(1,1) (-p; O,r'), u(l,l) (p;r,1)=Oei".u(1,1)(-p; 1-r,0). It follows that v(1, 1)jD I x 0[0, 1f = -2v(1, 1)JD I x {O} X [0,1] +2v(1, 1)JD I x [0,1] x {O}. Which shows that v(1, 1) is a Zrcycle. 214 Appendix. The Theorem of Lusternik and Schnirelmann Now assume that vEV(1, 1). This implies the existence of a map with nou=v, such that u(p; 1;r') = Bu( - p; O;r'), u(p; T, 1) = Bein . u( - p; 1 - T, O). For such a pair we again consider homotopies. By this we mean a continuous map satisfying the following conditions. Put Then and image u" EP M. If (u, v) = (uo , VO), then we also take (ul , Vi) to be an element of V(1,1). As a particular example of such a homotopy of a pair (u, n o U)EU(O, 1) x V(O, 1) we mention u(p; T+o"(2), 0~T~1-(J/2, h (J. . T - o( ,p,)-!Bu(-p; T-1+(J/2), 1-(J/2~T~1. The resulting pair is also denoted by (ut, vt). Similarly, we define, for (u, n o u)EU(1, 1) x V(1, 1), the homotopies u(p; T+(J/2,T'), 0~T~1-(J/2, h (J. . T T' - I( ,p, , )-!8U(-p;T-1+(J/2,T'), 1-(J/2~T~1, U(p;T,T'+(J'/2), 0~T'~1-(J'/2, h2 «(J;p; T,T')= !Be in . u(-p; 1-T,T'-1+(J'/2), 1-(J'/2~T'~1. We denote the resulting pairs by (ut, vi) and (u~, vn respectively. We now detine, for (i,j)E{(O,O),(O, 1),(1, 1) }, K(i,j) = inf. sup Elimage v. VEV(I,J) A.3.I Theorem (Lusternik and Schnirelmann [LS 1,2]). On the 2-dimensional sphere with an arbifrary Riemannian metric, there exist three closed geodesics without selj-intersections. A.3 The Theorem of Lustemik and Schnirelmann 215 Remark. As will follow from the proof, there are three such geodesics c(i,}) having length ~V 2K(1, 1) . The example of an ellipsoid with three different axes, all having approximately the same length, shows that generally there exist no more than three c10sed geodesics without multiple points. This was proved by Morse [Mor 2], cf. (5.1.2). If the ellipsoid has axes of strongly different length then there will exist in general more than three c10sed geodesics without self-intersections, cf. Viesel [Vi 2]. Proof From the definit ion of the K(i,jJ, it follows that K(O,O) ~K(O, 1)~K(1, 1). We first observe that K(O, O) > O. In fact, this is a special case of part of the proof of (A. 1. 5). Next we show that the set C of c1osed, non-self-intersecting geodesics c with E(C)=K(O,O) is non-empty. For simplicity, we write Ko instead K(O,O). If C=0, we have from (A.2.3) the existence of an Il> °such that Since there exists UEU(O,O) with image ucP"o+ u:(D 1 ,oD 1) x [O, 1]-+(PM,pO M) such that v=nou. Define the homotopy v", 0~a~1, of v=vo by nou", where u" is defined by u"(p; T)={Jj(2aj(-r),u(p; T») wheref:D 1 -+[0, 1] is a differentiable function withf(T) = 1 for IT-1/21~1/2-1l, some smallll>O, andf(0)=f(1)=0. Thus, under this homotopy, the boundary values of u remain unchanged. In order to obtain an effective deformation also on the boundary, we first replace (u,v) by (u~,v~), as defined above, and apply the same deformation as above to this pair. Actually, we could do the same with any pair occurring, for a fixed a, in the homotopy ho . Thisshowsthat,given(u, V)EU(O, 1) x Vea, 1), thereexists(ii, V)EU(O, 1) x V(0,1), homotopic to (u, v), such that E(v(p; T»)=E(ii(p; T»)~E({Jj(2,u(p; T»). Now assume that there are no c10sed geodesics without self-intersections at E-Ievel Ko=K(0,1). Then, from (A.2.3) and the application of the deformation 216 Appendix. The Theorem of Lusternik and Schnirelmann just defined to an element VEV(O, 1) with image vcf"o+ u:(D 1 ,oD 1 ) x [O,lf-+(PM,pOM) such that v=nou. Define v" by nou", O~u~l, and u" by u"(p; T,T')=§i(2uj(t,'t'),u(p; T,T'»). Here, J(T, T')E[O, 1],f(T, T') = 1 for (T, T')E[e, l-e] x [e, l-e], some small e> O, and JI8[0,1]2=0. Thus, under this homotopy the boundary of u remains unchanged. By going to pairs ofthe type (ur, vT) and (u!, vi), associated