Pseudo focal points along Lorentzian geodesics and Morse index
aa r X i v : . [ m a t h . DG ] A p r PSEUDO FOCAL POINTS ALONG LORENTZIAN GEODESICSAND MORSE INDEX
MIGUEL ´ANGEL JAVALOYES, ANTONIO MASIELLO, AND PAOLO PICCIONE
Abstract.
Given a Lorentzian manifold (
M, g ), a geodesic γ in M and atimelike Jacobi field Y along γ , we introduce a special class of instants along γ that we call Y -pseudo conjugate (or focal relatively to some initial orthogonalsubmanifold). We prove that the Y -pseudo conjugate instants form a finite set,and their number equals the Morse index of (a suitable restriction of) the indexform. This gives a Riemannian-like Morse index theorem. As special casesof the theory, we will consider geodesics in stationary and static Lorentzianmanifolds, where the Jacobi field Y is obtained as the restriction of a globallydefined timelike Killing vector field. Introduction
In the last years, there have been several attempts of stating a Morse indextheorem for stationary Lorentzian manifolds. Starting from the original resultsin [3, 4, 9], and aiming at establishing Morse theoretical results, several authorshave studied the relations between conjugate instants along a geodesic and itsindex form. With the development of new functional analytical and symplectictechniques, it has appeared naturally that the classical Riemannian statement ofthe theorem would not hold in the non positive definite case. In first place, itis easy to prove that, unless the geodesic is Riemannian, the index of its indexform is always infinite. On the other hand, the conjugate instants along a semi-Riemannian geodesic, unlike the Riemannian case, may accumulate. As a matterof fact, there are several pathological examples where the set of conjugate instantscan be arbitrarily complicated (see [19]). In order to obtain a meaningful statementof the Morse index theorem, one has to replace the notion of Morse index with themore general notion of spectral flow , which is an integer number associated to acontinuous path of Fredholm symmetric bilinear forms. Moreover, the count ofthe conjugate instants has to be interpreted as a suitable intersection number inthe Grassmannian of all Lagrangian subspaces in a finite dimensional symplecticspace; this number is called
Maslov index . The more general semi-RiemannianMorse index theorem (see for instance [18]) states that, given a semi-Riemannianmanifold (
M, g ) and a geodesic γ : [0 , → M , the spectral flow of the paths of Date : April 18th, 2008.2000
Mathematics Subject Classification.
Key words and phrases.
Geodesics, Lorentzian manifolds, Morse index theorem.First author was partially supported by Regional J. Andaluc´ıa Grant P06-FQM-01951, byFundaci´on S´eneca project 04540/GERM/06 and by Spanish MEC Grant MTM2007-64504. Thesecond author is supported by M.I.U.R. Research project PRIN07 ”Metodi Variazionali e Topo-logici nello Studio di Fenomeni Nonlineari” The third author is sponsored by Capes, Brazil, grantBEX 1509/08-0. symmetric forms I t , t ∈ ]0 , γ to theset of variational vector fields along γ | [0 ,t ] , equals the Maslov index of γ up to thesign.However, in the case of stationary Lorentzian manifolds, an alternative varia-tional principle is known for geodesics; among the main advantages, one provesthat each one of its critical points has finite index and, once again, its value equalsthe Maslov index of the corresponding geodesic. This alternative variational prin-ciple will be described with more details below. It is not known whether the set ofconjugate instants along a given geodesic is discrete in the stationary case. A verynatural conjecture would be that, under the stationarity assumption, conjugate in-stants do not accumulate, and that the Maslov index of a geodesic is equal to theirnumber, counted with multiplicity. This conjecture still remains an open problem,although it has been proven to hold in some special cases. For instance, in [13] theauthors prove that this is true in the case of semi-Riemannian Lie groups, endowedwith a left-invariant metric, whose dimension is less than or equal to 5. Otherthan this special example, basically nothing is known concerning the distributionof conjugate instants along a geodesic in a stationary manifold; the purpose of thepresent paper is to investigate in this direction.In [4] the authors establish a Riemannian-like Morse index theorem in a staticLorentzian manifold by considering a functional on the Riemannian base. Due toa technical gap in the proof, the result holds only under additional assumptions,although no counterexample to their general statement has been found so far. Re-cently, the more general case of stationary Lorentzian manifolds has been considered(see [10, 11]). The central idea is to consider the energy functional restricted to theset of curves γ : I → M satisfying the natural constraint g ( ˙ γ, Y ) = C γ , where C γ is a constant depending on γ , g is the Lorentzian metric on the stationary space-time M and Y is a timelike Killing field in ( M, g ). Such restriction has the samecritical points as the original geodesic action functional, but its second variation is essentially positive at each critical point. Thus, one has finite Morse index, and in[10] it is proven that this index is equal to the Maslov index.The main goal of this paper is to study in more detail the distribution of con-jugate or focal instants, and to formulate a Riemannian-like Morse index theorem.Rather than restricting to the fixed endpoints case, we will consider the more gen-eral case of geodesics whose endpoints are free to vary along two given smoothsubmanifolds. Our central result is the introduction of the class of Y -pseudo conju-gate, or ( P , Y )-pseudo focal instants related to the choice of a timelike Jacobi field Y ; these instants form a discrete set, and they carry all the information about thesecond variation of the geodesic action functional up to a correction term whichis either null or equal to 1. Although the notion of pseudo conjugate/focal pointdepends on the (existence and the) choice of an everywhere timelike Jacobi field, insome specific situations there is a canonical choice. This is the case, for instance, instationary Lorentzian manifolds with a distinguished timelike Killing vector field,which is the standard example we will refer to.Let us describe more precisely our result. Consider a Lorentzian manifold ( M, g ),two smooth nondegenerate submanifolds P , Q ⊂ M and a geodesic γ : [0 , → M with γ (0) ∈ P , γ (1) ∈ Q , ˙ γ (0) ∈ T γ (0) P ⊥ and ˙ γ (1) ∈ T γ (1) Q ⊥ . We will call such ageodesic {P , Q} -orthogonal; let Y be a timelike Jacobi field along γ , for instance, if( M, g ) is stationary, Y can be taken to be the restriction to γ of a globally defined SEUDO FOCAL POINTS 3 timelike Killing vector field. We will say that Y is admissible (Definition 3.1) if Y (0)and the covariant derivative Y ′ (0) are linearly independent. When the geodesic isspacelike or lightlike, given a Y non admissible, we can obtain an admissible timelikeJacobi field by perturbating the first one (see Lemma 3.2); on the other hand, Y will be called singular if Y and Y ′ are everywhere pointwise linearly dependent.For instance, if ( M, g ) is static, and γ is a geodesic orthogonal to the static Killingvector field Y , then the restriction of Y to γ is singular.A ( P , Y )-pseudo Jacobi field J is a smooth vector field along the geodesic γ satisfying g ( J ′ , Y ) − g ( J, Y ′ ) = 0 on [0 , J ′′ − R [ J ] is alinear combination of Y and Y ′ and it satisfies certain initial conditions (see Defi-nition 3.10). A ( P , Y )-pseudo focal instant t ∈ ]0 ,
1] is an instant such that thereexists a ( P , Y )-pseudo Jacobi field J along γ with J ( t ) = 0. In this situation, onecan consider the space H { γ, P , Q} of all variational vector fields V along γ satisfying V (0) ∈ T γ (0) P , V (1) ∈ T γ (0) Q and the linear constraint g ( V ′ , Y ) − g ( V, Y ′ ) = 0(we will suppress Q in the notations when Q reduces to a point); these cor-respond to variations of γ by a smooth family γ s , s ∈ ] − ε, ε [, by curves with γ s (0) ∈ P , γ s (1) ∈ Q and such that the quantity g ( ˙ γ s , Y ) is constant and equalto the constant C γ = g ( ˙ γ, Y ) for all s . Such space has codimension one in thespace H { γ, P , Q}∗ of all vector fields V satisfying the more general affine constraint g ( V ′ , Y ) − g ( V, Y ′ ) = constant.Consider the index form I { γ, P , Q} given by the second variation at γ of the geo-desic action functional on the space of curves with initial endpoint γ (0) in P andfinal endpoint γ (1) in Q . The difference between the indexes of the restrictions of I { γ, P , Q} to the spaces H { γ, P , Q} × H { γ, P , Q} and H { γ, P , Q}∗ × H { γ, P , Q}∗ , which is atmost one, is an invariant of the geodesic γ , that will be denoted by ǫ { γ, P , Q} . It isan intriguing question to determine which geodesics have non vanishing ǫ { γ, P , Q} ,and how this fact affects the distribution of P -focal instants along γ . As a specialexample, we will consider the case of geodesics in static manifolds, i.e., stationarymanifolds whose Killing field Y has integrable orthogonal distribution. In this case,each integral leaf of Y ⊥ is a totally geodesic submanifold of M , and those geodesicsthat are contained in one such integral submanifold have a purely Riemannianbehavior.The main results of the paper are the following. First, we show that ( P , Y )-pseudo focal instants are related with the kernel of the restriction of the index form(Proposition 3.11). The ( P , Y )-pseudo focal instants form a finite set, and theirnumber equals the index of the restriction of I { γ, P} to H { γ, P} ×H { γ, P} (Morse indextheorem, Theorem 4.9). When we consider I { γ, P , Q} defined in H { γ, P , Q} ×H { γ, P , Q} ,we must add to the number of ( P , Y )-pseudo focal instants, the index of a certainsymmetric bilinear form defined on a finite dimensional subspace. Moreover, inthe singular case the Morse index theorem holds in a stronger sense, in that therestrictions of I { γ, P} to H { γ, P}∗ × H { γ, P}∗ and to H { γ, P} × H { γ, P} have the sameindex, i.e., ǫ { γ, P} = 0 (Theorem 4.12). The last result is applied to horizontalgeodesics in static manifolds in Proposition 4.14. A discussion on the distributionof pseudo focal and focal points along