Morse Theory for the Travel Time Brachistochrones in Stationary Spacetimes

The travel time brachistochrone curves in a general relativistic framework are timelike curves, satisfying a suitable conservation law with respect to a an observer field, that are stationary points of the travel time functional. In this paper we develop a global variational theory for brachistochrones joining an event $p$ and the worldline of an observer $\gamma$ in a stationary spacetime $M$. More specifically, using the method of Lagrange multipliers, we compute the first and the second variation of the travel time functional, obtaining two variational principles relating the geometry of the brachistochrones with the geometry of geodesics in a suitable Riemannian structure. We present an extension of the classical Morse Theory for Riemannian geodesics to the case of travel time brachistochrones, and we prove a Morse Index Theorem for brachistochrones. Finally, using techniques from Global Analysis, we prove the Morse relations for the travel time functional and we establish some existence and multiplicity results for brachistochrones.


Introduction: the General Relativistic Brachistochrone Problem
The classical brachistochrone problem dates back to the end of the seventeenth century, when Johann Bernoulli challenged his contemporaries to solve the following problem.
If in a vertical plane two points A and B are given, then it is required to specify the orbit AM B of the movable point M , along which it, starting from A, and under the influence of its own weight, arrives at B in the shortest possible time. (Acta Eruditorum, June 1696) This problem attracted the attention of many important mathematicians of the time, including Newton, Leibniz, L'Hôpital, and Johann's brother, Jackob Bernoulli. The papers written on the subject may be considered the fundaments of a new field in mathematics, the Calculus of Variations. A beautiful historical exposition of the brachistochrone problem may be found in Reference [27], where the authors' thesis is that the brachistochrone problem also marks the birth of Optimal Control.
Still now the classical brachistochrone problem is very popular, and its importance is witnessed by the fact that there is hardly any book on Calculus of Variations that does not use this problem as a takeoff point. The well known solution to the brachistochrone problem is a cycloid, which is the curve described by a point P in a circle that rolls without slipping.
This problem has several generalizations, e.g., the homogeneous gravitational field could be replaced with an arbitrary Newtonian potential, and instead of releasing the particle from rest one could prescribe an arbitrary value for the initial speed, leaving the initial direction of the velocity undetermined.
In modern terminology, the Newtonian brachistochrone problem can be stated as follows. Given a manifold M 0 endowed with a Riemannian metric g 0 , to be interpreted as the state space, and a smooth function V : M 0 −→ IR, representing the gravitational potential, a brachistochrone of energy E > 0 between two points x 0 and x 1 of M is a curve x : [0, T x ] −→ M joining x 0 and x 1 that extremizes the travel time T x in the space of all curves y joining x 0 and 1991 Mathematics Subject Classification. 58E05, 53C22, 83Cxx. 1 x 1 and satisfying the conservation of energy law: 1 2 g(ẏ,ẏ) + V (y) ≡ E (1.1) (throughout this paper we will consider the motion of particles with unit mass). A well known variational principle states that a curve x joining x 0 and x 1 is a brachistochrone of fixed energy E if and only if x is a geodesic with respect to the conformal Riemannian metric φ E · g 0 , with conformal factor φ E = (E − V ) −1 .
The brachistochrone problem can also be formulated in the context of general relativity. We want to emphasize here that the original solution to the brachistochrone problem offered by Johann Bernoulli, which lacked mathematical rigor, can be made absolutely rigorous in a general relativistic context. Namely, the trajectory of a freely falling massive object, which is represented by a timelike geodesic in a Lorentzian manifold, is characterized by extremizing its arrival time measured by means of a smooth parameterization of the receiving observer. This is the so called general relativistic timelike Fermat Principle, suggested in [14] and rigorously proven in [6].
The first relativistic versions of the brachistochrone problem appear in [10] and [13]. V. Perlick (see [22]) has determined the brachistochrone equation in a stationary Lorentzian manifold of splitting type, and two of the authors, together with J. Verderesi, in [9] have generalized Perlick's result to the case of an arbitrary stationary manifold by reformulating the brachistochrone problem in the context of sub-Riemannian geometry. We recall that a stationary metric that satisfies the Einstein's equations describes a time-independent gravitational field in General Relativity.