to (u, v) as above, we see that given vEV(l,l) with v=nou, there exists vEV(l,l) such that E(v(P; T,T'»)~E(§i(2,u(p; T,T'»). We thus conclude that there exists a non-self-intersecting closed geodesic c(1,1) with E(c(1,1»)=K(1,1). We have therefore proved (A.3.1) for the case K(O, O) < K(O, 1) < K(1, 1). The possibility that we have equality in the relation K(O,O)~K(O,l)~K(l,l) remains to be discussed. It is at this point that the theory of subordinated homology classes comes into play. Since we do not wish to appeal to an extensive topological machinery, we discuss this possibility using only some ad hoc geometric con• structions. First assume that K(O,O)=K(O,1)=(briefly) Ko. We wish to show that there still exist two non-self-intersecting closed geodesics in fM with E-value ~Ko. If there is any non-self-intersecting closed geodesic c' on M with E(c') Consider the open set We take a continuous curve A.3 The Theorem of Lusternik and Schnirelmann 217 Then voh is a l-cycle mod 2 of fM. It is homologous mod 2 to vlD I x {O}modulo fO M. To see this, consider Then u o aH=u o HI{1} x [O, 1l-uo HI[0,1l x {1} +uoHI[0,1l x {O}=uoh +{ -eu(Â(p0-1)+po;0); -1~Â~O} + {u(.l.(po+ 1)-1; O); O~Î.~ 1} mod pOM. We claim that h meets (!J. Indeed, otherwise we can assume (after possibly applying to (u,v) some further E-decreasing deformations) that uohEPoM. This implies that I Each summand is a cycle mod pOM. Now recall from (A.1) that ulD x {T p } determines a mapping S2~S2 homotopic to the identity map. Thus, one of the two summands in (*) determines a mapping S2~S2 of odd degree (say the first one) and the other a mapping S2~S2 of even degree. This holds simultaneously for all pE[O, 1]. Thus, in particular, u I[PI , 1] x {1 } = eu I[ -1 ,Pol x {O} determines a map S2~S2 of even degree whereas the degree of the map determined by u I[ -1 ,Pol x {O} is odd, which is clearly a contradiction. This last argument did not use the hypothesis K (O, O) = K (0,1). It shows that fM mod fO M possesses a non-trivial Zz-cycle in dimension 1. Thus we have proved that h always meets the set (!J. Since D 1 x [0,1] is mapped under v like a M6bius strip, any two non-triviall-cycles in general position meet in an odd number of points. Thus, (!J carries a 1-cycle, i. e., there exists a map h with image h c (!J. I. e. there exists a continuous map Now assume that C possesses only finitely many S-orbits S . c*. Then we can choose OU such that it consists offinitely many pairwise disjoint open neighborhoods OU(S. c*) of such orbits. Moreover, we can choose these neighborhoods so small that they do not contain at the same time a curve d and its inverse ed. It follows that the cyc1e (**) lies entirely in one of these OU(S . c*), which is c1early impossible. Thus, nC is infinite. Finally, we discuss the case K(0,1)=K(1, 1) = (briefly) KI' We only indicate how to prove that, in this case, either there exist two non-self-intersecting c10sed geodesic with E-value < K1 , or that the set ofunparameterized non-self-intersecting c10sed geodesic of E-value K 1 is infinite. 218 Appendix. The Theorem of Lusternik and Schnirelmann Denote by C the set of non-self-intersecting closed geodesics c with E(c) =K1. Choose an open neighborhood o/i of C in CO(8, M). Then there exists an Il> O and a pair (u, v)eU(1, 1) x V(1, 1) with Put (I}=u- 1 (o/i)cD 1 x [0,1f. Consider the mapping ho: (p, p')e[O, 1]2 f-+(O, p, p')eD 1 x [0,1]2. Then v o ho = v I{O} x [0,1 f is a 2-cyc1e; its domain consists of the great circ1es on 8 2 • Let be such that voh defines a 2-cyc1e homotopic to voho. This presupposes that h satisfies certain boundary conditions, e. g. P1,p'= -Po,p' ,Pp,l = -Pp,o,f1,p,=1,fo,p,=0,fp,1 =1,fp ,o=0. 