GEODESICS AND JACOBI FIELDS OF PSEUDO-FINSLER MANIFOLDS

arXiv:1401.8149v2 [math.DG] 31 Mar 2014

GEODESICS AND JACOBI FIELDS OF PSEUDO-FINSLER MANIFOLDS MIGUEL ANGEL JAVALOYES AND BRUNO LEARTH SOARES Abstract. In this paper, we obtain the first and the second variation of the energy functional of a pseudo-Finsler metric using the family of affine connections associated to the Chern connection. This allows us to accomplish the computations with the free-coordinate methods of Modern Differential Geometry. We also introduce the index form using the formula for the second variation and give some properties of Jacobi fields. Finally we prove that lightlike geodesics and its focal points are preserved up to reparametrization by conformal changes.

1. Introduction Geodesics and Jacobi fields are probably the most important geometrical elements associated to a Finsler metric. Even though they can be defined without using any connection, choosing a connection associated to the Finsler metric can make it easier to get some properties of them. In particular, the main goal of this paper is to use the Chern connection to deduce some of these properties under the approach developed by H.-H. Matthias in [18, Definition 2.5], where the Chern connection is interpreted as a family of affine connections, namely, for every vector field V in an open subset Ω ⊂ M , non-zero everywhere, we get an affine connection ∇V . This affine connection is torsion-free and almost g-compatible, meaning that the derivative of the fundamental tensor is an expression in terms of the Cartan tensor (see subsection 2.3). Both properties allow one to make the computation of the first and second variation of the energy functional with a coordinate-free method. In this process, we will also use the further developments given in [13], where a satisfactory relation between the curvature of the affine connection and the Chern curvature is obtained. As one of our main goals is to promote the study of Finsler geometry between researchers of Riemannian background, we have made an special effort to make the paper selfcontained with an unusual amount of details and in some cases repeating computations already known in literature, with the purpose of providing in some cases proofs in a coordinate-free way or establishing the results in the very general setting of pseudo-Finsler metrics, apparently, the most general case where the Chern connection can be defined. In particular the square of a Finsler metric is a pseudo-Finsler metric and the notions of indefinite Finsler metrics [4, 5] and Finsler spacetimes [22] can fit into this definition. This work was partially supported by MINECO (Ministerio de Econom´ıa y Competitividad) and FEDER (Fondo Europeo de Desarrollo Regional) project MTM2012-34037, and Regional J. Andaluc´ıa Grant P09-FQM-4496. 2000 Mathematics Subject Classification: Primary 53C22, 53C50, 53C60, 58B20 Key words: Finsler, geodesics, Chern connection, Jacobi fields. 1

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M. A. JAVALOYES AND B. L. SOARES

The main goal of this paper is to provide a computation of the first and the second variation of the energy functional of a pseudo-Finsler metric (see Propositions 3.1 and 3.2). As a first step, we prove that geodesics are the critical points of the energy functional when we consider curves with fixed endpoints or more generally with endpoints in two submanifolds P and Q (see Corollary 3.7). Moreover, the second variation allows us to define the index form and the Jacobi fields (see subsection 3.7), and with our approach to Chern connection we can deduce straightforward some basic properties of Jacobi fields (see subsection 3.4) and to characterize the kernel of the index form as the (P, Q)-Jacobi fields along γ (see Proposition 3.11). Finally we use the characterization of geodesics as critical points of the energy functional and (P, Q)-Jacobi fields as the kernel of the index form to prove that lightlike geodesics and its P -focal points of a pseudo-Finsler metric are preserved by conformal transformations (see Proposition 4.4 and Theorem 4.7) generalizing a classical semi-Riemannian result (see [19, Section 2.6]). The paper is structured as follows. Section 2 contains some basic results about pseudo-Finsler metrics including some properties of its fundamental tensor, the Cartan tensor and the Chern connection. We also introduce several basic notions: parallelism of a vector field along a curve, geodesics, namely, curves having parallel tangent vector field, and the exponential map associated to the geodesics. In the last subsection we recall some properties of the Chern curvature obtained in [13]. In Section 3 we compute the first and the second variation of the energy functional (see Propositions 3.1 and 3.2) and then we get the index form when the boundary conditions are given by two submanifolds. As a previous step, in subsection 3.1 we introduce some definitions and properties of submanifolds of pseudoFinsler manifolds. Then in subsection 3.4, we give some properties of Jacobi fields. Finally, in Section 4 we study the effects of a conformal transformation in lightlike geodesics. We first obtain the first and second variation of the energy functional of the metric λL in terms of the Chern connection of the metric L (see Proposition 4.1 and 4.5). Then we prove that lightlike geodesics and its P -focal points are preserved by conformal changes (see Proposition 4.4 and Theorem 4.7). 2. Pseudo-Finsler metrics and its Chern connection 2.1. Preliminaries in pseudo-Finsler metrics. Let us introduce the most general notion of Finsler metric that admits a Chern connection. Let M be a manifold and denote by π : T M → M the natural projection of T M into M . Let A ⊂ T M \0 be a conic open subset of T M , namely, for every v ∈ A and λ > 0, λv ∈ A and such that π(A) = M . We say that a smooth function L : A → R is a (conic, two homogeneous) pseudo-Finsler metric if (i) L is positive homogeneous of degree 2, that is, L(λv) = λ2 L(v) for every v ∈ A and λ > 0, (ii) for every v ∈ A, the fundamental tensor of L defined as 1 ∂2 L(v + tu + sw)|t=s=0 , 2 ∂t∂s for any u, w ∈ Tπ(v) M , is nondegenerate. gv (u, w) :=

It follows straightforward from definition that the fundamental tensor is bilinear and symmetric.

GEODESICS AND JACOBI FIELDS OF PSEUDO-FINSLER MANIFOLDS

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Remark 2.1. In the following we will omit conic and two-homogeneous whenever there is no misunderstanding. In [15] the same name of pseudo-Finsler metrics is used for a concept somewhat different. In the cited reference, pseudo-Finsler metrics are not allowed to be non-positive away from the zero section and they are positive homogeneous of degree one. Moreover, its fundamental tensor is not necessarily nondegenerate. Nevertheless, if F : A1 ⊂ T M → [0, +∞) is a conic pseudo-Finsler metric on M as in [15] and A˜1 = {v ∈ A1 \ 0 : the fundamental tensor gv of F is nondegenerate}, then L = F 2 |A˜1 : A˜1 → (0, +∞) fits in our definition of pseudo-Finsler metric. Moreover, if L : A2 ⊂ T M \ 0 → R is a pseudo-Finsler metric on M as defined p ˜ above and A2 = {v ∈ A2 : L(v) 6= 0}, then F = |L| : A˜2 → (0, +∞) is a conic ˜2 A

pseudo-Finsler metric on M as in [15] (extending F continuously to the zero section if necessary). Finally, observe that the definition of pseudo-Finsler metrics of [15] is particularly convenient when one is interested in studying distance properties, whereas the definition above is the most general context where Chern connection can be defined. Remark 2.2. We have assumed that the conic subset A does not intersect the zero section. As A is required to be open, the only case in that this assumption can be a limitation is when A = T M \ 0. But in such a case it is easy to see that L can be extended continuously to the zero section and the definition coincides with the classical definition of Finsler metrics. Let us discuss several particular cases of pseudo-Finsler metrics: (i) if A = T M \ 0 and the fundamental tensor is√ positive definite, then L is positive away from the zero section and F = L is what traditionally has been called a Finsler metric, (ii) if A ( √T M \0, but the fundamental tensor is positive, then the positive square F = L is called a conic Finsler metric in [15], (iii) if the fundamental tensor has index one, then L is called a Lorentzian Finsler metric (see [11, 16]). This is also the case of Finsler spacetimes where some authors ask L to be defined in the whole T M [4, 5, 22]. Remark 2.3. Even if sometimes the domain of definition can change (see Remark 2.1), from now on with abuse of notation we will omit the subset A when fixing a (conic) pseudo-Finsler manifold (M, L), assuming that L is defined in A. Let us enumerate some other properties that follow from positive homogeneity. Proposition 2.4. Given a pseudo-Finsler metric L and v ∈ A, the fundamental tensor gv is positive homogeneous of degree 0, that is, gλv = gv for λ > 0. Moreover ∂ gv (v, v) = L(v) and gv (v, w) = 21 ∂z L (v + zw) |z=0 . Proof. Let us begin by showing the positive homogeneity of degree zero: λ2 ∂ 2 1 1 1 ∂2 L(λv + tu + sw)|t=s=0 = L(v + t u + s w)|t=s=0 2 ∂t∂s 2 ∂t∂s λ λ 1 1 = λ2 gv ( u, w) = gv (u, w) λ λ

gλv (u, w) =

4


for every u, w ∈ Tπ(v) M . Moreover, gv (v, v) =

1 ∂2 1 ∂2 L(v + sv + tv)|t=s=0 = (1 + s + t)2 |t=s=0 L(v) = L(v) 2 ∂s∂t 2 ∂s∂t

and 1 ∂2 L(v + sv + tw)|t=s=0 2 ∂t∂s t 1 ∂2 2 (1 + s) L v + w = 2 ∂t∂s 1+s t=s=0 1 ∂ t ∂ = w (1 + s)2 L v + 2 ∂s ∂t 1+s t=s=0 1 ∂ ∂ 1 ∂ = (1 + s) L (v + zw) L (v + zw) |z=0 , = 2 ∂s ∂z 2 ∂z z=s=0

gv (v, w) =

for any w ∈ Tπ(v) M , where z = t/(1 + s) and ∂z/∂s = −z/(1 + s).