a geodesic is discussed in Section 5.The proof of the main results is obtained by functional analytical techniques,involving the study of the nullity and the variation of the index for a smooth familyof Fredholm bilinear forms with varying domains. Establishing the smoothness of M. A. JAVALOYES, A. MASIELLO, AND P. PICCIONE the domains is a surprisingly non trivial fact (Proposition 4.3), complicated by theoccurrence of the singular case. The kernel of the restriction of the index form I { γ, P} to H { γ, P} × H { γ, P} is studied in Section 3. In order to get the Morse indextheorem, in Section 2 we prove an abstract Morse Index Theorem in the spirit of[21] (see also [7, 8]). As to the plethora of abstract Morse index theorems appearingin the literature, few remarks are in order. When dealing with a family of closedsubspaces, it is customary to make two assumptions: • monotonicity of the family, to guarantee monotonicity of the index function; • continuity of the family, to guarantee the semi-continuity of the index func-tion.These two assumptions are not totally independent; for instance, monotonicity isnot compatible with continuity in the norm operator topology (see Definition 2.3and Lemma A.1). For the result aimed in this paper, we cannot apply directly[21, Theorem 1.11], because we cannot guarantee any kind of continuity for ourmonotonic family of closed subspaces; however, continuity in the norm operatortopology is obtained by considering a family of deformations (more precisely repa-rameterizations , see Proposition 4.3) of the subspaces, but this operation does notpreserve monotonicity. The abstract index theorem proved here, Proposition 2.9,deals with this situation.The authors gratefully acknowledge an important contribution to the final ver-sion of the paper given by the anonymous referee, who pointed out a mistakecontained in the original version of the manuscript.2. An abstract Morse index theorem
The main result of this section (Proposition 2.9) gives an abstract version ofthe Morse index theorem for continuous families of bounded symmetric bilinearforms on varying domains. Very likely, some of the preliminary results are alreadyknown in the literature, but for the reader’s convenience we give complete proofs ofevery statement. Basic bibliography for the topics of this section are the classicaltextbooks [6, 14].Let H be a (real) Hilbert space, with inner product h· , ·i . A bounded symmetricbilinear form B : H × H → R is said to be Fredholm if it is represented by a (self-adjoint) Fredholm operator T : H → H , i.e., B = h T · , ·i . Note that the operatorthat represents B depends on the choice of the inner product, but the notion ofFredholmness does not. A symmetric Fredholm bilinear form is nondegenerate ifKer( B ) = { x ∈ H : B ( x, y ) = 0 ∀ y ∈ H } = Ker( T ) is trivial; this implies that T is an isomorphism. Observe that Ker( B ) is finite dimensional if B is Fredholm.A subspace Z ⊂ H is B -isotropic (or simply isotropic) if B | Z × Z is null. Given aself-adjoint Fredholm operator T , there exists an orthogonal decomposition: H = V − ( T ) ⊕ Ker( T ) ⊕ V + ( T )into T -invariant closed subspaces such that B = h T · , ·i is negative definite (resp.,positive definite) on V − ( T ) (resp., on V + ( T )). The index of B = h T · , ·i denotedby n − ( B ), is the dimension of V − ( T ); equivalently, n − ( B ) is the dimension of amaximal subspace of H on which B is negative definite. Observe that if Z is anisotropic subspace, then Z ∩ V − ( T ) = Z ∩ V + ( T ) = { } . If X ⊂ H is a subspace,we set: X ⊥ B = (cid:8) y ∈ H : B ( x, y ) = 0 ∀ x ∈ X (cid:9) ; SEUDO FOCAL POINTS 5 assume that X is closed, then if B and B | X × X are nondegenerate, B | X ⊥ B × X ⊥ B isnondegenerate, and H = X ⊕ X ⊥ B . In this case:n − ( B ) = n − (cid:0) B | X × X (cid:1) + n − (cid:0) B | X ⊥ B × X ⊥ B (cid:1) . Lemma 2.1.
Let B be a Fredholm bilinear form and Z ⊂ H be a B -isotropicsubspace such that Z ∩ Ker( B ) = { } . Then, n − ( B ) ≥ dim( Z ) .Proof. First, we observe that we can assume Ker( B ) = { } . Namely, should thisnot be the case, one can consider the quotient H = H/ Ker( B ), endowed with theinduced Fredholm bilinear form B , that has the same index as B . If π : H → H isthe projection, since Z ∩ Ker( B ) = { } , then setting Z = π ( Z ), we get a B -isotropicsubspace of H with the same dimension as Z . This shows that it suffices to considerthe case that Ker( B ) = { } .Assume Ker( B ) = { } ; consider the representative T of B and the decomposition H = V − ( T ) ⊕ V + ( T ). If dim( Z ) > dim (cid:0) V − ( T ) (cid:1) , then it would be Z ∩ V + ( T ) = { } ,which contradicts the assumption that Z is isotropic. This concludes the proof. (cid:3) Let us now prove the following:
Proposition 2.2.
Let X ⊂ H be a closed subspace, and let B be a Fredholm bilinearform on H . Assume that X ∩ Ker( B ) = { } ; then: n − ( B ) ≥ n − (cid:0) B | X × X (cid:1) + dim (cid:2) Ker (cid:0) B | X × X (cid:1)(cid:3) . Proof.
As in Lemma 2.1, we can assume Ker( B ) = { } . Let V ⊂ X be a maximalsubspace on which B | X × X is negative definite, so that n − ( B | X × X ) = dim( V ), B | V ⊥ B × V ⊥ B is nondegenerate and H = V ⊕ V ⊥ B . Clearly, the kernel Ker (cid:0) B | X × X (cid:1) is an isotropic subspace of V ⊥ B , thus, by Lemma 2.1:n − ( B ) = n − (cid:0) B | V × V (cid:1) + n − (cid:0) B | V ⊥ B × V ⊥ B (cid:1) = dim( V ) + n − (cid:0) B | V ⊥ B × V ⊥ B (cid:1) ≥ dim( V ) + dim (cid:2) Ker (cid:0) B | X × X (cid:1)(cid:3) . This concludes the proof. (cid:3)
We will denote by L ( H ) the algebra of all bounded linear operators on H .The Grassmannian G ( H ) of all closed subspaces of H , endowed with the distancedist( X, Y ) = k P X − P Y k , is a complete metric space, where P Z : H → H denotesthe orthogonal projection onto Z ∈ G ( H ) and k · k is the operator norm. Definition 2.3.
A family { H s } s ∈ [ a,b ] of closed subspaces of H is said to be a con-tinuous family of closed subspaces if the map [ a, b ] ∋ s H s ∈ G ( H ) is continuous.Weaker notions of continuity may also be considered (see Appendix A).Given a projection P ∈ L ( H ), we will denote by Im( P ) the image P ( H ), whichis a closed subspace of H . The following lemma can be found in [5, Lemma 4.7]. Lemma 2.4.
Let
P, Q be projections in L ( H ) with k P − Q k < . Then, therestriction e P : Im( Q ) → Im( P ) of P is an isomorphism. By a projection , we mean an operator P ∈ L ( H ) such that P = P ; by an orthogonalprojection we mean a self-adjoint projection. M. A. JAVALOYES, A. MASIELLO, AND P. PICCIONE
A self-adjoint operator T in L ( H ) is said to be essentially positive if it is ofthe form P + K , where P is a positive isomorphism of H , that is, a self-adjointisomorphism satisfying that h P x, x i > x ∈ H \ { } , and K is a com-pact (self-adjoint) operator on H . In particular, an essentially positive operator isFredholm. A symmetric bilinear form B will be called essentially positive if it isrepresented by an essentially positive operator. Also this notion does not dependon the choice of an inner product. An essentially positive operator has finite index;moreover, the restriction to any closed subspace of an essentially positive form isessentially positive. Lemma 2.5.
Let B : H × H → R be an essentially positive symmetric bilinearform. Then B has finite index.Proof. Since B is essentially positive, the self-adjoint operator T associated to B can be expressed as P + K , with P a positive isomorphism and K a compact self-adjoint operator on H . Considering the equivalent scalar product h· , ·i = h P · , ·i ,the self-adjoint operator associated to B can be expressed as I + P − K , where I is the identity in H and P − K is compact. Note that P − K is self-adjointrelatively to the inner product h· , ·i . The index of B is the sum of the dimensionsof the eigenspaces of the self-adjoint compact operator P − K corresponding to itseigenvalues λ < −
1; this is a finite number. (cid:3)
The following characterization of essentially positive symmetric bilinear formswill be useful:
Lemma 2.6.
Let B be a bounded symmetric bilinear form on H . Then, B isessentially positive if and only if there exists a closed finite codimensional subspace V of H such that: inf x ∈ V k x k =1 B ( x, x ) > . (1) Proof.
Assume that a subspace V as in the statement of the Lemma exists. Let e P : V → V be the positive isomorphism such that B | V × V = h e P · , ·i , and define P : H → H by setting P ( x ) = e P ( x ) for x ∈ V and P ( x ) = x for x ∈ V ⊥ . Clearly, P is a positive isomorphism of H . Moreover, the difference B −h P · , ·i is representedby a finite rank (hence compact) operator K of H ; namely, K ( V ) ⊂ V ⊥ , and so K ( H ) ⊂ V ⊥ + K ( V ⊥ ), which is a finite dimensional subspace of H . Thus, B isessentially positive.Conversely, assume that B is essentially positive, and set B = h ( P + K ) · , ·i ,where P is a positive isomorphism of H and K is a compact self-adjoint operatoron H . There exists a positive constant c > h P x, x i ≥ c k x k for all x ∈ H . Since K is compact, there exists also a finite codimensional closed space V of H such that |h Kx, x i| ≤ c k x k for all x ∈ V . Namely, V can be taken to be theclosure of the direct sum of the eigenspaces of K corresponding to all the eigenvalues λ of K with | λ | ≤ c . Now, for x ∈ V , B ( x, x ) = h P x, x i + h Kx, x i ≥ c k x k . Thisconcludes the proof. (cid:3) Let L sa ( H ) be the closed subspace of L ( H ) consisting of all self-adjoint operators,and let B s ( H ) denote the space of bounded symmetric bilinear forms on H . Once aninner product is fixed on H , one has a natural identification of these two spaces by L sa ( H ) ∋ T
7→ h T · , ·i ∈ B s ( H ); we will consider B s ( H ) endowed with the inducedtopology. SEUDO FOCAL POINTS 7
Lemma 2.7.
Let B be a bounded symmetric bilinear form on H , let V ⊂ H be aclosed subspace such that (1) holds. If P is a projection in L ( H ) which is sufficientlyclose to the orthogonal projection P V onto V and e B ∈ B s ( H ) is close enough to B ,then inf x ∈ P ( V ) k x k =1 e B ( x, x ) > . Proof.
It is a consequence of the fact that convergence in B s ( H ) means uniformconvergence on the unit sphere of H and the following inequality:11 + k P − P V k ≤ k y k ≤ − k P − P V k for every y ∈ V such that k P ( y ) k = 1. (cid:3) Corollary 2.8.