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Enseignement du 30m• cycle de la Physique en Suisse Romande. Ecole Polytechnique Federale, Lausanne 1975. [Zi] Zilier, W.: Geschlossene Geodătische auf global symmetrischen und homogenen Răumen. Bonn, Math. Schr. Nr. 85, 1976. [Zo] Zol!, O.: Uber FIăchen mit Scharen geschlossener geodătischer Linien. Math. Ann. 57, 108 -133 (1903). Index Abraham, R. 114 critical submanifold 58 action-angular variables 182 -, non-degenerate 58 Alber, S. 1. 50, 51, 53 critical value of a q.,-family 33 1X0 -bundle 16 critical point in AM/S 42 IX'-bundle 16 cuplenth 39 angular momentum 186 Anosov, D. V. 195 differential graded-commutative algebra, DGA Anosov type, manifold of 189 156 AS", space of parameterized great circles on S" -, minimal 156 47 LIS", space of great circles on S" 48 a verage index 164 Du Bois-Raymond, Lemma of 21 Duistermaat, J.J. 74 Baire space 108 Berger, M. 182 E, energy integral 21 Birkhoff, G. 101,104 eigenvalue, of the index from 57 Birkhoff Normal Form of a symplectic trans- eigenvalue, principal, of a symplectic map 100 formation 101 Ehresmann, C. 49 Birkhoff-Lewis, Fixed Point Theorem of 104 Eliasson, H. 30, 70 Bolt, R. 91, 95, 99 elliptic type, manifold of 177 BS", space of parameterized circles on S" 47 energy integral E 21 bumpy metric 114 Busemann, H. 201 Fermi coordinates 108 Fet's Theorem 130 CO-maps, CO(S. M) 8 C"-maps, C"(S, M) 8 generic property 108 canonical transformation 77 geodesic, almost invariant 199 Cartan, E. 178 -, almost periodic 199 characteristic manifold at c 133 -, closed 21 Chem, S. S. 49 -, closed non-degenerate 58 center manifold 105 -, c10sed orientable 192 center Poincare map 107 -, minimizing 199 Christoffel symbol 2 geodesic flow 81 circle on S" 48 geodesic spray 81 -, parameterized on S" 47 gradient field grad E 25 -, unparameterized on S" 48 Green, L. 181 circle action 41 Gromoll, D 90,140 Clairaut's Theorem 186 Gromoll-Meyer, Theorem of 140 closed geodesic 21 r S", space of circles on S" 48 Comparison Theorem for Large Triangles 198 completely integra bie Hamiltonian systems 182 H'-maps, H'(S, M) 8 condition (C) 25 Hamiltonian equations 79 connection 2 Hamiltonian flow 79 conjugate point 194 Hamiltonian function 79 convariant derivative 4 Hamiltonian system 79 Cr A 27 -, completely integrable 182 226 Index Hamiltonian vector field 79 normal bundle of a non-degenerate critical Hilbert manifold t submanifold 123 horizontal subspace 3 normal bundle of a c10sed geodesic 66, 132 hyperbolic Poincare map 96 nullity of a closed geodesic 57 hyperbolic closed geodesic 97 nullity of non-degenerate critical submanifold hyperbolic type, manifold of 197 58 hypothesis (GM)p 140 orientable closed geodesic 192 index form at a critical point 55 orientation reversing map O 42 index of a closed geodesic 57 index of a non-degenerate critical submanifold Palais, R. 11, 15, 25, 60 58 Palais, Lemma of 11 Integral of a Hamiltonian system 182 parabolic type, manifold of 201 Integrals in involntion 182 parameterized circle 8 invariant manifolds, theorem on 105 parameterized (great) circle 47 isotropy group of a closed curve 41 period of a torus flow 183 isotropic su bspace 86 periodic point of the Poincare map 104 periodic orbit 83, 104 Jacobi field 82 1>-family 32 llM(ÎIM), space of unparameterized (oriented) Karcher, H, 30,45 closed curves on M 43 (42) kinetic energy 79 Poincare map 84 positive definite form 5 AM, space of closed curves on M 15 prime c10sed geodesic 42, 84 AKM,AK-M 25 prime period 83 Lagrangian su bspace 92 prime closed curve 42 Levi-Civita connection 6 principal eigenvalue of a linear symplectic map Lewis, D,C. 104 100 linear symplectic transformation 86 principal part of a local representation 2 Liouville line element 184 property ('cx) 163 local transversal hypersurface 84 ",-family 46 Lusternik and Schnirelmann 38 -, Theorem of 214 quasi-periodic torus flow 183 Lyusternik and Fet, Theorem of 36, 207 real subspace of a complex vector space 86 Maslov index 93 residual set 108 Meyer, W, 60,90, 140 p-index 90 m-fold covering of a c10sed curve 42 p-index form 90 Michel, R. 181 p-index Theorem 95 minimal model 143 p-nullity 90 minimizing geodesic 199 Riemannian connection 5 Morse, M. 59,70, 175, 179, 194,200 Riemannian manifold 5 Morse complex 70 Riemannian metric 5 -, full 70 Riemannian O (2)-vector bundle 59 Morse function 59 Morse Lemma, generalized 59 Sacks, J. 156 Moser,1. 