2.2. Cartan tensor. In Finsler geometry, unlike the Riemannian setting, we need to consider the third derivatives of the metric in order to define a connection. This information is contained in the Cartan tensor, which is defined as the trilinear form ! 3 X ∂3 1 , (1) si wi L v+ Cv (w1 , w2 , w3 ) = 4 ∂s3 ∂s2 ∂s1 i=1

s1 =s2 =s3 =0

for v ∈ A and w1 , w2 , w3 ∈ Tπ(v) M . Observe that Cv is symmetric, that is, its value does not depend on the order of w1 , w2 and w3 .

Remark 2.5. Let πA : A → M be the restriction to A of the natural projection ∗ π : T M → M . Now let πA (T ∗ M ) be the fiber bundle over A induced by the natural projection of the cotangent bundle π ∗ : T ∗ M → M through πA . Observe ∗ that the fundamental tensor is a symmetric section of the fiber bundle πA (T ∗ M ) ⊗ ∗ ∗ πA (T M ). Moreover, the Cartan tensor is a symmetric section of the fiber bundle ∗ ∗ ∗ πA (T ∗ M ) ⊗ πA (T ∗ M ) ⊗ πA (T ∗ M ). Furthermore, the Cartan tensor can be obtained from the fundamental tensor as 1 ∂ Cv (w1 , w2 , w3 ) = gv+zw1 (w2 , w3 ) . 2 ∂z z=0

∗ ∗ If g is an arbitrary symmetric section of πA (T ∗ M ) ⊗ πA (T ∗ M ) that is positive homogeneous of degree zero (gv = gλv for λ > 0) we define its Cartan tensor as above.

Proposition 2.6. The Cartan tensor is homogeneous of degree −1, that is, Cλv = 1 λ Cv for any v ∈ A and λ > 0. Moreover, Cv (v, w1 , w2 ) = 0 for every w1 , w2 ∈ Tπ(v) M . Proof. Recall that gv is homogeneous of degree zero in v (see Proposition 2.4). Then, for the homogeneity of Cv : 1 ∂ 1 ∂ gλv+zw1 (w2 , w3 ) gv+z wλ1 (w2 , w3 ) = Cλv (w1 , w2 , w3 ) = 2 ∂z 2 ∂z z=0 z=0 1 = Cv (w1 /λ, w2 , w3 ) = Cv (w1 , w2 , w3 ), λ


for any w1 , w2 , w3 ∈ Tπ(v) M . For the second property, ∂ ∂ Cv (v, w1 , w2 ) = g(1+z)v (w1 , w2 ) = gv (w1 , w2 ) = 0. ∂z ∂z z=0 z=0

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∗ Proposition 2.7. An arbitrary (non-degenerate) symmetric section g of πA (T ∗ M )⊗ ∗ πA (T ∗ M ) that is positive homogeneous of degree zero comes from a pseudo-Finsler metric if and only if its Cartan tensor is symmetric.

Proof. One implication is trivial. For the other one, assume that g is a symmetric ∗ ∗ section of πA (T ∗ M ) ⊗ πA (T ∗ M ) that is positive homogeneous of degree zero and such that its associated Cartan tensor is symmetric. Define L(v) = gv (v, v) for any v ∈ A. Observe that L : A → R is positive homogeneous of degree two and smooth. Then, we have to prove that 1 ∂ 2 L(v + tu + sw) gv (u, w) = 2 ∂t∂s t=s=0

for every v ∈ A and u, w ∈ Tπ(v) M . We have that ∂ ∂ L(v + tu + sw) = gv+tu+sw (v + tu + sw, v + tu + sw) ∂s ∂s s=0 s=0 ∂ = gv+tu+sw (v + tu, v + tu) ∂s s=0 ∂ +2 (sgv+tu+sw (v + tu, w)) ∂s s=0 ∂ 2 (s gv+tu+sw (w, w)) + ∂s s=0 = 2Cv+tu (w, v + tu, v + tu) + 2gv+tu (v + tu, w) = 2gv+tu (v + tu, w),

where in the last equation we have used the symmetry of Cv+tu and Proposition 2.6. Now using the last equation we get 1 ∂ 2 L(v + tu + sw) ∂ gv+tu (v + tu, w) = 2 ∂t∂s ∂t t=0 t=s=0 ∂ ∂ gv+tu (v, w) (tgv+tu (u, w)) + = ∂t ∂t t=0 t=0 = 2Cv (u, v, w) + gv (u, w) = gv (u, w), using again the symmetry of Cv and Proposition 2.6, which concludes.

Remark 2.8. The result in Proposition 2.7 is well-known. It appears for example ∗ ∗ in [2, Theorem 3.4.2.1]. An arbitrary symmetric section g of πA (T ∗ M ) ⊗ πA (T ∗ M ) that is positive homogeneous of degree zero is usually known as a generalized metric and it was introduced by A. Moor in 1956 [20] and popularized by R. Miron’s school (Iasi, Romania) and J. Kern [17]. Unfortunately, the Chern connection is not welldefined for generalized metrics unless they come from a pseudo-Finsler metric. This is because the following remark is essential to prove existence of a connection which is torsion-free and almost metric compatible (see for example [13, Proposition 2.3]).

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Remark 2.9. In the case of a pseudo-Finsler manifold (M, L), the Cartan tensor is symmetric. This together with Proposition 2.6 means that Cv (v, w1 , w2 ) = Cv (w1 , v, w2 ) = Cv (w1 , w2 , v) = 0 for any v ∈ A and w1 , w2 ∈ Tπ(v) M . 2.3. Chern connection and covariant derivative. Assume that (M, L) is a pseudo-Finsler manifold with domain A ⊂ T M and denote by X(Ω) the space of smooth vector fields on an open subset Ω ⊂ M . We say that V ∈ X(Ω) is Ladmissible if V (p) ∈ A for every p ∈ Ω. Then associated to any L-admissible V ∈ X(Ω) we can define an affine connection ∇V determined by the following properties (i) ∇V is torsion-free: ∇VX Y − ∇VY X = [X, Y ], (ii) ∇V is almost g-compatible: X(gV (Y, Z)) = gV (∇VX Y, Z) + gV (Y, ∇VX Z) + 2CV (∇VX V, Y, Z)

for every X, Y, Z ∈ X(Ω). We will say that ∇V is the Chern connection of (M, L) associated to the L-admissible vector field V in Ω ⊂ M . It is easy to see that this connection is positive homogeneous of degree zero in V , in the sense that ∇λV = ∇V . Assume that dim M = n and fix a coordinate system ϕ : Ω → ϕ(Ω) ⊂ Rn , ∂ ∂ with ϕ(p) = (x1 (p), x2 (p), . . . , xn (p)) for every p ∈ Ω. If we denote by ∂x 1 , . . . , ∂xn the partial vectors associated to the system, then the formal Christoffel symbols of ϕ associated to V , Γkij (V ), are determined by the system X n ∂ ∂ V = Γkij (V ) k , ∇ ∂ j i ∂x ∂x ∂x k=1

for i, j = 1, . . . , n. Given a smooth curve γ : [a, b] → M , let us denote by γ ∗ (T M ) the vector bundle over [a, b] induced by the map π : T M → M through γ, and X(γ) the space of smooth sections of this vector bundle. We will say that W ∈ X(γ) is L-admissible if W (t) ∈ A for every t ∈ [a, b]. Then we can define a covariant derivative along γ on M for every L-admissible W ∈ X(γ). Namely, if X ∈ X(γ), then n n X X dX i ∂ ∂ DγW X = + X i (t)γ˙ j (t)Γkij (W (t)) k , (2) i dt ∂x ∂x i=1 i,j,k=1

1

n

1

n

where (X , . . . , X ) and (γ˙ , . . . , γ˙ ) are respectively the coordinates of X and γ˙ in ϕ (see [13, Proposition 2.6]). Moreover, this covariant derivative is almost g-compatible, meaning that if X, Y ∈ X(γ), then d gW (X, Y ) = gW (DγW X, Y ) + gW (X, DγW Y ) + 2CW (DγW W, X, Y ). dt

(3)

2.4. Parallel vector fields. Once we have defined the covariant derivative, we can introduce the concept of parallelism along a curve. In principle, we can choose any vector field along the curve as a reference vector to compute the covariant derivative, but there is one distinguished reference vector, which is the velocitiy of the curve. Nevertheless, this is not always possible, since the velocity of the curve could not belong to A. We will introduce a special class of curves. We will say that a curve γ : [a, b] → M is piecewise smooth if there exist a partition a = t0 < t1 . . . < tn−1 < tn = b such that γ is continous on [a, b] and smooth in [ti , ti+1 ] for i = 0, . . . , n − 1. The instants ti , i = 1, . . . , n − 1 will be called the break points of γ.


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Definition 2.10. Let (M, L) be a pseudo-Finsler manifold. We say that a piecewise smooth curve γ : [a, b] ⊂ R is L-admissible if γ(t) ˙ ∈ A for any t ∈ [a, b] (in the break points, both velocities must belong to A). Definition 2.11. Let (M, L) be a pseudo-Finsler manifold and X a vector field along a smooth L-admissible curve γ : [a, b] → M . We say that X is parallel if Dγγ˙ X = 0. Remark 2.12. In the following we will assume that the image of γ : [a, b] → M is contained in a system of coordinates. This is not restrictive since we can find a partition a = t0 < t1 < . . . < tn = b such that γ([ti , ti+1 ]) is contained in the domain of a system of coordinates. Then we can apply the parallel transport to every segment to get the final result. Proposition 2.13. Let (M, L) be a pseudo-Finsler manifold and γ : [a, b] → M an L-admissible smooth curve. Then for every w ∈ Aγ(a) = Tγ(a) M ∩ A there is a ˙ ˙ unique parallel vector field X along γ such that X(a) = w. P ∂ is γ-parallel if and Proof. In a system of coordinates, X(t) = ni=1 X i (t) ∂x i γ(t) only if n X dX i =− γ˙ j X k Γijk (γ) ˙ dt j,k=1

for i = 1, . . . , n, and by the results of existence and uniqueness of ordinary differential equations, there exists a unique X satisfying last equation such that X(a) = w (recall Remark 2.12).