The set: A = (cid:8) ( B, V ) ∈ B s ( H ) × G ( H ) : B | V × V is essentially positive (cid:9) is open in B s ( H ) × G ( H ) . The map A ∋ ( B, V ) n − (cid:0) B | V × V (cid:1) + dim [Ker ( B | V × V )] ∈ N (2) is upper semi-continuous, and the map A ∋ ( B, V ) n − (cid:0) B | V × V (cid:1) ∈ N (3) is lower semi-continuous.Proof. The openness of A follows immediately from Lemma 2.6 and Lemma 2.7.Namely, if B V × V is essentially positive, then there exists a closed subspace W ⊂ V having finite codimension in V such that inf x ∈ W k x k =1 B ( x, x ) >
0. Then, if P isan orthogonal projection sufficiently close to P V , P ( W ) is a finite codimensionalsubspace of P ( V ), and if e B ∈ B s ( H ) is sufficiently close to B by Lemma 2.7inf x ∈ P ( W ) k x k =1 e B ( x, x ) >
0. Thus, by Lemma 2.6, e B | P ( W ) × P ( W ) is essentially positive.Given ( B, V ) ∈ A , the quantity n − (cid:0) B | V × V (cid:1) + dim [Ker ( B | V × V )] is equal to thecodimension in V of a maximal closed subspace W ⊂ V on which B is positivedefinite. Given one such W , an orthogonal projection P sufficiently close to P V and a symmetric bilinear form e B sufficiently close to B , then by Lemma 2.7 e B ispositive definite on P ( W ), and, by Lemma 2.4, the codimension of P ( W ) in P ( V )equals the codimension of W in V . This proves that n − (cid:0) B | V × V (cid:1) + dim [Ker ( B | V × V )] ≥ n − (cid:0) e B | P ( V ) × P ( V ) (cid:1) + dim h Ker (cid:16) e B | P ( V ) × P ( V ) (cid:17)i , i.e., the upper semi-continuity of the map (2).Similarly, if ( B, V ) ∈ A , the quantity n − (cid:0) B | V × V (cid:1) is equal to the dimension ofa maximal closed subspace W ⊂ V on which − B is positive definite. Such W isnecessarily finite dimensional, hence inf x ∈ W k x k =1 − B ( x, x ) >
0. Given one such W , anorthogonal projection P sufficiently close to P V and a symmetric bilinear form e B sufficiently close to B , then by Lemma 2.7 − e B is positive definite on P ( W ), and, M. A. JAVALOYES, A. MASIELLO, AND P. PICCIONE by Lemma 2.4, the dimension of P ( W ) is equal to the dimension of W . This provesthat n − (cid:0) B | V × V (cid:1) ≤ n − (cid:0) e B | P ( V ) × P ( V ) (cid:1) , i.e., the lower semi-continuity of the map (3). (cid:3) We can therefore prove the following:
Proposition 2.9 (Abstract Morse Index Theorem) . Let B s : H × H → R with s ∈ [ a, b ] be a continuous family of bounded symmetric bilinear forms and let { H s } s ∈ [ a,b ] be a continuous family of closed subspaces of H such that the restriction B s : H s × H s −→ R is essentially positive for all s . Assume that for every s, t ∈ [ a, b ] with s < t , thereexists an injective linear map ϕ { s,t } : H s → H t with closed range such that (1) B t (cid:0) ϕ { s,t } ( V ) , ϕ { s,t } ( W ) (cid:1) = B s ( V, W ) for V, W ∈ H s ; (2) Ker (cid:0) B t | H t × H t (cid:1) ∩ ϕ { s,t } ( H s ) = { } .Assume also that B a | H a × H a is non degenerate. Then: (a) the map [ a, b ] ∋ s n − (cid:0) B s | H s × H s (cid:1) ∈ N is nondecreasing; (b) the set of instants s ∈ ] a, b [ such that Ker (cid:0) B s | H s × H s (cid:1) = { } is finite; (c) n − (cid:0) B b | H b × H b (cid:1) = n − ( B a | H a × H a ) + P s ∈ ] a,b [ dim (cid:2) Ker (cid:0) B s | H s × H s (cid:1)(cid:3) . Proof.
Part (a) is obvious, since by (1), the restriction of B to ϕ { s,t } ( H s ) ⊂ H t has the same index than the restriction of B to H s . By Lemma 2.5, we know thatn − (cid:0) B s | H s × H s (cid:1) is finite for all s . Proposition 2.2 and assumptions (1) and (2) implythat if t > s n − ( B t | H t × H t ) ≥ n − ( B s | H s × H s ) + dim (cid:2) Ker (cid:0) B s | H s × H s (cid:1)(cid:3) . A repeated use of this inequality shows that if there existed an infinite numberof instants s ∈ ] a, b [ at which B s | H s × H s degenerates, then n − ( B | H b × H b ) wouldbe infinite. This is absurd, and proves (b). Corollary 2.8 says that if there isno s ∈ [ c, d ] such that B s | H s × H s degenerates, then n − (cid:0) B s | H s × H s (cid:1) is constant on[ c, d ]; namely, if there is no s ∈ [ c, d ] such that B s | H s × H s degenerates, then thefunction s n − (cid:0) B s | H s × H s (cid:1) is both lower and upper semi-continuous on [ c, d ], i.e.,continuous and therefore constant. Using (2) and Proposition 2.2, the jumps of themap n − (cid:0) B s | H s × H s (cid:1) at a degeneracy instant are at least equal to the dimension ofKer (cid:0) B s | H s × H s (cid:1) . On the other hand, Corollary 2.8 says that the value of this jumpis at most equal to the dimension of Ker (cid:0) B s | H s × H s (cid:1) , from which the equality in (c)follows. (cid:3) Remark . The reader will find several analogies between the result of Proposi-tion 2.9 and several other abstract Morse index theorems appearing in the literature,most notably, [21, Theorem 1.11] (see also [7, 8]). All these results originated froma celebrated index theorem due to Smale [20] which holds for a strongly ellipticself-adjoint differential operator L of even order defined on the sections of a Rie-mannian vector bundle E over a compact manifold with boundary M . In order toobtain Smale’s result, one considers the following setup: H is (a suitable closureof) the space C ∞ ( E ) of smooth sections of E vanishing on ∂M , B is the bilinearform B ( u, v ) = R M h Lu, v i d M , and H s is the space of sections of E | M s vanishingon ∂M s , corresponding to a smooth deformation of M by a filtration of compact SEUDO FOCAL POINTS 9 submanifolds M s ⊂ M , s ∈ [ a, b ]. The strong ellipticity assumption gives that B is essentially positive. The assumption that L has the uniqueness property forthe Cauchy problem, i.e., that if u ∈ C ∞ ( E ) satisfies Lu = 0 and u vanishes ona nonempty open subset implies u ≡
0, gives assumption (2) in Proposition 2.9.In this setup, the family H s is not continuous in the sense of Definition 2.3 (seeAppendix A), but only in a weaker sense. Nonetheless, an index theorem is provedin this context using the fact that the family H s is increasing , i.e., H s ⊂ H t when s ≤ t , in which case it suffices to require that the family of orthogonal projectionsonto H s is continuous relatively to the strong operator topology. This is the basicidea in the results of [7, 8, 21]. In the present paper we will consider a situationwhere the weak continuity of a given increasing family of closed subspaces may fail,and [21, Theorem 1.11] does not apply.3. Pseudo focal points and Morse-Sturm systems
Stationary Lorentzian manifolds and geodesics.
Let (
M, g ) be a sta-tionary Lorentzian manifold, ∇ the associated Levi-Civita connection, P a smoothsubmanifold of M and Y a timelike Killing field on M (see [2, 12, 16] for de-tails). Given a geodesic γ : [0 , → M , the equation g ( ∇ ˙ γ ˙ γ, Y ) = 0 integrates as g ( ˙ γ, Y ) = C γ , where C γ is a real constant. In [11], it was proposed the space of H -curves N p,q M joining p and q in M and satisfying the condition g ( ˙ γ, Y ) = C γ almost everywhere to study the energy functional in a stationary Lorentzian mani-fold. This can be generalized as in [10] to the situation in that the curves depart notfrom a point, but from an orthogonal initial submanifold P . Let S P ˙ γ (0) denote thesecond fundamental form of P at the orthogonal direction ˙ γ (0). Recall that S P ˙ γ (0) is the symmetric bilinear form on T γ (0) P defined by S P ˙ γ (0) ( v, w ) = g (cid:0) ˙ γ (0) , ∇ v W (cid:1) ,where v, w ∈ T γ (0) P and W is any extension of w to a local vector field along P .It is a convenient assumption that P be nondegenerate at γ (0), i.e., that therestriction of the Lorentzian metric tensor g to T γ (0) P be nondegenerate. Thisassumption has two basic consequences:(a) there are no P -focal points on a sufficiently short initial portion of γ ;(b) S P ˙ γ (0) can be written in terms of the shape operator of P , which is a g -symmetric linear endomorphism, also denoted by S P ˙ γ (0) , defined as the linearoperator associated to the second fundamental form S P ˙ γ (0) relatively to therestriction of g to T γ (0) P .The subset N {P ,q } M is a submanifold of the manifold Ω {P ,q } M consisting of all H -curves from P to q in M satisfying g ( ˙ γ, Y ) = C γ . It is not difficult to showthat the tangent space to N {P ,q } M is given by the H -vector fields V along γ with V (0) ∈ T γ (0) P , V (1) = 0 and g ( V ′ , Y ) − g ( V, Y ′ ) = C V (4)a. e. on [0 ,
1] for any constant C V (in the following we will use the upper index ′ to denote covariant differentiation along γ or derivation depending on the context).Moreover, if we consider the energy functional E ( γ ) = Z g ( ˙ γ, ˙ γ ) d s, restricted to N {P ,q } M , its critical points are the geodesics from P to q that departorthogonally from P . Along this section we are going to consider the subspace of T γ (cid:0) N {P ,q } M (cid:1) putting C V = 0. The idea is to restrict the tangent of N {P ,q } M tothe tangent of the subset of curves having the same constant C γ . We observe thatthis subset may fail to be a submanifold of N {P ,q } M and when it is, the criticalpoints of the energy functional restricted to it may not be geodesics. Anyway, itwill be of a great help to study the index form, which can be written as I { γ, P} ( V, W ) = Z (cid:2) g ( V ′ , W ′ ) + g (cid:0) R ( ˙ γ, V ) ˙ γ, W (cid:1)(cid:3) d t − g ( S P ˙ γ (0) [ V (0)] , W (0)) , (5)where R is the curvature tensor of M chosen with the sign convention R ( X, Y ) =[ ∇ X , ∇ Y ] − ∇ [ X,Y ] . Recall that a Jacobi field along γ (see [16]) is a vector field J along γ satisfying the Jacobi equation J ′′ = R ( ˙ γ, J ) ˙ γ ;then using that the restriction of Y to γ is a Jacobi field, it is easy to prove that J satisfies Eq. (4). We say that t ∈ ]0 ,
1] is a focal instant of the geodesic γ withrespect to P , if there exists a non null Jacobi field J satisfying J (0) ∈ T γ (0) P , J ′ (0) + S P ˙ γ (0) [ J (0)] ∈ (cid:0) T γ (0) P (cid:1) ⊥ , and J ( t ) = 0.3.2. Morse-Sturm systems and Jacobi fields.