104,106,115 slice 44 multiplicity of a closed curve 42 Smale, S. 25 splitting numbers of a linear sympletic map 99 natural atlas of AM 13 sta bIe manifold 67,73, 105, 106 natural chart based at cEAM 13, 20 -, slrong 67,73, 105, 106 N-elementary symplectic transformation slable bundles 189, 193 100 Stăckelline element 183 non-degenerate closed geodesic 58 subordinated homology classes 38, 46 non-degenerate critical submanifold 58 Sullivan, D. 141, 143 non-degenerate subspace of a symplectic vector Sullivan (co-)homology classes 143 space 86 super bumpy metric 163 Index 227 Svarc, A. S. 128 underlying prime c10sed curve 42 symmetric spaces of rank 1 177 unparameterized c10sed curve 43 symplectic atlas 77 unparameterized, oriented c10sed curve 42 symplectic manifold 77 unparamctcrized (great) circ1e 48 symplectic transformations 77 unstable bundles . 189, 193 -, linear 86 unstable manifold 67, 73, 105, 106 symplectic vector space 86 -, strong 67, 73, 105, 106 Takens,F. 111,115 vertical su bspace Three Closed Geodesics, Theorem of the 170 Vigue, M. 141 torsion 6 torsion free connection 6 Wasserman, A. 60 transformation formula 6 Weinstein, A. 180 twist type, a symplectic diffeomorphism of 103, Wiedersehensflăche 181 107 type number 135, 138 Zilier, W. 72,98, 142 -, singular 135 Zoll, O. 180 Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Beriicksichtigung der Anwendungsgebiete Eine Auswahl 23. Pasch: Vorlesungen iiber neuere Geometrie 41. Steinitz: Vorlesungen iiber die Theorie der Polyeder 45. Alexandroff/Hopf: Topologie. Band 1 46. Nevanlinna: Eindeutige analytische Funktionen 63. Eichler: Quadratische Formen und orthogonale Gruppen 102. Nevanlinna/Nevanlinna: Absolute Analysis 114. Mac Lane: Homology 123. Yosida: Functional Analysis 127. Hermes: Enumcrability, Decidability, Computability 131. Hirzebruch: Topological Methods in Algebraic Geometry 135. Handbook for Automatic Computation. VoI. 1/Part a: Rutishauser: Description of ALGOL 60 136. Greub: Multilinear Algebra 137. Handbook for Automatic Computation. VoI. I/Part b: Grau/Hill/Langmaack: Translation of ALGOL 60 138. Hahn: Stability of Motion 139. Mathematische Hilfsmittel des Ingenieurs. 1. Teil 140. Mathematische Hilfsmittel des Ingenieurs. 2. Teil 141. Mathematische Hilfsmittel des Ingenieurs. 3. Teil 142. Mathematische Hilfsmittel des Ingenieurs. 4. Teil 143. Schur/Grunsky: Vorlesungen iiber lnvariantentheorie 144. Weil: Basic Number Theory 145. Butzer/Berens: Semi-Groups of Operators and Approximation 146. Treves: Locally Convex Spaces and Linear Partial Differential Equations 147. Lamotke: Semisimpliziale algebraische Topologie 148. Chandrasekharan: Introduction to Analytic Number Theory 149. Sario/Oikawa: Capacity Functions 150. losifescu/Theodorescu: Random Processes and Learning 151. Mandl: Analytical Treatment ofOne-dimensional Markov Processes 152. Hewitt/Ross: Abstract Harmonic Analysis. VoI. 2: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups 153. Federer: Geometric Measure Theory 154. Singer: Bases in Banach Spaces 1 155. Miiller: Foundations of the Mathematical Theory of Electromagnetic Waves 156. van der Waerden: Mathematical Statistics 157. Prohorov/Rozanov: Probability Theory. Basic Concepts. Urnit Theorems. Random Processes 