2.5. Geodesics and the exponential map. We can now introduce the notion of geodesic. Definition 2.14. We say that a smooth curve γ : [a, b] → M is a geodesic of the pseudo-Finsler manifold (M, L) if γ˙ is parallel along γ. Proposition 2.15. Let (M, L) be a pseudo-Finsler manifold. For every v ∈ A there exists a unique geodesic γv : [0, b) → M such that γ˙ v (0) = v, b ∈ (0, +∞]. Proof. Let x1 , . . . , xn be a coordinate system in Ω ⊂ M , such that π(v) ∈ Ω. Then γ is a geodesic if and only if n X dγ˙ i =− γ˙ j γ˙ k Γijk (γ) ˙ (4) dt j,k=1

for i = 1, . . . , n. By existence and uniqueness of solutions of ordinary differential equations, there exists a unique geodesic γv with the initial conditions γv (0) = π(v) and γ˙ v (0) = v. Remark 2.16. Observe that if γ : [a, b] → M is a geodesic of a pseudo-Finsler manifold (M, L), then L(γ) ˙ is constant along γ, since applying that L(γ) ˙ = gγ˙ (γ, ˙ γ), ˙ d L(γ) ˙ = the covariant derivative is almost g-compatible and Remark 2.9, we get ds 2gγ˙ (Dγγ˙ γ, ˙ γ) ˙ = 0. In particular, we will say that a geodesic is lightlike when L(γ) ˙ = 0. Definition 2.17. Given p ∈ M , let us define the subset U ⊂ Ap = A ∩ Tp M as the vectors v ∈ Ap such that if γv : [0, b) → M is the unique geodesic such that

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γv (0) = v, and the interval [0, b) is its maximal interval of definition to the right, then b > 1. Then the exponential map expp : U ⊂ Ap → M is defined as expp (v) = γv (1) for every v ∈ U . Proposition 2.18. The domain of expp is an open subset of Ap and expp is smooth in Ap . Moreover, when Ap = Tp M \ 0, the exponential is defined in an open subset of 0p , putting expp (0p ) = p. Proof. It follows again from (4) and the smooth dependence of the solutions of ordinary differential equations with respect to parameters. Up to the last statement, observe that for every p ∈ M and i = 1, . . . , n, the functions Tp M \ 0 ∋ v → v j v k Γijk (v) ∈ R are positive homogeneous of degree 2, and then they can be extended as C 1 functions to zero, since a homogeneous function of positive degree can be extended continuously to zero as zero and the differential of a homogeneous function of degree 2 is a homogeneous function of degree 1 (see Proposition 2.4). In fact they will be C 1 in T Ω. 2.6. Jacobi operator and flag curvature. If we fix an L-admissible vector field V in Ω ⊂ M , the Chern connection ∇V is an affine conection and we can define a curvature tensor given by RV (X, Y )Z = ∇VX ∇VY Z − ∇VY ∇VX Z − ∇V[X,Y ] Z for every X, Y, Z ∈ X(Ω). We can also compute the ∇V -covariant derivative of the Cartan tensor CV , obtaining a (0, 4) tensor defined by ∇VX CV (Y, Z, W ) = X(CV (Y, Z, W )) − CV (∇VX Y, Z, W )

− CV (Y, ∇VX Z, W ) − CV (Y, W, ∇VX W ),

for every X, Y, Z, W ∈ X(Ω). It is straightforward to check that ∇VX CV is trilinear, symmetric and ∇VX CV (V, Z, W ) = −CV (∇VX V, Z, W ).

(5)

Moreover, the curvature tensor has the following symmetries: (i) RV (X, Y ) = −RV (Y, X), (ii) gV (RV (X, Y )Z, W ) + gV (RV (X, Y )W, Z) = 2B V (X, Y, Z, W ), where B V (X, Y, Z, W ) = ∇VY CV (∇VX V, Z, W ) − ∇VX CV (∇VY V, Z, W )

+ CV (RV (Y, X)V, Z, W ),

(iii) RV (X, Y )Z + RV (Y, Z)X + RV (Z, X)Y = 0, and gV (RV (X, Y )Z, W ) − gV (RV (Z, W )X, Y ) =

B V (Z, Y, X, W ) + B V (X, Z, Y, W ) + B V (W, X, Z, Y ) + B V (Y, W, Z, X) + B V (W, Z, X, Y ) + B V (X, Y, Z, W ) (6)


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for every X, Y, Z, W ∈ X(Ω) (see [13, Proposition 3.1]). We can also define that the Jacobi operator Rγ along an L-admissible curve γ : [a, b] ⊂ R → M . Recall that the curve γ is L-admissible if γ˙ belongs to A. Then if u, w are vectors in Tγ(t) M , V Rγ (γ(t), ˙ u)w := Rγ(t) (V, U )W,

(7)

where V is an L-admissible extension of γ˙ along γ : (t − ε, t + ε) → M for ε > 0 small enough in a neighborhood of γ(t), and U, W are extensions of u, w in such a neighborhood. Obseve that the Jacobi operator depends only on the curve γ. Indeed, it depends only on γ˙ and Dγγ˙ γ˙ evaluated in t ∈ [a, b] (see [13, Theorem 3.4]). In particular, when γ is a geodesic, the Jacobi operator in (7) coincides with the one in [3, Subsection 7.2] and the flag curvature can be computed as Kv (u) =

gv (Rγv (v, u)u, v) , L(v)gv (u, u) − gv (v, u)2

see [13, Corollary 3.5]. 3. Variation of the energy Let us define the energy functional associated to a pseudo-Finsler manifold (M, L) for any L-admissible piecewise smooth curve γ : [a, b] ⊂ R → M as Z 1 b L(γ)ds. ˙ (8) E(γ) = 2 a We will see that the geodesics of (M, L) are the critical points of the energy functional. With this goal, we will compute the variations of E. Let γ : [a, b] → M be an L-admissible piecewise smooth curve and consider a piecewise smooth variation Λ : [a, b] × (−ε, ε) → M , with breaks t0 = a < t1 < t2 < . . . < th < th+1 = b, namely, Λ is continuous in its domain and smooth in [ti , ti+1 ] × (−ε, ε) for any i = 0, . . . , h. We will denote by γs0 : [a, b] → M the curve defined as γs0 (t) = Λ(t, s0 ) for every t ∈ [a, b] and by βt0 : (−ε, ε) → M the curve defined as βt0 (s) = Λ(t0 , s) for every s ∈ (−ε, ε). Moreover, we will use the notation Λt (t, s) = γ˙ s (t) and Λs (t, s) = β˙ t (s) and we will denote by Λ∗ (T M ) the vector bundle over [a, b] × (−ε, ε) induced by π : T M → M through Λ. Then the space of smooth sections of Λ∗ (T M ) will be denoted as X(Λ). Observe that a vector field V ∈ X(Λ) induces vector fields in X(γs0 ) and X(βt0 ) for every s0 ∈ (−ε, ε) and t0 ∈ [a, b]. We will say that V is L-admissible if V (t, s) ∈ A for every (t, s) ∈ [a, b] × (−ε, ε). When Λ lies in the domain of a coordinate system x1 , . . . , xn , we will denote Λi = xi ◦ Λ. Observe that when we have a variation of curves (or more generally a two parameters map), as the Chern connection is free of torsion, we have the following property: DγVs β˙ t = DβVt γ˙ s ,

(9)

(see also [13, Proposition 3.2]). We say that the variation Λ is L-admissible if γs is L-admissible for every s ∈ (−ε, ε). Moreover, we will denote by W the variational vector field of Λ along γ, namely, W (t) = Λs (t, 0) for any t ∈ [a, b] and γ(t ˙ + i ) and − γ(t ˙ i ) the right and left derivatives of γ in the break ti , i = 1, . . . , h. Finally let us recall the definition of the Legendre transformation of the pseudo-Finsler metric, namely, the map LL : A → T M ∗ , where LL (v) is defined as the one-form given by LL (v)(w) = gv (v, w) for every w ∈ Tπ(v) M and g is the fundamental tensor of L.