The results we are going toobtain hold in the more general context of
Morse-Sturm systems , i.e., differentialsystems of the form: V ′′ ( t ) − R ( t )[ V ( t )] = 0 , (6)where V ∈ H ([0 , R n ) and R : [0 , → L ( R n ) is a continuous map for every t ∈ [0 ,
1] taking values in the space of all endomorphisms of R n that are symmetricrelatively to a given nondegenerate symmetric bilinear form g on R n . To obtain aMorse-Sturm system from the geometrical setup, it is enough to consider a parallelorthonormal frame along the geodesic γ , so that the Jacobi equation of the geodesicsbecomes a Morse-Sturm system in R n . We will need some additional data. Let g bea bilinear form with index 1 in R n × R n (that in the stationary context representsthe Lorentzian metric). For every t ∈ [0 ,
1] we ask R ( t ) to be a g -symmetric linearmap, that is, g ( R ( t )[ x ] , y ) = g ( x, R ( t )[ y ]) for every x, y ∈ R n . Let Y be a map Y : [0 , → R n such that g ( Y ( t ) , Y ( t )) < Y ′′ ( t ) = R ( t )[ Y ( t )] , let P be a g -nondegenerate subspace of R n ( P represents the tangent space T γ (0) P ),and S : P → P a g -symmetric linear map (that represents the shape operator S P ˙ γ (0) of P at γ (0) in the normal direction ˙ γ (0)). We observe that the symbol ⊥ willdenote the orthogonal subspace with respect to g . The initial conditions of theMorse-Sturm system (6) are given by V (0) ∈ P and V ′ (0) + S [ V (0)] ∈ P ⊥ , (7)and the associated index form of the problem is defined as I ( V, W ) = Z (cid:2) g ( V ′ , W ′ ) + g (cid:0) R ( t )[ V ] , W (cid:1)(cid:3) d t − g (cid:0) S [ V (0)] , W (0) (cid:1) . (8)Summing up, we will assume the initial data ( g, R, Y, P, S ) defined above, we willrefer to the solutions of (6) as Jacobi fields , and we will say that t ∈ ]0 ,
1] is a focal
SEUDO FOCAL POINTS 11 instant of the given data if there exists a non null Jacobi field satisfying the initialdata (7) and such that J ( t ) = 0. It is easy to see that a Jacobi field V satisfies g ( V ′ , Y ) − g ( V, Y ′ ) = C V . (9)3.3. Admissible and singular Jacobi fields.
In order to establish the results weaim to, we will need some additional properties of the Jacobi field Y . In particular,the following definitions will be useful. Definition 3.1.
We say that Y is admissible if Y (0) and Y ′ (0) are linearly indepen-dent and singular when Y ( s ) and Y ′ ( s ) are linearly dependent for every s ∈ [0 , m ( Y )( s ) = Y ( s ) g (cid:0) Y ( s ) , Y ( s ) (cid:1) ! ′ + Y ′ ( s ) g ( Y ( s ) , Y ( s )) , (10)then Y is admissible iff m ( Y )(0) = 0, and singular iff m ( Y )( s ) = 0 for every s ∈ [0 , g ( Y, Y ) < m ( Y )( s ) = 0 isequivalent to Y ′ ( s ) g ( Y ( s ) , Y ( s )) − Y ( s ) g ( Y ( s ) , Y ′ ( s )) = 0, and the last equality isequivalent to Y ( s ) and Y ′ ( s ) being linearly dependent whenever g ( Y ( s ) , Y ( s )) = 0.We are especially interested in the case where the data comes from a geomet-rical setup. In fact, the initial data can be obtained from a more general contextthan stationary manifolds, that is, when considering a geodesic γ in a Lorentzianmanifold, a submanifold P orthogonal to γ through γ (0) and a timelike Jacobi fieldalong γ . In this case the notion of admissible and singular Jacobi fields can bebrought in the obvious way.Even if we find a timelike Jacobi field Y along γ , it might not be admissible orsingular. To overcome this situation we can consider the family of Jacobi fields e Y = Y + ( a + b t ) ˙ γ for a, b ∈ R small enough and look for a Jacobi field with therequired properties. Lemma 3.2.
Let ( M, g ) be a Lorentzian manifold, γ a geodesic in M and Y atimelike Jacobi field along γ . Then: ( i ) If γ is timelike, ˙ γ is a singular Jacobi field along γ . ( ii ) Consider a Jacobi field ¯ Y ( a, b ) = Y + ( a + b t ) ˙ γ for some a, b ∈ R , such that ¯ Y ( a, b ) is timelike (for example when a, b ∈ R are small enough). If γ islightlike or spacelike, then there exist a and b in R such that ¯ Y = Y +( a + bt ) ˙ γ is admissible. ( iii ) If γ is spacelike, then the Jacobi field ¯ Y = Y − g ( Y , ˙ γ ) E − γ ˙ γ , where E γ = g ( ˙ γ, ˙ γ ) , is timelike and orthogonal to the geodesic γ .Proof. The first assertion is obvious and ( iii ) can be shown by a straightforwardcomputation. Let us prove ( ii ). Assume that Y is not admissible, that is, thereexists α such that Y ′ (0) = α Y (0). Then ¯ Y ′ (0) = Y ′ (0) + b ˙ γ (0), so that ¯ Y is notadmissible when there exists β satisfying Y ′ (0) + b ˙ γ (0) = β ( Y (0) + a ˙ γ (0)). Thisimplies that ( α − β ) Y (0) = ( βa − b ) ˙ γ (0) . As Y (0) is timelike and ˙ γ (0) does not,it follows that β = α and βa = b , but as α is fixed, we can choose a and b smallenough such that βa − b = αa − b = 0 and ¯ Y is timelike. (cid:3) In the following lemma we are going to give a geometric characterization ofsingularity for a vector field related to γ . Lemma 3.3.
Let ( M, g ) be a Lorentzian manifold and Y a timelike Jacobi fieldalong a spacelike geodesic γ : [0 , → M . If we assume that Y is orthogonal to ˙ γ ,which is not restrictive by Lemma 3.2, then Y is singular if and only if there existsa ( n − -tuple of parallel orthogonal vector fields F = { E , . . . , E n − } along γ suchthat { Y ( s ) } ⊥ = span { E ( s ) , . . . , E n − ( s ) } for every s ∈ [0 , .Proof. If Y is singular, it is easy to see that there exists α : [0 , → R \ { } suchthat the vector field t → α ( t ) Y ( t ) is parallel. Indeed, if Y ′ ( t ) = β ( t ) Y ( t ), we canchoose α ( t ) = e − R t β ( s )d s . Then considering an orthonormal frame of Y (0) ⊥ andmaking the parallel transport along γ we obtain the family F . The other side canbe shown as follows. We know that g ( Y ( t ) , E i ( t )) = 0 for every t ∈ [0 ,
1] and E i are parallel along γ , so that g ( Y ′ ( t ) , E i ( t )) = 0 for i = 1 , . . . , n − Y ′ ( t ) hasto be linearly dependent to Y ( t ) for every t ∈ [0 , (cid:3) Remark . Lemma 3.3 gives a relation between the geodesics admitting a singularJacobi field and those that are contained in a totally geodesic hypersurface. It isclear that when the geodesic is contained in a totally geodesic spacelike hypersurfaceand there exists a timelike Jacobi field orthogonal to the hypersurface, then thereexists a frame as in Lemma 3.3 and a singular Jacobi field Y along γ .3.4. Functional analytical setup.