158. Constantinescu/Cornea: Potential Theory on Harmonic Spaces 159. K5the: Topological Vector Spaces I 160. Agrest/Maksimov: Theory of Incomplete Cylindrical Functions and their Applications 161. Bhatia/Szeg5: Stability Theory of Dynamical Systems 162. Nevanlinna: Analytic Functions 163. Stoer/Witzgall: Convexity and Optimization in Finite Dimensions I 164. Sario/Nakai: Classification Theory of Riemann Surfaces 165. Mitrinovic/Vasic: Analytic Inequalities 166. Grothendieck/Dieudonne: Elements de Geometrie Algebrique I 167. Chandrasekharan: Arithmetical Functions 168. Palamodov: Linear Differential Operators with Constant Coefficients 169. Rademacher: Topics in Analytic Number Theory 170. Lions: Optimal Control of Systems Governed by Partial Differential Equations 171. Singer: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces 172. Biihlmann: Mathematical Methods in Risk Theory 173. Maeda/Maeda: Theory of Symmetric Lattices 174. Stiefel/Scheifele: Linear and Regular Celestial Mechanics: Perturbed Two-body Motion-Numerical Methods-Canonical Theory 175. Larsen: An Introduction to the Theory of Multipliers 176. Grauert/Remmert: Analytische Stellenalgebren 177. Fliigge: Practical Quantum Mechanics I 178. Fliigge: Practical Quantum Mechanics II 179. Giraud: Cohomologie non abelienne 180. Landkof: Foundations of Modem Potential Theory 181. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications I 182. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications II 183. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications III 184. Rosenblatt: Markov Processes. Structure and Asymptotic Behavior 185. Rubinowicz: Sommerfeldsche Polynommethode 186. Handbook for Automatic Computation. VoI. 2. Wilkinson/Reinsch: Linear Algebra 187. Siegel/Moser: Lectures on Celestial Mechanics 188. Wamer: Harmonic Analysis on Semi-Simple Lie Groups I 189. Wamer: Harmonic Analysis on Semi-Simple Lie Groups II 190. Faith: Algebra: Rings, Modules, and Categories I 191. Faith: Algebra II: Ring Theory 192. Mal'cev: Aigebraic Systems 193. P.6lya/Szego: Problems and Theorems in Analysis I 194. 19usa: Theta Functions 195. Berberian: Baeu-Rings 196. Athreya/Ney: Branching Processes 197. Benz: Vor1esungen iiber Geometrie der Algebren 198. Gaal: Linear Analysis and Representation Theory 199. Nitsche: Vor1esungen liber Minima1f1ăchen 200. Dold: Lectures on Aigebraic Topology 201. Beck: Continuous Flows in the Plane 202. Schmetterer: Introduction to Mathematical Statistics 203. Schoeneberg: E1liptic Modular Functions 204. Popov: Hyperstability of Control Systems 205. Nikol'skii: Approximation of Functions of Several Variables and Imbedding Theorems 206. Andn!: Homologie des Algebres Commutatives 207. Donoghue: Monotone Matrix Functions and Analytic Continuation 208. Lacey: The Isometric Theory of Classical Banach Spaces 209. Ringel: Map Color Theorem 210. Gihman/Skorohod: The Theory of Stochastic Processes I 211. Comfort/Negrepontis: The Theory of Ultrafilters 212. Switzer: Aigebraic Topology-Homotopy and Homology 213. Shafarevich: Basic Aigebraic Geometry 214. van der Waerden: Group Theory and Quantum Mechanics 215. Schaefer: Banach Lattices and Positive Operators 216. P6Iya/Szego: Problems and Theorems in Analysis II 217. Stenstrom: Rings of Quotients 218. Gihman/Skorohod: The Theory of Stochastic Processes II 219. Duvaut/Lions: Inequalities in Mechanics and Physics 220. Kirillov: Elements of the Theory of Representations 221. Mumford: Algebraic Geometry 1: Complex Projective Varieties 222. Lang: Introduction to Modular Forms 223. Bergh/Lofstrom: Interpolation Spaces. An Introduction 224. GilbargfTrudinger: Elliptic Partial Differential Equations of Second Order 225. Schlitte: Proof Theory 226. Karoubi: K-Theory 227. Grauert/Remmert: Theorie der Steinschen Răume