The variational principle proven in [9] was then used in [7] to prove some results concerning the existence and the multiplicity of relativistic brachistochrones with a given value of energy between a fixed event and a fixed observer of a stationary spacetime.
We formulate the general relativistic brachistochrone problem for the travel time as follows. Let (M, g) be a 4-dimensional Lorentzian manifold, i.e., an arbitrary spacetime in the sense of general relativity and fix a timelike smooth vector field Y on M. For simplicity, we assume that Y is complete, i.e., its integral lines are defined over the entire real line. The integral curves of Y can be interpreted as the worldlines of observers. Please note that we do not require Y to be normalized, i.e., in general the worldlines of our observers are not parameterized by proper time. The reason is that in the stationary case, i.e., if (M, g) admits a timelike Killing vector field, it is convenient to choose this Killing vector field for Y and not a renormalized version of it.
To formulate the brachistochrone problem with respect to our arbitrarily chosen observer field Y , we fix a point p in M, a (maximal) integral curve γ : IR −→ M of Y and a real number k > 0. The trial paths for our variational problem are all timelike smooth curves σ : [0, 1] −→ M which are nowhere tangent to Y and satisfy the following conditions: g(σ(0), Y (σ(0))) = −k − g(σ,σ) 1/2 ; (1.4) g(∇σσ,σ) = 0; (1.5) g(∇σσ, Y ) = 0. (1.6) Here ∇ denotes the Levi-Civita connection of the Lorentzian metric g. We denote by B p,γ (k) the set of trial paths; in the rest of the paper we will be working with suitable completions of this space.
If we interpret each integral curve of Y as a "point in space", (1.2) and (1.3) mean that all trial paths connect the same two points in space, where the starting time is fixed whereas the arrival time is not. Condition (1.4) says that all trial paths start with the same speed with respect to the observer field Y .
Observe that, in order to simplify the mathematics, we have chosen to parameterize our trial curves on the interval [0, 1], rather than using a proper time parameterization over intervals varying with the curves. By condition (1.5), the quantity T σ defined by −T σ 2 = g(σ,σ) is a constant for each trial path σ (but takes different values for different trial paths). This implies that the curve parameter t along σ is related to proper time τ by an affine transformation, τ = T σ t + const. As a consequence, the 4-velocity along each trial path is given by T σ −1σ , whereas the 4-acceleration is given by T σ −2 ∇σσ. Hence, conditions (1.5) and (1.6) require the 4-acceleration to be perpendicular to the plane spanned byσ and Y . In other words, with respect to the observer field Y there are only forces perpendicular to the direction of motion. Such forces can be interpreted as constraint forces supplied by a frictionless slide which is at rest with respect to the observer field Y .
The brachistochrone problem can now be formulated in the following way. Among all trial paths that satisfy the above-mentioned conditions, we want to find those curves for which the travel time is minimal or, more generally, stationary.
A different general relativistic brachistochrone problem can be formulated, by requiring that the solutions be stationary points for the arrival time functional, given by AT (σ) = γ −1 (σ(1)). In other words, AT (σ) is the value of the proper time of the receiver at the arrival event. In physical terms, the two brachistochrone problems differ by the way of measuring time: in the first case the time is measured by a watch traveling along the trajectory of the mass, in the second case the time is measured by the observer that receives the mass at the end of its trajectory. The two variational problems are essentially different; in this paper we stick to the first problem, while the "arrival time brachistochrones" are the subject of a followup paper.
If (M, g) is a stationary spacetime and Y is a Killing vector field, i.e., the flow of Y preserves the metric g, then the condition (1.6) means that the product g(σ, Y ) is constant along σ. The value of this constant can be easily computed using condition (1.4), that gives g(σ, Y ) ≡ −kT σ . Hence, in the stationary case, the conditions (1.4) and (1.6) can be resumed in the condition: Again, observe that the value of the travel time T σ appears in formula (1.7) because of our choice of the parameterization on the interval [0, 1] of our trial curves. The condition (1.7) is the relativistic counterpart of the energy conservation law (1.1) in the Newtonian case. Although physically meaningful, the mathematical approach to the general relativistic brachistochrone problem in the non stationary case presents difficulties of higher order than in the stationary case. For instance, it is not even clear whether the non stationary brachistochrones are solutions to a second order differential equation; in Reference [23], the authors used a Lagrange multiplier technique to derive a system of differential equations for the brachistochrones and for the Lagrangian multipliers. Unfortunately, it does not seem to be possible to eliminate the Lagrangian multipliers from the system without introducing integrals, unless in the stationary case. Thus, it looks as if the brachistochrones in the non-stationary case are not determined by a second-order differential equation, but rather by an integro-differential equation.