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Proposition 3.1. Let Λ be an L-admissible piecewise smooth variation of γ. With the above notation Z b d b ′ E(γs ) |s=0 = − gγ˙ (W, Dγγ˙ γ)dt ˙ + [gγ˙ (W, γ)] ˙ a E (0) = ds a +

h X i=1

LL (γ(t ˙ + ˙ − i ))(W (ti )) − LL (γ(t i ))(W (ti )) . (10)

Proof. As the variation is piecewise smooth, we get Z 1 b d d E(γs ) = gγ˙ (γ˙ s , γ˙ s )dt ds 2 a ds s Z b gγ˙ s (Dβγ˙ st γ˙ s , γ˙ s ) + Cγ˙ s (Dβγ˙ st γ˙ s , γ˙ s , γ˙ s ) dt = a

=

Z

a

b

gγ˙ s (Dγγ˙ ss β˙ t , γ˙ s )dt,

(11)

where we have used first that the Chern connection is almost g-compatible and then Remark 2.9 and (9). Moreover, applying again that the Chern connection is almost g-compatible, we get gγ˙ (Dγγ˙ W, γ) ˙ =

d gγ˙ (W, γ) ˙ − gγ˙ (W, Dγγ˙ γ), ˙ dt

(12)

because Cγ˙ (Dγγ˙ γ, ˙ W, γ) ˙ = 0 (again by Remark 2.9), and substituting (12) in (11) with s = 0 and integrating, we get finally (10). Let us denote by CL (M, [a, b]) the subset of L-admissible piecewise smooth curves γ : [a, b] → M and for any γ ∈ CL (M, [a, b]), define Tγ CL (M, [a, b]) as the subset of L-admissible continuous piecewise smooth vector fields along γ with the same breaks as γ. Proposition 3.1 allows us to define formally the differential of E in γ as the map dEγ : Tγ C(M, [a, b]) → R given by dEγ (W ) = −

Z

a

b

gγ˙ (W, Dγγ˙ γ)dt ˙ + [gγ˙ (W, γ)] ˙ ba +

h X i=1

LL (γ(t ˙ + ˙ − i ))(W (ti )) − LL (γ(t i ))(W (ti )) ,

for any W ∈ Tγ CL (M, [a, b]). In fact, observe that given W , we can choose a variation Λ : [a, b] × (−ε, ε) → M having W as a variation vector field. If Λ is C 1 , being A an open subset and [a, b] compact, we can choose a smaller ε if necessary in such a way that Λ is L-admissible. When Λ is piecewise smooth, we can do the same thing in every interval [ti , ti+1 ] for i = 0, . . . , h obtaining an εi > 0 such that Λ : [ti , ti+1 ] × (−εi , εi ) → M is L-admissible. Finally we consider ε = min{ε0 , . . . , εh } to get the L-admissible variation of γ. From now on, given a smooth curve γ : [a, b] → M of a pseudo-Finsler manifold and W ∈ X(γ), we will denote W ′ = Dγγ˙ W . As we will see later, the critical points of the energy functional are geodesics with some boundary conditions. Let us compute the second variation for an arbitrary geodesic.


11

Proposition 3.2. Let γ : [a, b] → M be a geodesic of (M, L) and consider an L-admissible smooth variation Λ. Then with the above notation d2 E(γs ) |s=0 ds2 Z b ib h ˙ (−gγ˙ (Rγ (γ, ˙ W )W, γ) ˙ + gγ˙ (W ′ , W ′ )) ds + gγ˙ (Dβγ˙ t β˙ t |s=0 , γ) = , (13)

E ′′ (0) =

a

a

where

Dβγ˙ t β˙ t |s=0

is the transverse acceleration vector field of the variation.

Proof. We will use Remark 2.9 along the proof without further comment. Using (11) and the almost g-compatibility of the Chern connection, we get Z b Z b d d2 d γ˙ s ˙ E(γ ) = gγ˙ s (Dγγ˙ ss β˙ t , γ˙ s )dt β , γ ˙ )dt = (D g s t s γ˙ ds2 ds a s γs ds a Z b = gγ˙ s (Dβγ˙ st Dγγ˙ ss β˙ t , γ˙ s ) + gγ˙ s (Dγγ˙ ss β˙ t , Dβγ˙ st γ˙ s ) dt. a

Now using that d2 E(γs ) = ds2

Dβγ˙ st Dγγ˙ ss β˙ t

Z

a

b

=

Dγγ˙ ss Dβγ˙ st β˙ t

− Rγs (γ˙ s , β˙ t )β˙ t (see (7)),

gγ˙ s (Dγγ˙ ss Dβγ˙ st β˙ t − Rγs (γ˙ s , β˙ t )β˙ t , γ˙ s )dt +

Z

a

b

gγ˙ s (Dγγ˙ ss β˙ t , Dβγ˙ st γ˙ s )dt.

(14)

d (gγ˙ s (Dβγ˙ st β˙ t , γ˙ s )) for Finally, as γ0 = γ is a geodesic, we have gγ˙ s (Dγγ˙ ss Dβγ˙ st β˙ t , γ˙ s ) = dt s = 0, and using this in (14), integrating and recalling (9), we get (13).

Observe that the transverse acceleration Dβγ˙ t β˙ t |s=0 depends not only on the vector W along γ, but on the variation Λ. We will see later that the dependence on the variation disappears when we put certain boundary conditions. 3.1. Submanifolds and second fundamental form. We refer the reader to [21] for the basic notions and notation on submanifolds in semi-Riemannian manifolds. Let us assume that (M, L) is a pseudo-Finsler manifold and P ⊂ M a submanifold of M . We denote the tangent bundle of P as T P and define the normal bundle T P ⊥ of P as the vectors v ∈ A such that π(v) ∈ P and gv (v, w) = 0 for every w ∈ Tπ(v) P . We also denote Tp P ⊥ = T P ⊥ ∩ Tp M for every p ∈ P , which is a conic subset, namely, if v ∈ Tp P ⊥ , then λv ∈ Tp P ⊥ for every λ > 0. Let us see that even though T P ⊥ is not necessarily a fiber bundle over P , it admits a structure of submersion. Let us denote P0 = {p ∈ P : ∃v ∈ T P ⊥ , π(v) = p}, r = dim P and recall that n = dim M . Lemma 3.3. The subset T P ⊥ , if not empty, is a submanifold of T M of dimension n, P0 is an open subset of M and the map π : T P ⊥ → P0 , is a submersion, where, with abuse of notation, we call π the restriction of the natural projection π : T M → M . In particular, for every p ∈ P , Tp P ⊥ is a submanifold of Tp M of dimension n − r.

12


Proof. Assume that E1 , . . . , Er are vector fields in an open subset Ξ of P which, at every point p ∈ P , form a basis of Tp P . Define ϕ : A ∩ π(Ξ)−1 → Rr as ϕ(v) = (gv (v, E1 ), . . . , gv (v, Er )).Observe that if h : (−ǫ, ǫ) → Tp M is a curve such ˙ that h(0) = v and h(0) = u and w ∈ Tp M then using the covariant derivative along the constant curve equal to p, (3) and Remark 2.9, we get d gh(t) (h(t), w)|t=0 = gv (u, w) + Cv (u, v, w) = gv (u, w) dt and then the fiber derivative of ϕ is given by Df ϕv (u) = (gv (u, E1 ), . . . , gv (u, Er )) for every u ∈ Tπ(v) M . As gv is a non-degenerate metric and E1 , . . . , Er are linearly independent, this ensures that the map ϕ is a submersion. Then T P ⊥ ∩ π −1 (Ξ) = ϕ−1 (0), which, if not empty, is a submanifold of T M of dimension n = dim M and T P ⊥ ∩ Tp M is a submanifold of Tp M of dimension n − r. This implies that P0 = π(T P ⊥ ) is open and π : T P ⊥ → P0 a submersion. We will denote by X(P ) the space of smooth sections of the fiber bundle T P over P , by X(P )⊥ the smooth sections of π : T P ⊥ → P0 and by F (P ) the subset of smooth real functions on P . Given N ∈ X(P )⊥ we will denote by X(P )⊥ N the subset of smooth sections W of π : i∗ (T M ) → P , where i : P → M is the inclusion, with i∗ (T M ) the pulled-back tangent bundle of π : T M → M , and such that for every p ∈ P , W (p) is gN -orthogonal to Tp P . Observe that in particular N ∈ X(P )⊥ N. Then if gN |Tp P ×Tp P is nondegenerate, we have the decomposition Tp M = Tp P ⊕ (Tp P )⊥ N,

(15)

where (Tp P )⊥ N is the subspace of Tp M of gN -orthogonal vectors to Tp P . Then for every smooth section V of π : T M → P , we can define tanN (V ) (resp. norN (V )) as the vector field in X(P ) obtained in every p ∈ P projecting V (p) to Tp P (resp. (Tp P )⊥ N ) through the decomposition (15). Definition 3.4. Fix N ∈ X(P )⊥ and suppose that gN |Tp P ×Tp P is nondegenerate for every p ∈ P . Then

(i) we define the second fundamental form of P in the direction of N as the P P N function SN : X(P ) × X(P ) → X(P )⊥ N given by SN (U, W ) = norN ∇U W , P P ˜ (ii) we define the normal second fundamental form SN : X(P ) → X(P ) as SÑ (U ) = N tanN ∇U N .

P Proposition 3.5. With the above notation, SN is F (P )-bilinear and symmetric P ˜ and SN is F (P )-linear. Moreover, P P gN (SN (U, W ), N ) = −gN (SÑ (U ), W )

(16)

for every U, W ∈ X(P ). P Proof. Let us see that SN is F (P )-bilinear. This is immediate for the first variable. For the second one, let f ∈ F (P ) and U, W ∈ X(P ), then P N N SN (U, f W ) = norN ∇N U (f W ) = norN (U (f )W + f ∇U (W )) = f norN (∇U W ),

since W is tangent to P . For the symmetry, P P N SN (U, W ) − SN (W, U ) = norN (∇N U W − ∇W U ) = norN [U, W ] = 0,


13

P since [U, W ] is tangent to P . Again it is straightforward to check that SÑ is F (P )linear. For (16), using that Chern connection is almost g-compatible, gN (N, W ) = 0 and Remark 2.9, we get P gN (SN (U, W ), N ) = gN (∇N U W, N ) N ˜P = −gN (W, ∇N U N ) − 2CN (∇U N, W, N ) = −gN (SN (U ), W ),

as required.