In this subsection we will introduce several L -spaces and will state some density results, that will be used in the next subsec-tion to compute the kernel of a restriction of the index form. Let us consider theHilbert space L ([ a, b ]; R n ) of Lebesgue integrable functions from [ a, b ] to R n andthe Sobolev space H ([ a, b ]; R n ) of all absolutely continuous maps from [ a, b ] to R n vanishing in the endpoints and having derivatives in L ([ a, b ]; R n ). Analogously, H ([ a, b ]; R n ) is the space of C maps, with an absolutely continuous first derivativeand whose second derivative is in L ([ a, b ]; R n ). Moreover, H P ([ a, b ]; R n ) consistsof the functions V ∈ H ([ a, b ]; R n ) such that V ( a ) ∈ P and V ( b ) = 0, being P asubspace of R n .Using Y , we can define a smooth family of positive definite inner products g ( r ) t on R n as g ( r ) t ( V, W ) = g ( V, W ) − g ( V, Y ( t )) · g ( W, Y ( t )) g ( Y ( t ) , Y ( t )) . (11)We observe that there is a smooth family A : [0 , → L ( R n ) of g ( r ) t -symmetricoperators such that g ( V, W ) = g ( r ) t ( A ( t )[ V ] , W )for every V, W ∈ R n . We also define the following inner product in the Hilbertspace L ([0 , σ ]; R n ): R σ ( V, V ) = Z σ g ( r ) t ( V, V )d t. (12)We will now introduce two subspaces of L ([0 , σ ]; R n ), that reproduce the L -version of the space T γ N { p,q } M in the geometrical setup and a one-codimensionalsubspace obtained by setting C V = 0: K ( σ ) = n V ∈ L ([0 , σ ]; R n ) : g (cid:0) V ( t ) , Y ( t ) (cid:1) = − tσ Z σ g ( V, Y ′ ) d s + 2 Z t g ( V, Y ′ ) d s a. e. on [0 , σ ] o SEUDO FOCAL POINTS 13 and K ( σ ) = n V ∈ L ([0 , σ ]; R n ) : g (cid:0) V ( t ) , Y ( t ) (cid:1) = 2 Z t g ( V, Y ′ ) d s a. e. on [0 , σ ]and Z σ g ( V, Y ′ ) d s = 0 o . The spaces K ( σ ) and K ( σ ) can also be described as follows: K ( σ ) = n V ∈ L ([0 , σ ]; R n ) : g ( V, Y ) ∈ H ([0 , σ ]; R )and ∃ C V ∈ R such that dd t g ( V, Y ) = C V + 2 g ( V, Y ′ ) o , K ( σ ) = n V ∈ L (cid:0) [0 , σ ]; R n (cid:1) : g ( V, Y ) ∈ H (cid:0) [0 , σ ]; R (cid:1) anddd t g ( V, Y ) = 2 g ( V, Y ′ ) o . Moreover, it is easy to see that K ( σ ) and K ( σ ) are closed subpaces of L ([0 , σ ]; R n ).In order to simplify notations, we will omit the argument σ when unnecessary. Wewant to show that K ( σ ) ∩ H (cid:0) [0 , σ ]; R n (cid:1) is dense in K ( σ ). The proof of this factis based in the following abstract result. Lemma 3.5 (Density criteria) . Let H be a Hilbert space and let R ⊂ H be a denselinear subspace. • If H ⊂ H is a closed subspace such that there exists a projection P (notnecessarily self-adjoint) from H onto H with P ( R ) ⊂ R , then, R ∩ H isdense in H . • If H ⊂ H is a closed subspace with finite codimension in H , then R ∩ H is dense in H .Proof. Fix x ∈ H and let r n ∈ R be a sequence with lim r n = x . Then, P ( r n ) ∈ R ∩ H , because P ( R ) ⊂ R , and lim P ( r n ) = P ( x ) = x , which proves that R ∩ H is dense in H . For the second statement, first note that H + R is closed, because it contains H , and dense, because it contains R ; thus H + R = H . We cantherefore find a finite dimensional complement H to H such that H ⊂ R . Then,the projection P onto the first factor H = H ⊕ H → H satisfies P ( R ) ⊂ R , andby the first density criterion R ∩ H is dense in H . (cid:3) Proposition 3.6. K ( σ ) ∩ H (cid:0) [0 , σ ]; R n (cid:1) is dense in K ( σ ) .Proof. By Corollary 3.2 in [15] we know that K ( σ ) ∩ H (cid:0) [0 , σ ]; R n (cid:1) is dense in K ( σ ). Then, the thesis is obtained easily from the second density criterion inLemma 3.5, applied to the Hilbert space H = K ( σ ), the dense linear subspace R = K ( σ ) ∩ H (cid:0) [0 , σ ]; R n (cid:1) , and the closed subspace H = K ( σ ), that has codimension1 in H (it is the kernel of the bounded linear functional K ( σ ) ∋ V C V ∈ R ). (cid:3) Recall that any subspace that contains a closed finite codimensional subspace is also closed.
The kernel of the restricted index form.
As a previous result to the com-putation of the kernel of the restricted index form in Proposition 3.11 we need a de-scription of the orthogonal space of K with respect to the Hilbert structure given by(12), that we denote K ⊥ . First, we observe that K can be described as intersectionof kernels of bounded linear operators between Hilbert spaces. Indeed, we considerthe operators L ([0 , σ ]; R n ) ∋ V → T ( V )( t ) = g (cid:0) V ( t ) , Y ( t ) (cid:1) − R t g ( V, Y ′ )d s ∈ L ([0 , σ ]; R ) and L ([0 , σ ]; R n ) ∋ V → T ( V )( t ) = R σ g ( V, Y ′ )d s ∈ L ([0 , σ ]; R )then K = T − (0) ∩ T − (0). Recall now the following abstract result in Banachspaces that can be found in [15, Lemma 3.4]. Lemma 3.7.
Let X and Y be Banach spaces and T : X → Y be a bounded linearoperator with closed image in Y . Then, Im( T ∗ ) = (Ker( T )) ; where (Ker( T )) isthe annihilator of Ker( T ) in X ∗ . In particular, Im( T ∗ ) is closed in X . Therefore, we have to compute the adjoint operators T ∗ , T ∗ : L ([0 , σ ]; R ) → L ([0 , σ ]; R n )with respect to the usual product in L ([0 , σ ]; R ) and the product defined in (12)in L ([0 , σ ]; R n ). It is easily seen that T ∗ and T ∗ can be expressed as T ∗ ( φ )( t ) = φ ( t ) · ( A ( t )[ Y ( t )]) − A ( t )[ Y ′ ( t )]) · Z σt φ ( s )d s. (13)and T ∗ ( φ )( t ) = Z σ φ ( s ) d s A [ Y ′ ( t )] . Lemma 3.8.
The image
Im( T ) is closed in L ([0 , σ ]; R ) .Proof. Consider the map e T : L ([0 , σ ]; R ) → L ([0 , σ ]; R ) defined as e T ( µ ) = T ( µ · Y ). By the definition of T , the operator e T is the sum of the isomorphism µ → µ · g ( Y, Y ) and a compact operator on L ([0 , σ ]; R ), so that by the Fred-holm’s alternative we conclude that Im( e T ) is closed and has finite codimension in L ([0 , σ ]; R ). Finally, Im( e T ) ⊂ Im( T ) implies that Im( T ) is closed, because itcontains a closed subspace with finite codimension. (cid:3) Moreover, as the image of T is the subset of constant functions, then it is alsoclosed. Hence, by Lemma 3.7 and Lemma 3.8, determining K ( σ ) ⊥ is equivalentto obtaining a description of Im( T ∗ ) + Im( T ∗ ). With this in mind, we observe thatgiven a function φ ∈ L ([0 , σ ]; R ) there exists a unique h φ ∈ H ([0 , σ ]; R ) such that h φ ( σ ) = 0 and h ′ φ = φ . The following corollary follows straightforward. Corollary 3.9.
The orthogonal space K ( σ ) ⊥ in L ([0 , σ ]; R n ) is K ( σ ) ⊥ = (cid:8) h ′ · A [ Y ] + 2 h · A [ Y ′ ] : h ∈ H ([0 , σ ]; R ) (cid:9) . (14)Let us consider the following symmetric bilinear form on H P ([0 , σ ]; R n ) given by I σ ( V, W ) = Z σ h g ( V ′ , W ′ ) + g ( R [ V ] , W ) i d t − g (cid:0) S [ V (0)] , W (0) (cid:1) (15)and let us denote W P ( σ ) = { V ∈ H P ([0 , σ ]; R n ) : g ( V ′ , Y ) − g ( V, Y ′ ) = 0 a. e. on [0 , σ ] } . In order to describe the kernel of I σ we will introduce the following generalizationof Jacobi fields. SEUDO FOCAL POINTS 15
Definition 3.10.
We will say that V ∈ H ([0 , σ ]; R n ) is a Y -pseudo Jacobi field ifthere exists λ ∈ R such that V ′′ − R ( t )[ V ] = λ m ( Y ) , (see (10)) and g ( V ′ , Y ) − g ( V, Y ′ ) = 0. Moreover, we say that V is a ( P, Y ) -pseudoJacobi field when in addition it holds the initial conditions V (0) ∈ P and V ′ (0) + λ Y (0) g ( Y (0) , Y (0)) + S [ V (0)] ∈ P ⊥ , (when Y is singular, we take λ = 0).When the choice of Y is clear by the context we will say just pseudo Jacobi or P -pseudo Jacobi fields. Proposition 3.11.
A vector V σ ∈ W P ( σ ) belongs to the kernel of the restrictionof I σ to W P ( σ ) × W P ( σ ) if and only if V σ is a P -pseudo Jacobi field.Proof. If V σ ∈ W P ( σ ) belongs to the kernel of I σ restricted to W P ( σ ) × W P ( σ ),then using a standard boot-strap argument one proves that V σ is differentiable. Byapplying integration by parts we obtain that I σ ( V σ , W ) = Z σ g ( − V ′′ σ + R [ V σ ] , W )d s for every W ∈ H ([0 , σ ]; R n ) ∩ W P ( σ ) = H ([0 , σ ]; R n ) ∩ K ( σ ). In particular, itfollows that − A [ V ′′ σ ] + A [ R [ V σ ]] ∈ (cid:0) K ( σ ) ∩ H ([0 , σ ] , R n ) (cid:1) ⊥ = K ( σ ) ⊥ , (16)where ⊥ is taken with respect to the scalar product (12). This is because K ( σ ) ∩ H ([0 , σ ]; R n ) is dense in K ( σ ) (Proposition 3.6). From Eq. (16) and Corollary 3.9we deduce that there exists a function h ∈ H ([0 , σ ]; R n ) such that − V ′′ σ + R [ V σ ] = h ′ · Y + 2 h · Y ′ . (17)Then multiplying by Y with the g -scalar product we get g ( − V ′′ σ , Y ) + g ( R [ V σ ] , Y ) = ( h · g ( Y, Y )) ′ . Observing that V σ ∈ W P ( σ ) satisfies g ( V ′′ σ , Y ) = g ( V σ , Y ′′ ), R is g -symmetric and Y is a Jacobi field, we deduce that ( h · g ( Y, Y )) ′ = 0. This implies that h = µ g ( Y, Y )for some real constant µ . Substituting in (17) we obtain that V σ is a pseudo Jacobifield. Applying again integration by parts to I σ ( V σ , W ), now with W ∈ W P ( σ ),and using that V σ is a pseudo Jacobi field, we obtain that I σ ( V σ , W ) = − g ( V ′ σ (0) + λ Y (0) g ( Y (0) , Y (0)) + S [ V σ (0)] , W (0)) . As there exists a vector field W ∈ W P ( σ ) such that W (0) = U for every U ∈ P ,we deduce that V ′ σ (0) + λ Y (0) /g ( Y (0) , Y (0)) + S [ V σ (0)] ∈ P ⊥ and therefore V σ isa P -pseudo Jacobi field. The other way is straightforward. (cid:3) The Morse Index Theorem in stationary spacetimes
Smooth family of Hilbert spaces.