For these technical reasons, in this paper we will only study the case of a manifold M with metric g which is stationary with respect to the observer field Y .
The purpose of this article is to present a complete variational theory for travel time brachistochrones in a stationary Lorentzian manifold and, in particular, it will be developed a fullfledged infinite dimensional Morse theory for the critical points of the travel time.
We present below a list of the main results proven in this paper: • the general-relativistic brachistochrone problem in a stationary Lorentzian manifold is presented in a context of Global Analysis on infinite dimensional Hilbertian manifolds (Section 2); • the travel time brachistochrones are smooth curves; they can be characterized as the only solutions of a second order differential equation (formula (3.21) and Proposition 4.1); • the brachistochrones can also be characterized as local minimizers for the travel time, and, equivalently, as curves whose spatial part is a geodesic with respect to a suitable Riemannian structure on M (Proposition 4.5); • it is computed a second order variation formula for the travel time functional, which is characterized by a Morse Index Theorem (Theorem 7.12). In analogy with the Riemannian geodesic problem, this theorem relates the nature of a stationary point for the travel time with some metrical properties of M and with the convexity of the timelike curve γ representing the observer and measured by the second fundamental form of γ; • under suitable completeness hypotheses for M, we prove the global Morse relations for the travel time functional in a completion of the space B p,γ (k) (Section 8); thanks to this relations one obtains estimates on the number of brachistochrones of fixed energy k between p and γ, according to the topology and the metric of M.
From a strictly mathematical point of view, the paper presents some technicalities that is worth discussing. The main difficulties in our variational problem are due to the presence of the double constraint given by (1.5) and (1.6) (or (1.7)), which are, respectively, quadratic and linear in the first derivative.
Due to this kind of constraint, in order to put a differentiable structure on the set B p,γ (k) of trial paths, one needs to consider a Hilbert space completion of B p,γ (k) made in a Sobolev space of curves having at least the C 1 -regularity, and thus one is forced to consider curves of class H 2 (see formula (2.16), Proposition 2.1 and Remark 2.7).
However, the H 2 -approach has the disadvantage of introducing new difficulties, especially for the following reasons: • the Riesz duality in the Hilbert spaces H i involves products of functions and also their derivatives, resulting in lengthy and complicated calculations when using the Lagrange multipliers method; • the arrival time functional does not satisfy good compactness properties in the space of H 2 -curves, like the Palais-Smale condition (see Appendix B), which is an essential tool for developing an infinite dimensional Morse Theory.
The problem of duality in Hilbert spaces of curves with high regularity is faced through the introduction of a suitable formalism based on the theory of distribution and generalized functions, whose technical details are worked out at the beginning of Section 3. Unavoidably, the results needed are stated and proven in a formal way, and this part of the paper turns out to be rather technical nature. Even though these results are essential from a formal point of view, the reader should not be intimidated by Proposition 3.1 and the few subsequent Lemmas, and should keep his/her attention to the main issue of the paper.
As to the problem of lack of compactness for the travel time functional, the crucial observation here is that, if one is only interested in a local differentiable structure, then around each smooth (C 2 ) curve σ in B p,γ (k) it can be defined a differentiable chart on the set of H 1 -curves that are uniformly close to σ (see Proposition 2.8). Since the solution to our variational problem are proven to be curves of class C 2 , then one can relax the requirement of convergence for the Palais-Smale sequences, which allows to prove the global Morse relations for the arrival time functional (Section 8).
The paper is organized according to the following outline. In Section 2 we discuss the variational setup, where we define our main function spaces and functionals, proving their differentiability in the setting of infinite dimensional Hilbertian manifolds.