Remark 3.6. Observe that from the homogeneity of Chern connection (∇λV = ∇V P P P for a positive function λ) it follows that SλN = SN and then from (16), that S˜λN = P P P P ˜ ˜ λSN . Moreover, we will interpret SN and SN as maps SN : Tp P × Tp P → (Tp P )⊥ N P and SÑ : Tp P → Tp P respectively even when X(P )⊥ is empty (think that P could be non-orientable), if Tp P ⊥ is not empty, then there is some open subset Ξ ⊂ P where X(Ξ)⊥ is not empty. 3.2. The endmanifold case. Consider now the space of curves CL (P, Q) ⊂ CL (M, [a, b]) joining two submanifolds P and Q of M , namely, CL (P, Q) = {γ ∈ CL (M, [a, b]) : γ(a) ∈ P, γ(b) ∈ Q}. When we consider a piecewise smooth (P, Q)-variation of γ ∈ CL (P, Q) by curves in CL (P, Q), the variational vector field is tangent to P and Q in the endpoints. Indeed, we will define Tγ CL (P, Q) = {W ∈ Tγ CL (M, [a, b]) : W (a) ∈ Tγ(a) P, W (b) ∈ Tγ(b) Q}. Moreover, we say that γ is a critical point of E|CL (P,Q) if dEγ (W ) = 0 for every W ∈ Tγ CL (P, Q). Corollary 3.7. Let γ ∈ CL (P, Q) and assume that the Legendre transformation LL is injective. Then γ is a critical point of the energy functional E|CL (P,Q) if and only if γ is a geodesic gγ˙ -orthogonal to P and Q. Proof. Let t0 ∈ (a, b) an instant where γ is smooth. As the scalar product gγ˙ is nondegenerate, if we assume that Dγγ˙ γ˙ 6= 0, using bumpy functions, we can choose a vector field W such that gγ˙ (Dγγ˙ γ, ˙ W ) > 0 in a neighborhood of t0 that does not contain breaks and zero everywhere else. Then using (10), we get a contradiction. Thus, γ must be a piecewise geodesic. Assume that γ(t ˙ + ˙ − i ) 6= γ(t i ) for some i = 1, . . . , h. As LL is assumed to be injective, we can choose a variational − vector field Wi such that LL (γ(t+ i ))(Wi ) − LL (γ(ti ))(Wi ) 6= 0 and it is zero in the other breaks. This gives a contradiction in (10), since γ is a critical point. Therefore, γ is a geodesic. Finally given w ∈ Tγ(a) P , construct a vector field W such that W (a) = w and W (b) = 0. Then (10) implies that gγ(a) (γ(a), ˙ w) = 0. ˙ Analogously, we show that for any v ∈ Tγ(b) Q, gγ(b) ( γ(b), ˙ v) = 0. The converse is ˙ trivial. Corollary 3.8. Let γ ∈ CL (P, Q) be a geodesic of (M, L) that is gγ˙ -orthogonal to P and Q in the endpoints and such that gγ(a) |P ×P and gγ(b) |Q×Q are nondegenerate. ˙ ˙

14


Consider a smooth L-admissible (P, Q)-variation. Then Z b ′′ (−gγ˙ (Rγ (γ, ˙ W )W, γ) ˙ + gγ˙ (W ′ , W ′ )) ds E (0) = a

Q P + gγ˙ (Sγ(b) (W, W ), γ(b)) ˙ − gγ˙ (Sγ(a) (W, W ), γ(a)), ˙ ˙ ˙

where W is the variational vector field of the variation along γ. Proof. It is a straightforward consequence of the definition of the second fundamental form in Definition 3.4 and (13). 3.3. The index form. When γ is a geodesic of a pseudo-Finsler manifold (M, L) such that it is gγ˙ -orthogonal to P and Q in the endpoints and such that gγ(a) |P ×P ˙ and gγ(b) |Q×Q are nondegenerate, we can define the (P, Q)-index form of γ as ˙ Z b γ IP,Q (V, W ) = (−gγ˙ (Rγ (γ, ˙ V )W, γ) ˙ + gγ˙ (V ′ , W ′ )) ds a

Q P (V, W ), γ(b)) ˙ − gγ˙ (Sγ(a) (V, W ), γ(a)), ˙ + gγ˙ (Sγ(b) ˙ ˙

where V, W ∈ Tγ CL (P, Q). Remark 3.9. Observe that when P (resp. Q) is a hypersurface of M , the nondegeneracy condition on gγ(a) |P ×P (resp. gγ(b) |Q×Q ) is equivalent to L(γ(a)) ˙ 6= 0 ˙ ˙ (resp. L(γ(b)) ˙ 6= 0).

Let us observe that the tensor B V , defined in subsection 2.6 for any L-admissible vector field V ∈ X(Ω), is well-defined along a curve γ whenever the first component is γ˙ (see (7)). Indeed, for X, Y, Z ∈ X(γ), we will denote by B γ (γ, ˙ X, Y, Z) the quantity obtained with any extension of X, Y, Z and γ. ˙ Lemma 3.10. With the above notation, if γ is a geodesic of (M, L), then B γ (γ, ˙ X, Y, Z) = 0 when at least one of the vector fields X, Y, Z is γ. ˙ Moreover, B V (X, Y, V, V ) = 0. Proof. Let us denote by V an extension of γ˙ and observe that, as B V is antisymmetric in the first two variables, B γ (γ, ˙ γ, ˙ Y, Z) = 0. Using the definition of B γ , Remark 2.9 and taking into account that γ is a geodesic, we get B γ (γ, ˙ X, γ, ˙ Z) = −∇γγ˙˙ Cγ˙ (∇VX V, γ, ˙ Z),

and using that ∇γγ˙˙ Cγ˙ is symmetric and (5),

B γ (γ, ˙ X, γ, ˙ Z) = Cγ˙ (∇γγ˙ γ, ˙ ∇VX V, Z) = 0,

and as B V is symmetric in the last two components we also have that B γ (γ, ˙ X, Z, γ) ˙ = 0. To check that B V (X, Y, V, V ) = 0, let us use Remark 2.9 and then the symmetry of ∇VX CV and ∇VY CV and (5) to get B V (X, Y, V, V ) = ∇VY CV (∇VX V, V, V ) − ∇VX CV (∇VY V, V, V ) as required.

= −CV (∇VY V, ∇VX V, V ) + CV (∇VX V, ∇VY V, V ) = 0,


15

γ Proposition 3.11. The kernel of IP,Q is given by the vector fields V ∈ Tγ CL (P, Q) such that V ′′ = Rγ (γ, ˙ V )γ˙ ′ P ′ ˜ and tanγ˙ V (a) = S (V (a)) and tanγ˙ V (b) = S˜Q (V (b)). γ˙

γ˙

γ

Proof. Observe that gγ˙ (R (γ, ˙ V )W, γ) ˙ = −gγ˙ (Rγ (γ, ˙ V )γ, ˙ W ) (recall the properties of the curvature tensor in subsection 2.6 and Lemma 3.10). Then using also that d (gγ˙ (V ′ , W )) − gγ˙ (V ′′ , W ), dt where we have used that the Chern connection is almost g-compatible and γ is a geodesic, the index form can also be expressed as Z b γ IP,Q (V, W ) = (gγ˙ (Rγ (γ, ˙ V )γ, ˙ W ) − gγ˙ (V ′′ , W )) ds gγ˙ (V ′ , W ′ ) =

a

b

Q P (V, W ), γ(b)) ˙ − gγ˙ (Sγ(a) (V, W ), γ(a)). ˙ + [gγ˙ (V ′ , W )]a + gγ˙ (Sγ(b) ˙ ˙

This means that V ∈ Tγ CL (P, Q) belongs to the kernel of the index form if and only if V ′′ − Rγ (γ, ˙ V )γ˙ = 0 and P (V, W ), γ(a)) ˙ = 0, gγ˙ (V ′ (a), W ) + gγ˙ (Sγ(a) ˙ Q gγ˙ (V ′ (b), W ) + gγ˙ (Sγ(b) (V, W ), γ(b)) ˙ = 0, ˙

in the endpoints. Using (16), we get that gγ(a) (V ′ (a) − S˜γP˙ (V (a)), u) = 0, ˙

gγ(b) (V ′ (b) − S˜γP˙ (V (b)), w) = 0 ˙

for every u ∈ Tγ(a) P and w ∈ Tγ(b) Q and then tanγ˙ V ′ (a) = S˜γP˙ (V (a)) and tanγ˙ V ′ (b) = S˜γQ ˙ (V (b)) as required. 3.4. Jacobi fields and conjugate and focal points. Definition 3.12. Given an arbitrary geodesic γ : [a, b] → M of (M, L), let us define a Jacobi field of γ as a vector field J along γ satisfying J ′′ = Rγ (γ, ˙ J)γ. ˙ Moreover, given a submanifold P such that γ(a) ∈ P and γ(a) ˙ is gγ(a) -orthogonal ˙ to P , we say that a Jacobi field is P -Jacobi if J(a) is tangent to P and tanγ˙ J ′ (a) = S˜γP˙ (J(a)), and that an instant t0 ∈ (a, b] is (i) conjugate if there exists a Jacobi field J along γ such that J(a) = J(t0 ) = 0, (ii) P -focal if there exists a P -Jacobi field J such that J(t0 ) = 0.