We have now enough information toprove a Morse index theorem for the index form in (8) in a suitable restriction byapplying the abstract theorem stated in Proposition 2.9. We will proceed studyingthe evolution of the index of I σ when σ goes to 1. As we mentioned in the intro-duction, we cannot assure any kind of continuity of the path σ → W P ( σ ), so thatwe will consider another one obtained as a reparametrization in the interval [0 , σ : H P ( σ ) −→ W P ( σ ) , (18)where H P ( σ ) = (cid:8) V ∈ H P ([0 , , R n ) : g ( V ′ ( t ) , Y ( σt )) − σg ( V ( t ) , Y ′ ( σt )) = 0 (cid:9) (19)and Φ σ ( V ) = e V is given by s → e V ( s ) = V ( sσ ), which is clearly one-to-one. Weobserve that H P ( σ ) can be extended to σ = 0 putting H P (0) = (cid:8) V ∈ H P ([0 , , R n ) : g (cid:0) V ′ ( t ) , Y (0) (cid:1) = 0 (cid:9) = (cid:8) V ∈ H P ([0 , , R n ) : g (cid:0) V ( t ) , Y (0) (cid:1) = 0 (cid:9) . Analogously, we define H ∗ P ( σ ) = (cid:8) V ∈ H P (cid:0) [0 , , R n (cid:1) : g (cid:0) V ′ ( t ) , Y ( σt ) (cid:1) − σg (cid:0) V ( t ) , Y ′ ( σt ) (cid:1) = C V ∈ R (cid:9) , for σ ∈ [0 , H P ( σ ) varies smoothly with σ . In order to formalize this fact, one needs to use the differentiable structure of theGrassmannian of all closed subspaces of a Hilbert space, see for instance reference[1]. In analogy with Definition 2.3 we give the following: Definition 4.1.
Let H be a Hilbert space, I ⊂ R an interval and {D t } t ∈ I bea family of closed subspaces of H . We say that {D t } t ∈ I is a C -family of closedsubspaces if the map I ∋ t
7→ D t ∈ G ( H ) is of class C .It is not hard to show that {D t } t ∈ I is a C -family of closed subspaces if forall t ∈ I there exist ε >
0, a C -curve α : ] t − ε, t + ε [ ∩ I
7→ L ( H ) and aclosed subspace D ⊂ H such that α ( t ) is an isomorphism and α ( t )( D t ) = D forall t ∈ ] t − ε, t + ε [. Moreover, the following criterion for the smoothness of afamily of closed subspaces can be found in [10, Lemma 2.9]. Proposition 4.2.
Let I ⊂ R be an interval, H , e H be Hilbert spaces and F : I ( H , e H ) be a C -map such that each F ( t ) is surjective. Then, the family D t =Ker (cid:0) F ( t ) (cid:1) is a C -family of closed subspaces of H . (cid:3) Proposition 4.3.
Assume that Y is singular and Y (0) is orthogonal to P or that Y (0) is not orthogonal to P . Then the family of closed subspaces H P ( σ ) with σ ∈ [0 , defined in (19) is a C -family of H P ([0 , R n ) . If Y (0) is orthogonal to P and Y is admissible, then (19) is a C -family of H P ([0 , R n ) in ]0 , .Proof. Consider the map F σ : H P ([0 , R n ) → L ([0 , R ) defined as F σ ( V )( t ) = g ( V ′ ( t ) , Y ( σt )) − σg ( V ( t ) , Y ′ ( σt ))for σ ∈ [0 , F σ : [0 , → L ( H P ([0 , R ) , L ([0 , R )) is C asin [10, Lemma 4.3]. Moreover, by [10, Lemma 4.4] we know that given a function SEUDO FOCAL POINTS 17 h ∈ L ([0 , R ) there exists ˆ f ∈ H ([0 , R ) such that if ˆ Y σ ( u ) = Y ( σu ), then F σ ( ˆ f · ˆ Y σ ) = h + C for a certain C ∈ R . In order to prove that F σ is surjective it is enough to find forany C ∈ R functions W ∈ H ([0 , R n ) and ˆ h ∈ H ([0 , R ) such that F σ ( W + ˆ h · ˆ Y σ ) = C. The last equation is equivalent to g ( W ′ ( u ) , Y ( σu )) − σg ( W ( u ) , Y ′ ( σu )) + ˆ h ′ ( u ) g ( Y ( σu ) , Y ( σu )) = C, so thatˆ h ( u ) = Z u Cg ( Y ( σs ) , Y ( σs )) d s + Z u − g ( W ′ ( s ) , Y ( σs )) + σg ( W ( s ) , Y ′ ( σs )) g ( Y ( σs ) , Y ( σs )) d s. If we find a function W ∈ H P ([0 , R n ) such that ˆ h (1) = 0 the result is proven.To this end it is enough to show that there exists W ∈ H P ([0 , R n ) such that Z − g ( W ′ ( s ) , Y ( σs )) + σg ( W ( s ) , Y ′ ( σs )) g ( Y ( σs ) , Y ( σs )) d s = 0 . By applying integration by parts this is equivalent to − g ( W (0) , Y (0)) g ( Y (0) , Y (0)) + σ Z g ( W ( s ) , m ( Y )( σs )) d s = 0 , (20)where m ( Y ) is defined in (10). When σ = 0, this condition is satisfied iff Y (0) is notorthogonal to P . For σ ∈ ]0 , W ∈ H P ([0 , R n )if and only if m ( Y )( σs ) = 0 for some s ∈ [0 ,
1] or Y (0) is not orthogonal to P .If Y is admissible then m ( Y )( σs ) = 0 for s small enough. Summing up, F σ issurjective for σ ∈ [0 ,
1] when Y (0) is not orthogonal to P and for σ ∈ ]0 ,
1] when Y is admissible. Applying Proposition 4.2 we conclude that H P ( σ ) is a C -family inboth cases. Assume now that Y is singular and Y (0) is orthogonal to P . We willsee that in this case H P ( σ ) = H ∗ P ( σ ) , so that we can apply [10, Corollary 4.5] toconclude that H P ( σ ) is a C -family. A function W ∈ H ∗ P ( σ ) satisfies that − g ( W ′ ( s ) , Y ( σs )) + σg ( W ( s ) , Y ′ ( σs )) = C W . Dividing by g ( Y ( σs ) , Y ( σs )), integrating between 0 and 1 and applying integrationby parts, we obtain − g (cid:0) W (0) , Y (0) (cid:1) g (cid:0) Y (0) , Y (0) (cid:1) + Z g (cid:0) W ( s ) , m ( Y ( σs )) (cid:1) d s = C W Z d sg (cid:0) Y ( σs ) , Y ( σs ) (cid:1) . The left term is zero and the right term is zero iff C W = 0, so that we concludethat the constant C W has to be zero and therefore, H P ( σ ) = H ∗ P ( σ ). (cid:3) Morse index and nullity.
We must find the counterpart of the index formin H P ( σ ). Using the map (18) and the index form I σ given in (15), we obtainˆ I σ ( V, W ) = I σ (Φ σ ( V ) , Φ σ ( W )); more explictly,ˆ I σ ( V, W ) = Z (cid:2) σ g (cid:0) V ′ ( t ) , W ′ ( t ) (cid:1) + σg (cid:0) R ( σt )[ V ( t )] , W ( t ) (cid:1)(cid:3) d t − g (cid:0) S [ V (0)] , W (0) (cid:1) . (21) We observe that C σ = σ ˆ I σ can be extended to σ = 0 in a continuous way as C ( V, W ) = Z g (cid:0) V ′ ( t ) , W ′ ( t ) (cid:1) d t. Proposition 4.4.
The Morse index of ˆ I σ | H P ( σ ) ×H P ( σ ) is finite for all σ ∈ ]0 , and the related operator to ˆ I σ is essentially positive.Proof. The first part of the proposition follows from the second one and Lemma 2.5.By using Eq. (11) and the identity g ( V ′ ( t ) , Y ( σt )) = σg ( V ( t ) , Y ′ ( σt )) for V ∈H P ( σ ), C σ = σ ˆ I σ can be expressed as C σ ( V, W ) = Z g ( r ) σt ( V ′ ( t ) , W ′ ( t ))d t + 2 σ Z g ( V ( t ) , Y ′ ( σt )) g ( W ( t ) , Y ′ ( σt )) g ( Y ( σt ) , Y ( σt )) d t + σ Z g ( R ( σt )[ V ( t )] , W ( t ))d t − g ( S [ V (0)] , W (0)) . (22)The first term gives the identity times a constant as associated operator with respectto the scalar product given by Z g ( r ) σt ( V ′ ( t ) , W ′ ( t ))d t, (23)for V, W ∈ H P ([0 , R n ), but the positivity of the related operator does not dependon the scalar product. The other terms give a continuous operator that has to becompact because H P ([0 , R n ) is compactly embedded in C ([0 , R n ). (cid:3) The case σ = 0 must be considered separately, because the path H P ( σ ) may notbe even continuous at that instant and when it is, we need to compute the indexof C to establish the Morse index theorem. Lemma 4.5. If Y (0) is orthogonal to P , then the index of ˆ I σ is zero if σ is smallenough. If Y (0) is not orthogonal to P , the index of C is zero.Proof. We first observe that if Y (0) is orthogonal to P , then n − ( g | P ) = 0, so thatby [10, Proposition 4.10 ] we know that ˆ I σ restricted to H ∗ P ( σ ) × H ∗ P ( σ ) is positivedefinite if σ is small enough. As H P ( σ ) ⊂ H ∗ P ( σ ), the thesis follows. Assume that Y (0) is not orthogonal to P . In [10, Proposition 4.10] it was shown that H ∗ P (0) canbe decomposed as a direct sum ( H ∗ P (0)) + ⊕ ( H ∗ P (0)) − , where( H ∗ P (0)) + = { V ∈ H ∗ P (0) : V (0) ∈ P + } , ( H ∗ P (0)) − = { V : [0 , → R n affine function : V (0) ∈ P − , V (1) = 0 } and P = P + ⊕ P − is a decomposition of P as a direct sum of a positive and anegative space, in such a way that C is positive definite in ( H ∗ P (0)) + and negativedefinite in ( H ∗ P (0)) − . On other hand, if V ∈ H P (0), then V (0) ∈ { Y (0) } ⊥ ∩ P ;moreover as g is positive definite on { Y (0) } ⊥ ∩ P , we can choose a decomposition P = P + ⊕ P − in such a way that { Y (0) } ⊥ ∩ P ⊂ P + . Then H P (0) ⊂ ( H ∗ P (0)) + and C is positive definite on H P (0). (cid:3) Definition 4.6.
An instant t ∈ (0 ,
1] is said (
P, Y ) -pseudo focal if there exists a( P, Y )-pseudo Jacobi field such that V ( t ) = 0.Finally we can get a Morse index theorem of Riemannian type. SEUDO FOCAL POINTS 19
Theorem 4.7.
Assume that Y (0) is not orthogonal to P or that Y is either ad-missible or singular. Then, the Morse index of I | H P (1) ×H P (1) coincides with thenumber of ( P, Y ) -pseudo focal points counted with multiplicity.Proof. It follows from the Abstract Morse Index Theorem given in Proposition 2.9by taking H s = H P ( s ) ⊂ H ([0 , R n ) and B s = C s : H ([0 , R n ) × H ([0 , R n ) → R (see Eq. (21) and the following paragraph) defined for s ∈ [0 ,
1] when Y (0) is notorthogonal to P and on [ ε,
1] with ε > I s is positive definitein H P ( s ) × H P ( s ) when s ∈ (0 , ε ] in the other case. It is easy to prove that C s iscontinuous. Moreover, we choose ϕ { s,t } : H s → H t with s < t defined as follows.Let E { s,t } : H P ([0 , s ]; R n ) → H P ([0 , t ]; R n ) be the map that carries an element of H P ([0 , s ]; R n ) to its extension to zero in [ s, t ]; this gives an element in H P ([0 , t ]; R n ).Then ϕ { s,t } ( V ) = Φ − t · E { s,t } · Φ s ( V ). It is a straightforward computation to verifythat ϕ { s,t } satisfies the hypothesis in Proposition 2.9. By Proposition 4.3 the path s → H P ( s ) is smooth, and by [5, Proposition 4.9] this implies the continuity asclosed subspaces in the sense of Section 2; by Proposition 4.4 the symmetric bilinearforms C s are essentially positive and by Lemma 4.5 the initial contribution is alwayszero, so that the index theorem follows for ˆ I and of course for I = I restrictedto W P (1) × W P (1) (we observe that W P (1) = H P (1)). By Proposition 3.11 thedimensions of the kernel of I s restricted to W P ( s ) × W P ( s ) coincides with thenumber of ( P, Y )-pseudo focal points counted with multiplicity. (cid:3)
The case of two variable endpoints.
We will use the idea of [17, TheoremII.6] to extend the Morse index theorem to the situation in that the two endpointsare variable. In the context of Morse-Sturm systems we have to add to the initialdata ( g, R, Y, P, S ) (after Eq. (8)) another subspace Q of R n and a g -symmetriclinear map S Q : Q → Q . Moreover, we rename S as S P . Thus, we have the initialdata ( g, R, Y, P, Q, S P , S Q ) and the index form I { P,Q } ( V, W ) = Z (cid:2) g ( V ′ , W ′ ) + g (cid:0) R ( t )[ V ] , W (cid:1)(cid:3) d t + g (cid:0) S Q [ V (1)] , W (1) (cid:1) − g (cid:0) S P [ V (0)] , W (0) (cid:1) , (24)defined for V and W in H { P,Q } ([0 , R n ), that is, the functions V in H ([0 , R n ),such that V (0) ∈ P and V (1) ∈ Q . Denote H { P,Q } = (cid:8) V ∈ H { P,Q } ([0 , R n ) : g ( V ′ ( t ) , Y ( t )) − g ( V ( t ) , Y ′ ( t )) = 0 (cid:9) and H ∗{ P,Q } = (cid:8) V ∈ H { P,Q } ([0 , R n ) : g ( V ′ ( t ) , Y ( t )) − g ( V ( t ) , Y ′ ( t )) = C V ∈ R (cid:9) . Furthermore, let J ∗ Q be the subspace of P -Jacobi fields contained in H ∗{ P,Q } , J Q = J ∗ Q ∩H { P,Q } and F the symmetric bilinear form obtained as the restriction of I { P,Q } to J ∗ Q . By applying integration by parts we obtain that F ( J , J ) = g (cid:0) S Q [ J (1)] , J (1) (cid:1) + g (cid:0) J ′ (1) , J (1) (cid:1) , where J and J are in J ∗ Q . Finally, J [ t ] = { J ( t ) ∈ R n : J is a P -Jacobi field and C J = 0 } and J ∗ [ t ] = { J ( t ) ∈ R n : J is a P -Jacobi field } . Adapting [17, Theorem II.6] to this situation we obtain the following.
Theorem 4.8.
Assume, that J [1] ⊇ Q . Then, the index of I { P,Q } restricted to H { P,Q } × H { P,Q } is equal to the sum of the index of I | H P (1) ×H P (1) and the index of F | J Q ×J Q .Proof. Denote J = { J ∈ J Q : J [1] = 0 } and choose a complementary subspace J in J Q such that J Q = J ⊕J . As J [1] ⊇ Q , we have that H { P,Q } = H P (1) ⊕J and it is easy to see that this decomposition is I { P,Q } -orthogonal. It follows that theindex of I { P,Q } | H { P,Q } ×H { P,Q } is equal to the index of I | H P (1) ×H P (1) plus the indexof F | J ×J . The theorem follows from the observation that J is contained in thekernel of F , so that the index of F | J ×J coincides with the one of F | J Q ×J Q . (cid:3) The Morse index theorem in the geometrical setup.
As it was ob-served in Section 3.1, the index form associated to a geodesic in a Lorentzianmanifold (
M, g ) can be reduced to the index form of a Morse-Sturm system. Whenwe consider the energy functional defined in the manifold Ω {P , Q} M of H -curvesjoining two given submanifolds P and Q of M , the critical points are geodesics γ : [0 , → M orthogonal to P and Q in the endpoints. Furthermore, the associ-ated index form is defined for V and W in T γ (cid:0) Ω {P , Q} M (cid:1) and it is given by I { γ, P , Q} ( V, W ) = Z (cid:2) g ( V ′ , W ′ ) + g (cid:0) R ( ˙ γ, V ) ˙ γ, W (cid:1)(cid:3) d t − g (cid:0) S P ˙ γ (0) [ V (0)] , W (0) (cid:1) + g (cid:0) S Q ˙ γ (1) [ V (1)] , W (1) (cid:1) , (25)where S P ˙ γ (0) and S Q ˙ γ (1) are the second fundamental forms of P and Q in the directionsof ˙ γ (0) and ˙ γ (1) respectively. We observe that the tangent space to Ω {P , Q} M in γ can be described as T γ (cid:0) Ω {P , Q} M (cid:1) = { V : V is a H -vector field along γ , V (0) ∈ T γ (0) P and V (1) ∈ T γ (1) Q} . Assume that there exists a timelike Jacobi field Y along γ . In order to establishthe Morse index theorem we consider the subspaces H { γ, P , Q}∗ = (cid:8) V ∈ T γ (cid:0) Ω {P , Q} M (cid:1) : g ( V ′ , Y ) − g ( V, Y ′ ) = const. (cid:9) and H { γ, P , Q} = (cid:8) V ∈ T γ (cid:0) Ω {P , Q} M (cid:1) : g ( V ′ , Y ) − g ( V, Y ′ ) = 0 (cid:9) . We observe that we just suppress Q in all the notations when it is a point. Weknow that the index of I { γ, P , Q} given in (25) restricted to H { γ, P , Q}∗ × H { γ, P , Q}∗ and H { γ, P , Q} × H { γ, P , Q} is finite. Moreover the difference between the two restrictionsis 1 or 0, because H { γ, P , Q} is a codimensional-one subspace of H { γ, P , Q}∗ , and wecall this difference ε { γ, P , Q} . If we fix a parallel orthonormal frame along γ , we canget the initial data ( g, R, Y, P, Q, S P , S Q ) as the corresponding coordinate versionof ( g, R ( ˙ γ, · ) ˙ γ, Y , T γ (0) P , T γ (1) Q , S P ˙ γ (0) , S Q ˙ γ (1) ) in such a way that the index (24) isobtained from (25) when considering the coordinates in the parallel orthonormalframe. Obviously, P -focal points of γ are in correspondence with P -focal pointsof the data ( g, R, Y, P, Q, S P , S Q ), so that we can bring the Morse index theorem SEUDO FOCAL POINTS 21 for Morse-Sturm systems to the geometrical setup. We extend all the definitionsrelated to P -Jacobi fields and ( P, Y )-pseudo Jacobi fields of a Morse-Sturm system(see Subsections 3.2 and 3.3 and Definition 3.10) to P -Jacobi fields and ( P , Y )-pseudo Jacobi fields in the obvious way, so as the Definition 4.6 of ( P , Y )-pseudofocal points. Furthermore, we will use J Q , J ∗Q and F to denote the geometricalobjects correponding to J Q , J ∗ Q and F defined in Subsection 4.3, and the samenotation to the geometrical counterpart of J [ t ] and J ∗ [ t ]. In the next results weassume that Y (0) is not orthogonal to P or that Y is singular or admissible. Theorem 4.9.
Assume that J [1] ⊇ T γ (1) Q . Then the Morse index of I { γ, P , Q} given in (25) restricted to H { γ, P , Q} ×H { γ, P , Q} coincides with the sum of the numberof ( P , Y ) -pseudo focal points counted with multiplicity of the geodesic γ and theindex of F| J Q ×J Q .Proof. It follows from Theorems 4.7 and 4.8. (cid:3)
Corollary 4.10.
The number of ( P , Y ) -pseudo focal points counted with multiplic-ity is finite. (cid:3) Corollary 4.11.
The Morse index of I { γ, P} restricted to H { γ, P}∗ × H { γ, P}∗ coin-cides or it is one unit larger than the number of ( P , Y ) -pseudo focal points countedwith multiplicity of the geodesic γ . When there is a singular timelike Jacobi field along the geodesic we can obtaina stronger Riemannian Morse index theorem.
Theorem 4.12.
Assume that the timelike Jacobi field Y is singular, P is orthogonalto Y (0) and J ∗ [1] ⊇ T γ (1) Q . Then the Morse index of I { γ, P , Q} in (25) restricted to H { γ, P , Q}∗ ×H { γ, P , Q}∗ coincides with the sum of the number of P -focal points countedwith multiplicity of the geodesic γ and the index of F| J ∗Q ×J ∗Q .Proof. It is enough to observe that when Y is singular and Y (0) is orthogonal to P , then H { γ, P , Q}∗ = H { γ, P , Q} (this can be shown as in the end of Proposition 4.3)and also J Q = J ∗Q . Moreover, as m ( Y ) = 0, ( P , Y )-pseudo focal points coincidewith P -focal instans, therefore the thesis follows from Theorem 4.9. (cid:3) The last theorem gives the Morse index theorem for timelike geodesics by taking Y = ˙ γ . Furthermore it can be used to compute the Morse index of a horizontalgeodesic in a static spacetime as we will see later. Corollary 4.13.
If the geodesic γ admits a singular timelike Jacobi field Y , thenthere only exists a finite number of conjugate instants along the geodesic. Moreover,if P is orthogonal to Y (0) there only exists a finite number of P -focal instants along γ . (cid:3) Static manifolds.