Observe that given a geodesic γ : [a, b] → M of a pseudo-Finsler manifold, we can choose an orthonormal parallel basis of vector fields along γ. Fix an orthonormal basis e1 , e2 , . . . , en of (Tγ(a) M, gγ(a) ), namely, a basis satisfying that gγ(a) (ei , ej ) = ˙ ˙ εi δij , where ε2i = 1, δij is the Kronecker’s delta and i, j = 1, . . . , n. Then define the parallel vector fields E1 , E2 , . . . , En along γ such that Ei (a) = ei for every i = 1, . . . , n. The fact that γ is a geodesic implies that gγ˙ (Ei , Ej ) = εi δij , since d gγ˙ (Ei , Ej ) = gγ˙ (Ei′ , Ej ) + gγ˙ (Ei , Ej′ ) + 2Cγ˙ (Dγγ˙ γ, ˙ Ei , Ej ) = 0, dt

16


where we have used that the Chern connection is almost g-compatible, γ is a geodesic and Ei and Ej are parallel along γ. We say that a variation is geodesic when it is given by geodesics. Proposition 3.13. Given a geodesic γ of (M, L), the vector field J along γ of a geodesic variation is a Jacobi field. Proof. Using (9) and (7) and that γs is a geodesic, we get = Rγ (γ, ˙ J)γ, ˙ = Dγγ˙ ss Dβγ˙ st γ˙ s J ′′ = Dγγ˙ ss Dγγ˙ ss β˙ t s=0

s=0

as required.

Lemma 3.14. Let γ : [a, b] → M be a geodesic of (M, L) with γ(a) = p. Then for any v, w ∈ Tp M , there exists a unique Jacobi field such that J(a) = v and J ′ (a) = w. Proof. Let E1 , E2 , . . . , En be a parallel orthonormal frame Pn field of γ as above. Then any vector field along γ can be expressed as Y = i=1 y i Ei for certain smooth functions yi : [a, b] → R, i = 1, . . . , n. Then the Jacobi equation is equivalent to n

d2 (y m ) X m j Rj y = dt2 j=1

(17)

with m = 1, . . . , n, where the coefficients are determined by the system Rγ (γ, ˙ Ej )γ˙ =

n X

Rjm Em

m=1

for j = 1, . . . , n. Moreover, the initial conditions are equivalent to y i (a) = v i i i i i and dy dt (a) = w , where v and w are respectively the coordinates of v and w in the orthonormal basis E1 (a), . . . , En (a). By the theory of ordinary differential equations, the system in (17) is uniquely determined by these initial conditions and the solution exists in the whole interval [a, b]. Proposition 3.15. Let p be a point of a pseudo-Finsler manifold (M, L) and v ∈ Tp M ∩ A that belongs to the domain of expp . For any w ∈ Tv (Tp M ), we have d expp (v)[w] = J(1), where J is the unique Jacobi field on γv such that J(0) = 0 and J ′ (0) = w. ˜ s) = t(v + sw) for 0 ≤ t ≤ 1 and s small enough Proof. Consider the variation Λ(t, and define ˜ s)) = γv+sw (t). Λ(t, s) = expp (Λ(t, By Lemma 3.14, the variational vector field J(t) = Λs (t, 0) is a Jacobi field. Moreover, as the curve s → Λ(0, s) = p is constant, J(0) = 0 and if we denote by β the constant curve with value p and γs = γv+sw , from (9), we get J ′ (0) = Dβγ˙ s γ˙ s (0) = w, since γ˙ s (0) = v + sw.


17

Given a geodesic γ : [a, b] → M of a pseudo-Finsler manifold, denote by γ˙ ⊥ = {v ∈ Tγ˙ M : gγ˙ (γ, ˙ v) = 0}. When L(γ) ˙ 6= 0, we have the decomposition Tγ˙ M = span(γ) ˙ ⊕ γ˙ ⊥ .

(18)

Moreover, if Y ∈ X(γ), let us denote by tanγ (Y ) and norγ (Y ) the first and second projection in the decomposition (18). Lemma 3.16. With the above notation, if L(γ) ˙ 6= 0, (tanγ (Y ))′ = tanγ (Y ′ ) and ′ ′ (norγ (Y )) = norγ (Y ). Proof. It is enough to prove that (tanγ (Y ))′ = tanγ (Y ′ ), since the other equality comes then from Y = tanγ (Y ) + norγ (Y ). Observe that gγ˙ (Y, γ) ˙ = gγ˙ (tanγ (Y ), γ), ˙ and using that γ is a geodesic and the Chern connection is almost g-compatible, we get that gγ˙ (Y ′ , γ) ˙ = gγ˙ ((tanγ (Y ))′ , γ), ˙ which concludes because gγ˙ (γ, ˙ γ) ˙ = L(γ) ˙ 6= 0. Lemma 3.17. Consider a vector field J along a geodesic γ : [a, b] → M . Then if J is a Jacobi field: (i) J is tangent to γ if and only if J = (a1 s + a2 )γ˙ with a1 , a2 ∈ R, (ii) the following statements are equivalent: (a) gγ˙ (γ, ˙ J) = 0, (b) there exist a, b such that gγ˙ (γ(a), ˙ J(a)) = gγ˙ (γ(b), ˙ J(b)) = 0 , (c) there exists a such that gγ˙ (γ(a), ˙ J(a)) = gγ˙ (γ(a), ˙ J ′ (a)) = 0. Moreover, if γ is nonnull, that is, L(γ) ˙ 6= 0, then J is a Jacobi field if and only if norγ J and tanγ J are Jacobi fields. Proof. For (i), observe that Rγ (γ, ˙ γ) ˙ = 0 because Rγ is antisymmetric in the first 2 two variables. Then, for J = f γ, ˙ the Jacobi equation reduces to df dt2 = 0. For (ii), observe that since γ is a geodesic, d2 (gγ˙ (J, γ)) ˙ = gγ˙ (J ′′ , γ) ˙ = gγ˙ (Rγ (γ, ˙ J)γ, ˙ γ), ˙ dt2 and observe that from the second symmetry of the curvature tensor in subsection 2.6, we deduce that the last term is zero, because the B term which appears there is zero (see Lemma 3.10). Hence gγ˙ (J, γ) ˙ = C1 t + C2 , where C1 and C2 are real constants, and gγ˙ (J ′ , γ) ˙ = C1 , thus (ii) follows. For the last statement, observe that Rγ (γ, ˙ tanγ J) = 0 and then Rγ (γ, ˙ J) = Rγ (γ, ˙ norγ J). Using again γ that gγ˙ (R (γ, ˙ J)γ, ˙ γ) ˙ = 0 and Lemma 3.16, the Jacobi equation splits into the two equations (tanγ J)′′ = 0

and

(norγ J)′′ = Rγ (γ, ˙ norγ J)γ. ˙

Finally applying part (i) we conclude. Proposition 3.18. If J1 and J2 are Jacobi fields along a geodesic γ, then gγ˙ (J1 , J2′ ) − gγ˙ (J1′ , J2 ) is constant.

18


Proof. Observe that using that γ is a geodesic, that the Chern connection is almost g-compatible and J1 and J2 satisfy the Jacobi equation, we obtain d (gγ˙ (J1 , J2′ ) − gγ˙ (J1′ , J2 )) =gγ˙ (J1′ , J2′ ) + gγ˙ (J1 , J2′′ ) dt − gγ˙ (J1′ , J2′ ) − gγ˙ (J1′′ , J2 )

=gγ˙ (J1 , Rγ (γ, ˙ J2 )γ) ˙ − gγ˙ (Rγ (γ, ˙ J1 )γ, ˙ J2 ),

which is zero (see (6) and Lemma 3.10).

3.5. Remarks about Morse theory. Let us observe that in principle, there should not be further obstructions to prove that geodesics of a pseudo-Finsler metric are critical points of the energy functional in a suitable infinite dimensional H 1 -Sobolev space, for example, by generalizing the proof in [9, Proposition 2.1]. But in order to make Morse theory available, we need to overcome several problems. The first one is that Palais-Smale condition only holds in general when the pseudo-Finsler metric is in fact a Finsler metric (with positive definite fundamental tensor), since it is well-known that Palais-Smale condition fails for semi-Riemannian metrics. The second problem is the differentiability of the energy function in the H 1 Sobolev space, because it is C 2 only when the pseudo-Finsler metric is semiRiemannian (see [1, Proposition 3.2] and [7]). This has been overcome in the case of Finsler metrics using that the energy functional is C 2 in the C 1 -topology (see [8, 10]). The third problem is that when A is strictly contained in T M \ 0, the space of L-admissible curves can be non-complete, thus it seems interesting to study conditions of completeness in the pseudo-Finsler metric to guarantee the validity of the Morse theory as in [8, 10]. In [11] some results of geodesic connectedness of conic Finsler metrics are deduced using Causality of spacetimes endowed with a Killing vector. In the general case, when the fundamental tensor is allowed to have any signature, Lemma 3.18 is the key point to develop a relation between the spectral flow of a certain path of operators and the Maslov index of conjugate points as in [23]. In the presence of a Killing vector field, more precise results have been obtained in the Lorentzian realm [6, 12, 14] and it is expectable to get similar results for Lorentzian Finsler metrics. 4. Lightlike geodesics In this section, we examine the effects of conformal transformations on lightlike curves, that is, curves γ such that L(γ) ˙ = 0. We prove that some key geometric properties of these curves (such as being a geodesic, and having conjugate or focal points) are preserved up to reparametrization by such transformations (see also Remark 2.16). 4.1. First variation of the energy. Let (M, L) be a pseudo-Finsler manifold and λ : M → (0, +∞) an arbitrary positive smooth function. We want to consider variations of the energy functional of the pseudo-Finsler metric λL. Given a piecewise smooth curve γ : [a, b] → M , let us denote Z 1 b Eλ (γ) = λ(γ)L(γ)dt. ˙ 2 a

Throughout this section we will always use the Chern covariant derivative Dγ along a smooth curve γ associated to L. Indeed, in the following we will try to express


19

the first and second variation of the energy Eλ in terms of Dγ rather than using the covariant derivative associated to λL. Observe that we will use the same notation for variations as in Section 3 and in particular LL denotes the Legendre transform of L. Moreover, ∇v will denote the gradient with respect to the metric gv . Proposition 4.1. Suppose that γ : [a, b] → M is an L-admissible piecewise smooth curve having an L-admissible piecewise smooth variation Λ : [a, b] × (−ε, ε) → M . Then, with the notation of Section 3, Eλ′ (0) =

d Eλ (γs ) |s=0 ds Z b 1 b γ˙ gγ˙ W, L(γ)(∇ = ˙ λ) − Dγγ˙ (λ(γ)γ) ˙ dt + [gγ˙ (W, λ(γ)γ)] ˙ a 2 a +

h X i=1

λ(γ(ti )) LL (γ(t ˙ + ˙ − i ))(W (ti )) − LL (γ(t i ))(W (ti )) . (19)

Proof. Observe that for any s ∈ (−ε, ε), we have Z 1 b d d Eλ (γs ) = (λ(γs )gγ˙ s (γ˙ s , γ˙ s )) dt ds 2 a ds Z Z 1 b d 1 b d = λ(γs ) gγ˙ s (γ˙ s , γ˙ s ) dt + λ(γs ) gγ˙ s (γ˙ s , γ˙ s ) dt. (20) 2 a ds 2 a ds

The first term on the right-hand side of the equation above is equal to Z Z 1 b 1 b dλγs (t) β˙ t gγ˙ s (γ˙ s , γ˙ s ) dt = gγ˙ (β˙t , ∇γ˙ s λ)gγ˙ s (γ˙ s , γ˙ s ) dt 2 a 2 a s Z b 1 = gγ˙ s β˙ t , L(γ˙ s )(∇γ˙ s λ) dt. (21) 2 a Furthermore, the second term on the right-hand side of (20) is computed using similar arguments to those in the proof of Proposition 3.1, namely, almost gcompatibility of Chern connection, the identity (9) and properties of the Cartan tensor, which gives Z b gγ˙ s (Dγγ˙ ss β˙ t , λ(γs )γ˙ s )dt a

=

Z

a

b

d gγ˙ (β˙ t , λ(γs )γ˙ s )dt − dt s

Z

a

b

gγ˙ s β˙ t , Dγγ˙ ss (λ(γs )γ˙ s ) dt . (22)

Computing the last terms in (21) and (22) in s = 0, and substituting in (20), we get (19). Last proposition allows us to obtain the geodesic equation for λL.

Proposition 4.2. The geodesics of λL are determined by 1 γ˙ L(γ)(∇ ˙ λ) − Dγγ˙ (λ(γ)γ) ˙ = 0. (23) 2 Proof. Along the proof, we consider curves with fixed endpoints and smooth Ladmissible variations. Reasoning as in Corollary 3.7, it is possible to prove that geodesics of λL are the critical points of Eλ . Observe that, since we consider smooth

20


variations, we do not need the injectivity of Legendre transform. From Proposition 4.1, we can prove that the critical points of Eλ are given by (23) analogously to the proof of Corollary 3.7. This allows us to conclude that a lightlike curve γ is a geodesic of λL if and only if Dγγ˙ (λ(γ)γ) ˙ = 0, (24) which implies that lightlike geodesics are preserved by conformal transformations up to reparametrization. Remark 4.3. Let us observe first that given a curve γ : [a, b] → M and a reparametrization γ˜ = γ ◦ ϕ with ϕ : [˜ a, ˜b] → [a, b], if V and W are vector fields ˜ ˜ along γ and V and W are the vector fields along γ˜ defined as V˜ (µ) = V (ϕ(µ)) and ˜ (µ) = W (ϕ(µ)) for any µ ∈ [˜ W a, ˜b], then ˜

˜ (µ) = DγV˜ W

dϕ (µ)DγV W (ϕ(µ)), dµ

for any µ ∈ [˜ a, ˜b], and DγV W = DγλV W for any function λ : [a, b] → (0, +∞) (recall subsection 2.3). Proposition 4.4. The curve γ is a lightlike geodesic of (M, λL), then γ ◦ ϕ is a lightlike geodesic of (M, L), where ϕ : [˜ a, ˜b] → [a, b] is the solution of the differential equation dϕ (µ) = λ(γ(ϕ(µ))). (25) dµ Proof. An easy consequence of (24) and Remark 4.3.

4.2. Second variation of the energy. Our next goal is to study the behavior of conjugate and focal points of lightlike geodesics under conformal transformations. We start by computing the second variation of the energy functional Eλ . Proposition 4.5. Let γ : [a, b] → M be a lightlike geodesic of (M, λL) and consider an L-admissible smooth variation Λ. Then, with the above notation, Eλ′′ (0) =

d2 Eλ (γs ) |s=0 ds2 Z b = λ(γ) − gγ˙ (Rγ (γ, ˙ W ) W, γ) ˙ + gγ˙ (W ′ , W ′ ) dt a

+2

Z

a

Dβγ˙ t β˙ t |s=0

b

ib h ˙ gγ˙ (W ′ , γ) ˙ gγ˙ W, ∇γ˙ λ dt + λ(γ)gγ˙ (Dβγ˙ t β˙ t |s=0 , γ) , (26) a

is the transverse acceleration vector field of the variation and Rγ where is the Chern curvature of L defined in (7).

Proof. Using (21) and (22), we get Z b d2 1 d γ˙ s ˙ dt E (γ ) = λ g L ( γ ˙ ) ∇ β , λ s γ˙ s s t ds2 2 a ds Z b d gγ˙ s Dγγ˙ ss β˙ t , λ (γs ) γ˙ s dt. (27) + a ds


21

The first term on the right-hand side of the above equation is equal to Z b Z b 1 d 1 γ˙ s d γ˙ s ˙ ˙ gγ˙ s βt , (∇ λ) dt L(γ˙ s ) L(γ˙ s ) gγ˙ s βt , (∇ λ) dt + ds 2 ds 2 a a Z b Z b d 1 L(γ˙ s ) gγ˙ s (Dγγ˙ ss β˙ t , γ˙ s )gγ˙ s (β˙t , ∇γ˙ s λ) dt + gγ˙ s β˙ t , (∇γ˙ s λ) dt, = ds 2 a a (28) d L(γ˙ s ) = 2gγ˙ s (Dγγ˙ ss β˙ t , γ˙ s ) (see (11)). As for the second where we have used that ds term on the right-hand side of (27), it equals Z b Z b d d γ˙ s ˙ λ (γs ) gγ˙ s Dγs βt , γ˙ s dt (λ (γs )) gγ˙ s Dγγ˙ ss β˙t , γ˙ s dt + ds a ds a Z b Z b d γ˙ s ˙ λ (γs ) gγ˙ s (Dγγ˙ ss β˙ t , γ˙ s ) gγ˙ s (β˙t , ∇γ˙ s λ) dt + = gγ˙ s Dγs βt , γ˙ s dt, ds a a (29)

and using the same arguments as in the proof of Proposition 3.2, we get that the last term above equals Z b λ(γs ) gγ˙ s Dγγ˙ ss Dβγ˙ st β˙ t − Rγs (γ˙ s , β˙ t )β˙t , γ˙ s + gγ˙ s Dγγ˙ ss β˙ t , Dγγ˙ ss β˙ t dt . (30) a

Substituting (28), (29) and (30) in (27), putting s = 0, observing that, since γ is a lightlike curve, the second term in (28) must equal 0, and using that

d λ(γs )gγ˙ s (Dγγ˙ ss Dβγ˙ st β˙ t , γ˙ s ) = (gγ˙ s (Dβγ˙ st β˙ t , λ(γs )γ˙ s )) dt for s = 0 (observe that γ0 = γ satisfies (24)), we get (26).

Recalling now the notation of subsection 3.2, let γ ∈ CL (P, Q) = CλL (P, Q) be a geodesic of (M, λL) which is gγ˙ -orthogonal to P and Q in the endpoints and P and Q be non-degenerate in γ(a) and γ(b) with respect to the metrics gγ(a) and ˙ 4.5 allows us to compute the index form of γ as gγ(b) respectively. Proposition ˙ γ,λ IP,Q (V, W )

= +

Z

b

λ(γ) (−gγ˙ (Rγ (γ, ˙ V )W, γ) ˙ + gγ˙ (V ′ , W ′ )) dt a

Z

a

b

gγ˙ (V ′ , γ)g ˙ γ˙ (W, ∇γ˙ λ) + gγ˙ (W ′ , γ)g ˙ γ˙ (V, ∇γ˙ λ) dt

Q P (V, W ), γ(b)) ˙ − λ(γ(a))gγ˙ (Sγ(a) (V, W ), γ(a)), ˙ + λ(γ(b))gγ˙ (Sγ(b) ˙ ˙

(31)

where V, W ∈ Tγ CL (P, Q) and S P and S Q are the fundamental forms of P and Q computed with L. This comes easily from Proposition 4.5, the equality Eλ′′ (0) = γ,λ IP,Q (W, W ) and the definition of second fundamental form. In the following we γ will call (P, Q)-Jacobi fields to the elements on the kernel of IP,Q (see Proposition 3.11). Our next goal is to show that the P -focal points of γ are preserved with multiplicity in γ˜ (recall the notation of Remark 4.3). Lemma 4.6. Let γ : [a, b] → M be a geodesic of (M, λL) and P and Q two submanifolds which are orthogonal to γ and non-degenerate in γ(a) and γ(b) with respect to the metrics gγ(a) and gγ(b) , respectively. Assume that J˜ is a (P, Q)-Jacobi ˙ ˙

22


˜ field of γ˜ with the metric L and J satisfies that J(ϕ(µ)) = J(µ) for every µ ∈ [˜ a, ˜b]. ˆ = J(t)+h(t)γ(t), Then there exists a function h : [a, b] → R such that J(t) ˙ t ∈ [a, b] is a (P, Q)-Jacobi field of γ with the metric λL. Proof. Observe that as J˜ is a Jacobi field of γ˜ , it holds ˙ ˙ ˜ γ˜˙ Dγγ˜˜ Dγγ˜˜ J˜ = Rγ˜ (γ˜˙ , J)

(32)

and γ ˜˙ (a)

tanγ˜˙ (a) (Dγ˜

˜ a)) = S˜P˙ (J(˜ ˜ a)), J(˜ γ ˜ (a)

˙ ˜ Now observe that Dγγ˜˜ J(µ) =

˙ ˜ ˜b)) = S˜Q (J( ˜ ˜b)). tanγ˜˙ (b) (Dγγ˜˜ J( γ ˜˙ (b)

(33)

dϕ γ˙ dµ (µ)Dγ J(ϕ(µ)),

˜ γ˜˙ )(µ) = λ(γ)2 (Rγ (γ, (Rγ˜ (γ˜˙ , J) ˙ J)γ)(ϕ(µ)) ˙ and recall (25), therefore, (32) and (33) can be rewritten as (λ(γ)J ′ )′ = λ(γ)Rγ (γ, ˙ J)γ, ˙

(34)

and recalling Remark 3.6, P (J(a)), tanγ(a) J ′ (a) = S˜γ(a) ˙ ˙

Q tanγ(b) J ′ (b) = S˜γ(b) (J(b)) ˙ ˙

(35)

Dγγ˙ ).