A Lorentzian manifold is said to be standard static if itcan be expressed as a product M × R endowed with a metric given by g ( x, t )[( ξ, τ ) , ( ξ, τ )] = g ( x )[ ξ, ξ ] − β ( x ) τ (26)where ( x, t ) ∈ M × R , ( ξ, τ ) ∈ T x M × R , g is a Riemannian metric in M and β a C ∞ positive function in M . Standard static spacetimes are always stationarywith Killing field given by Y = (0 ,
1) and we can prove a Morse index theorem forevery horizontal geodesic (in the following horizontal will mean orthogonal to thefibers of the natural projection π : M × R → M ). Proposition 4.14.
Let ( M × R , g ) be a standard static spacetime with g as in (26) , γ a horizontal geodesic in M × R , P a horizontal submanifold through γ (0) and orthogonal to ˙ γ (0) and I { γ, P} its related Morse index form. Then the index of I { γ, P} restricted to H { γ, P}∗ × H { γ, P}∗ coincides with the number of P -focal points of γ counted with multiplicity. Moreover, this index coincides with the index of I { γ, P} in the Riemannian manifold M .Proof. A horizontal geodesic [0 , ∋ s → ( x ( s ) , t ) ∈ M × R is always containedin the totally geodesic hypersurface M × { t } , so that by Remark 3.4 we concludethat Y = (0 ,
1) is singular. By applying Theorem 4.12, the first part of the thesisfollows. For the last part, it is enough to observe that
P ⊂ M × { t } and P -Jacobifields coincide with the P -Jacobi fields of γ in M . In order to see this, we observethat g ( J ′ (0) , Y (0)) − g ( J (0) , Y ′ (0)) = 0 and as J (0) is tangent to P and Y ′ (0)linearly dependent with Y (0) we deduce that J ′ (0) is tangent to M × { t } . As J (0) and J ′ (0) are tangent to M × { t } and this submanifold is totally geodesic,it can be deduced that J is also a P -Jacobi field in M ∼ = M × { t } . (cid:3) Evolution of the index functions and the distribution of focaland pseudo focal points
Let us finally discuss the question of distribution of focal points along a geodesicin a stationary Lorentzian manifold. Let us consider the geometrical setup intro-duced in Subsection 4.4; we will consider the only interesting case of a spacelike geodesic γ . Let us use the following notations: for all t ∈ ]0 , µ ( t ) the index of the bilinear form: I t ( V, W ) = Z t g ( V ′ , W ′ ) + g (cid:0) R ( ˙ γ, V ) ˙ γ, W (cid:1) d s − g (cid:0) S P ˙ γ (0) [ V (0)] , W (0) (cid:1) on the space H t of H -vector fields V along γ | [0 ,t ] satisfying V (0) ∈ T γ (0) P , V ( t ) = 0and g ( V ′ , Y ) − g ( V, Y ′ ) = C V a. e. on [0 , t ]. In the following we will asume that Y (0) is not orthogonal to P or that Y is singular or admissible. By µ ( t ) we willdenote the index of I t on the one-codimensional subspace H t of H t consisting ofvector fields V for which the constant C V vanishes. The functions µ and µ giveus information on the distribution of focal and pseudo focal instants. It is worthrecalling that a P -focal instant t ∈ ]0 ,
1] along γ is said to be nondegenerate if therestriction of the metric g to the space: J [ t ] = (cid:8) J ′ ( t ) : J is a P -Jacobi field and J ( t ) = 0 (cid:9) is nondegenerate. Nondegenerate P -focal instants are isolated in the set of all P -focal instants. Using the theory developed in this paper and some results in therecent literature, we can now summarize a few facts about the distribution of focaland pseudo focal instants along a geodesic.(a) 0 ≤ µ ( t ) − µ ( t ) ≤ t ∈ ]0 , H t ⊂ H t has codimension 1.(b) The function µ is nondecreasing in ]0 , t < t one has an injection H t ֒ → H t given by extension to 0 on [ t , t ],and that an I t -negative subspace of H t is mapped by such injection ontoan I t -negative subspace of H t . SEUDO FOCAL POINTS 23 (c) We cannot establish whether the function µ is nondecreasing. Note thatthere is no natural injection H t ֒ → H t that preserves (or decreases thevalues of) the index form, as in (b). Extension of vector fields to 0 is notallowed here, because of the constant g ( V ′ , Y ) − g ( V, Y ′ ) is not zero.(d) µ ( t ) is equal to the P -Maslov index of γ | [0 ,t ] , as proved in [10].(e) An instant t ∈ ]0 ,
1[ is a jump instant for the function µ if and onlyif it is a ( P , Y )-pseudo focal instant. This follows from the Morse indextheorem (Theorem 4.9); the value of the jump is precisely the multiplicitymul ( t ) of t as a pseudo focal instant. In particular, µ is constant onevery interval that does not contain pseudo focal instants.(f) If an instant t ∈ ]0 ,
1[ is a jump instant for µ , then t is a focal instantalong γ . This follows from the main result of [10], since the P -Maslov of γ has jumps only at the focal instants. Thus, µ is constant on every intervalthat does not contain focal instants. The contribution to the index function µ given by a nondegenerate P -focal instant is given by the signature of therestriction of g to the space J [ t ]. It is not known whether in the stationarycase such contribution may be null or negative.(g) The set of focal instants is closed, and contained in ] ε,
1] for some ε > µ . Letus call effective those focal instants that do determine a jump of the function µ . The jump of the function µ at an effective focal instant t ∈ ]0 ,
1[ is inabsolute value less than or equal to the multiplicity mul( t ) of t as a focalinstant. Note that the first effective focal instant t ∈ ]0 ,
1] must give apositive contribution to µ , because µ ≥ t ) >
1, then mul ( t ) >
0, i.e., a focal point of multiplicity largerthan 1 is pseudo focal. More precisely, mul ( t ) ≥ mul( t ) −
1. This followsfrom the fact that the space of P -Jacobi fields J along γ vanishing at 0 andsatisfying C J = g ( J ′ , Y ) − g ( J, Y ′ ) = 0 form a subspace of codimension 1 ofthe space of all P -Jacobi fields along γ vanishing at t . This implies thatfocal points with multiplicity larger than one do not accumulate.(j) If mul ( t ) >
1, then mul( t ) >
0, i.e., pseudo focal instants with mul-tiplicity larger than 1 are focal. More precisely, mul( t ) ≥ mul ( t ) − ( t ) = k > J , . . . , J k be a basis of ( P , Y )-pseudo Jacobi fields satisfying J s ( t ) = 0, with J ′′ s − R [ J s ] = λ s m ( Y ) forall s = 1 , . . . , k . Assume λ k = 0 (if all the λ s vanish, then all the J s are P -Jacobi fields vanishing at t , hence mul( t ) ≥ k ). Then, the vector fields W s = λ k J s − λ s J k , s = 1 , . . . , k −
1, are linearly independent P -Jacobi fieldsvanishing at t , thus mul( t ) ≥ k − t < t < t , t ]. Namely, if there were no effective focalinstant in [ t , t ], then lim t → t +2 µ ( t ) ≥ t → t − µ ( t ) ≥ t → t − µ ( t ) =1+lim inf t → t +2 µ ( t ), which gives a contradiction with the inequality µ ( t ) ≤ µ ( t ).(l) It is not clear whether effective focal instants are isolated. However, if t < t ≤ consecutive focal instants, i.e., there is no focal instantin ] t , t [, and if they give a positive contribution to µ , then there exists one pseudo focal instant in the interval [ t , t ]. Namely, if there were nopseudo focal instant in [ t , t ], then lim inf t → t +2 µ ( t ) ≥ t → t − µ ( t ) ≥ t → t − µ ( t ) = 2 + lim t → t +2 µ ( t ), which gives a contradiction with the inequality µ ( t ) ≤ µ ( t ) + 1.(m) If t ∈ ]0 ,
1] is the first pseudo focal point, then there exists an effective focalinstant t ∈ ]0 , t ] that gives a positive contribution to µ . For, otherwiseit would be lim t → t +0 µ ( t ) = mul ( t ) > t → t +0 µ ( t ), which contradicts theinequality µ ( t ) ≤ µ ( t ). Appendix A. Continuity and weak continuity of families of subspaces
We will discuss here a simple result showing that the abstract Morse indextheorem discussed in this paper and a similar result by Uhlenbeck (see [21, Theorem1.11]) are in fact independent. Recall that in [21, Theorem 1.11] it is considered anincreasing family of closed subspaces of a Hilbert space, satisfying the assumptionbelow.Let H be a Hilbert space and let ( H s ) s ∈ [ a,b ] be a family of closed subspaces of H ; denote by P s : H → H the orthogonal projection onto H s . We say that ( H s ) is continuous if the map s P s ∈ L ( H ) is continuous in the operator norm topology.A weaker notion of continuity can be introduced by considering the strong operatortopology (SOT) of L ( H ). Recall that this topology is the locally convex topologydefined by the family of semi-norms T
7→ k
T ξ k , where ξ ∈ H ; in other words, anet T α converges to T in the SOT if T α ξ → T ξ for all ξ ∈ H . We say that ( H s ) is weakly continuous if s P s is SOT-continuous. Lemma A.1.
Assume that the family ( H s ) is non decreasing, i.e., H s ⊂ H t when s ≤ t . Then: (1) ( H s ) is continuous if and only if it is constant. (2) ( H s ) is weakly continuous if and only if the following holds: [ s
Consider now the family ( K s ) of closed subspaces of H given by K s = H ⊥ s ; this isa non increasing family of subspaces, and the second equality in (27) is equivalent to S s>t K s = K t . By a totally analogous argument, the family of orthogonalprojections Q s = 1 − P s onto K s is SOT right- continuous; thus, P s is also rightcontinuous.Conversely, assume that P s is SOT continuous at t . Then, for all ξ ∈ H t ,lim s → t − P s ξ = P t ξ = ξ . Set ξ s = P s ξ ∈ H s , so that lim s → t − ξ s = ξ , hence the first equalityin (27) holds. By duality, the SOT continuity of the projections Q s = 1 − P s impliesthat also the second equality in (27) holds. (cid:3) References [1] A. Abbondandolo, P. Majer,
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Departamento de Geometr´ıa y Topolog´ıa.Facultad de Ciencias, Universidad de Granada.Campus Fuentenueva s/n, 18071 Granada, Spain
E-mail address : [email protected] Dipartimento di Matematica,Politecnico di Bari,Via Orabona 4, 70125, Bari, Italy
E-mail address : [email protected] Departamento de Matem´atica,Universidade de S˜ao Paulo,Rua do Mat˜ao 1010, S˜ao Paulo, Brasil
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