′

(recall that means to apply By Proposition 3.11, we know that (P, Q)Jacobi fields are the vector fields in the kernel of the index form. Reasoning as in Proposition 3.11 with the expression (31), we get that V ∈ CL (P, Q) is a (P, Q)Jacobi field along γ if and only if and

λ(γ)Rγ (γ, ˙ V )γ˙ − (λ(γ)V ′ )′ + gγ˙ (V ′ , γ)∇ ˙ γ˙ λ − (gγ˙ (V, ∇γ˙ λ)γ) ˙ ′=0 P (V (a)), tanγ(a) V ′ (a) = S˜γ(a) ˙ ˙

Q tanγ(b) V ′ (b) = S˜γ(b) (V (b)). ˙ ˙ ˙ ˜ γ˜˙ ) gγ˜˙ (Dγγ˜˜ J,

Now observe that as J˜ is a (P, Q)-Jacobi field for γ˜ , we have that (this follows easily from Lemma 3.17) and then 1 ˙ ˜ g ˙ (Dγ˜ J(µ), γ˜˙ (µ)) = 0. gγ(ϕ(µ)) (J ′ (ϕ(µ)), γ(ϕ(µ))) ˙ = ˙ λ(˜ γ (µ))2 γ˜ (µ) γ˜

(36) (37) =0

Using last equation, (24), (34), (35) and gγ˙ (γ, ˙ γ) ˙ = 0, we deduce that if V (t) = J(t) + h(t)γ(t) ˙ and h(a) = h(b) = 0, then V satisfies (36) and (37) if and only if −(λ(γ)(hγ) ˙ ′ )′ − (hλ˙ γ) ˙ ′ = (J(λ)(γ)γ) ˙ ′, ˙ where λ(t) =

d dt λ(γ(t)).

This equation is equivalent to

d2 1 d λ˙ h = − J(λ)(γ). (J(λ)(γ)) + dt2 λ(γ) dt λ(γ)2 It is easy to prove that there exists a unique solution h : [a, b] → R of the above differential equation such that h(a) = h(b) = 0. Then the vector field Jˆ = J + hγ˙ is a (P, Q)-Jacobi field along γ. Theorem 4.7. Assume that γ : [a, b] → M is a lightlike geodesic of (M, λL), γ˜ = γ ◦ ϕ is the reparametrization as a lightlike geodesic of (M, L) obtained in Proposition 4.4 and P an orthogonal submanifold passing through γ(a) and nondegenerate in that point with the metric gγ(a) . Then µ0 ∈ (˜ a, ˜b] is a P -focal point ˙ of γ˜ if and only if ϕ(µ0 ) is a P -focal point of γ with the same multiplicity.


23

Proof. It is a consequence of Lemma 4.6. Observe that if we choose Q = γ˜ (t0 ), ˜ 0 ) = 0 and P the lemma gives a map between P -Jacobi fields of γ˜ such that J(µ Jacobi fields of γ such that J(ϕ(µ0 )) = 0. Moreover, this map is injective, because ˜ if Jˆ = 0, then J(µ) = φ(µ)γ˜˙ (µ) for some smooth function φ : [˜ a, ˜b] → R, but ˜ from Lemma 3.17 it follows that J = 0. The injectivity of the map implies that mulγ˜ (µ0 ) ≤ mulγ (ϕ(µ0 )), namely, the multiplicity of µ0 as a P -focal point of γ˜ is less or equal to the multiplicity of ϕ(µ0 ) as a P -focal point of γ. Using Lemma 4.6 with the conformal change 1/λ and the metric λL we get the other inequality concluding that mulγ˜ (µ0 ) = mulγ (ϕ(µ0 )) as required. References [1] A. Abbondandolo and M. Schwarz, A smooth pseudo-gradient for the Lagrangian action functional, Adv. Nonlinear Stud., 9 (2009), pp. 597–623. [2] P. L. Antonelli, R. S. Ingarden, and M. Matsumoto, The theory of sprays and Finsler spaces with applications in physics and biology, vol. 58 of Fundamental Theories of Physics, Kluwer Academic Publishers Group, Dordrecht, 1993. [3] D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. [4] J. K. Beem, Indefinite Finsler spaces and timelike spaces, Canad. J. Math., 22 (1970), pp. 1035–1039. [5] J. K. Beem and M. A. Kishta, On generalized indefinite Finsler spaces, Indiana Univ. Math. J., 23 (1973/74), pp. 845–853. ´ nchez, Global hyperbolicity and Palais-Smale [6] A. M. Candela, J. L. Flores, and M. Sa condition for action functionals in stationary spacetimes, Adv. Math., 218 (2008), pp. 515– 536. [7] E. Caponio, The index of a geodesic in a Randers space and some remarks about the lack of regularity of the energy functional of a Finsler metric, Acta Math. Acad. Paedagog. Nyh´ azi. (N.S.), 26 (2010), pp. 265–274. ´ Javaloyes, and A. Masiello, Morse theory of causal geodesics in a [8] E. Caponio, M. A. stationary spacetime via Morse theory of geodesics of a Finsler metric, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 27 (2010), pp. 857–876. ´ Javaloyes, and A. Masiello, On the energy functional on Finsler [9] E. Caponio, M. A. manifolds and applications to stationary spacetimes, Math. Ann., 351 (2011), pp. 365–392. ´ Javaloyes, and A. Masiello, Addendum to“Morse theory of causal [10] E. Caponio, M. A. geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric”, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 27 (2010) 857–876, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 30 (2013), pp. 961–968. ´ Javaloyes, and M. Sa ´ nchez, Wind Finsler structures: from Zermelo’s [11] E. Caponio, M. A. navigation to the causality of spacetimes, preprint. [12] F. Giannoni and P. Piccione, An intrinsic approach to the geodesical connectedness of stationary Lorentzian manifolds, Comm. Anal. Geom., 7 (1999), pp. 157–197. ´ Javaloyes, Chern connection of a pseudo-Finsler metric as a family of affine connec[13] M. A. tions, to appear in Publicationes Mathematicae Debrecen. ´ Javaloyes, A. Masiello, and P. Piccione, Pseudo focal points along Lorentzian [14] M. A. geodesics and Morse index, Adv. Nonlinear Stud., 10 (2010), pp. 53–82. ´ Javaloyes and M. Sa ´ nchez, On the definition and examples of finsler metrics, [15] M. A. arXiv:1111.5066 [math.DG], to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. ´ Javaloyes and H. Vito ´ rio, Zermelo navigation and wind Finsler structures, preprint, [16] M. A. (2013). [17] J. Kern, Lagrange geometry, Arch. Math. (Basel), 25 (1974), pp. 438–443. [18] H.-H. Matthias, Zwei Verallgemeinerungen eines Satzes von Gromoll und Meyer, Bonner Mathematische Schriften [Bonn Mathematical Publications], 126, Universit¨ at Bonn Mathematisches Institut, Bonn, 1980. Dissertation, Rheinische Friedrich-Wilhelms-Universit¨ at, Bonn, 1980.

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´ nchez, The causal hierarchy of spacetimes, in Recent developments [19] E. Minguzzi and M. Sa in pseudo-Riemannian geometry, ESI Lect. Math. Phys., Eur. Math. Soc., Z¨ urich, 2008, pp. 299–358. ´ r, Entwicklung einer Geometrie der allgemeinen metrischen Linienelementr¨ [20] A. Moo aume, Acta Sci. Math. Szeged, 17 (1956), pp. 85–120. [21] B. O’Neill, Semi-Riemannian geometry, vol. 103 of Pure and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983. With applications to relativity. [22] V. Perlick, Fermat principle in Finsler spacetimes, Gen. Relativity Gravitation, 38 (2006), pp. 365–380. [23] P. Piccione, A. Portaluri, and D. V. Tausk, Spectral flow, Maslov index and bifurcation of semi-Riemannian geodesics, Ann. Global Anal. Geom., 25 (2004), pp. 121–149. ´ ticas, Departamento de Matema Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain E-mail address: [email protected] ´ tica, Computac ˜ o e Cognic ˜ o, Centro de Matema ¸a ¸a Universidade Federal do ABC (UFABC), e (SP), Brazil elia, 166 - 09210-170 - Santo Andr´ Rua Santa Ad´ E-mail address: [email protected]