In Section 3 we present a Lagrange multiplier approach to the brachistochrone problem, and we derive some conditions on the curves that are extrema for the travel time functional and their corresponding multipliers. Moreover, we obtain a differential equation that is satisfied by the brachistochrones. Section 4 is devoted to the proof of the variational principle for brachistochrones, that extends the principle proven in [9] for local minimizers of the travel time. We also prove that the differential equation determined in Section 3 is the equation obtained by the above variational principle.
In Section 5 we study the second variation of the travel time T at a given brachistochrone. We prove a second order variational principle for brachistochrones, that relates the Hessian H T to the Hessian of the energy functional of a suitable Riemannian metric on M.
In Section 6 we recall some known facts about the Morse Index Theorem for orthogonal geodesics between submanifolds in Riemannian geometry, and we prove a slightly different version of the theorem for the case of a manifold admitting a Killing vector field. This result (Theorem 6.9), which has some interest on its own and for this reason it is stated in a general form, is then used in the next section to prove a brachistochrone version of the Morse Index Theorem.
In Section 7, in analogy with the classical Morse theory for Riemannian geodesics, we define the notions of Jacobi fields and focal points along a brachistochrone, and we prove a version of the Morse Index Theorem for brachistochrones. Some immediate consequence of the theory concerning the local nature of the critical points of T are derived. Section 8 is dedicated to the proof of the global Morse relations, from which we obtain some results on the multiplicity of brachistochrones with a given value of the energy between an event and an observer.
Finally, the paper has two short appendices containing some side results. In Appendix A we show the explicit calculation of the second variation of the travel time functional at a given brachistochrone. In Appendix B we discuss a simple but instructive example to show that the travel time functional does not satisfy the Palais-Smale condition in the space of curves satisfying an H 2 -regularity condition.

The Functional Spaces and the Variational Setup
Throughout this paper we will denote by (M, g) a stationary Lorentzian manifold, with g a Lorentzian metric tensor on M, and Y is a smooth timelike Killing vector field on M, which is assumed to be complete.
The symbol ·, · will denote the bilinear form induced by g on the tangent spaces of M; the usual nabla symbol ∇ will denote the covariant derivative relative to the Levi-Civita connection of g. Given a smooth function φ on M, for q ∈ M we denote by ∇φ(q) the gradient of φ at q with respect to g, which is the vector in T q M defined by ∇φ(q), · = dφ(q)[ · ]; the Hessian H φ (q) of φ at q is the symmetric bilinear form on T q M given by We denote by ψ : M×IR −→ M the flow of Y , i.e., for q ∈ M and t ∈ IR, ψ(q, t) is the value γ q (t), where γ q is the maximal integral line of Y satisfying γ q (0) = q. The Killing property of Y , which is crucial in most of the results presented in this paper, will be used systematically in our computations through the following three facts: 1. the quantity Y, Y is constant along the flow lines of Y , 2. the differential d x ψ(q, t 0 ) : T q M −→ T ψ(q,t0) M of the map ψ(·, t 0 ) is an isometry for all t 0 , or, equivalently, for all t 0 the map q −→ ψ(q, t) is a local isometry of M; 3. ∇ v Y, w = − ∇ w Y, v for all pair of vectors v and w; in particular, for all v ∈ T M, we have ∇ v Y, v = 0. Observe that the second or the third condition above is in fact equivalent to the Killing property of Y (see [19,Proposition 9.25]).
We set: the physical interesting case is m = 4. We denote by R the curvature tensor of the Lorentzian metric g, with the following sign convention: for X, Y vector fields on M.
As customary, for 1 ≤ p ≤ +∞, L p ([0, 1], IR) will denote the space of Lebesgue p-integrable real functions; for n ∈ IN , H n ([0, 1], IR) will denote the Sobolev space of functions of class C n−1 and having weak n-th derivative in L 2 ([0, 1], IR). We introduce for convenience the auxiliary Riemannian metric g R on M, given by: for q ∈ M and v 1 , v 2 ∈ T q M. It is easy to see that Y is Killing also in the metric g R ; moreover, the restriction of g and g R on the orthocomplement of Y coincide.
We define the space L 2 ([0, 1], T M) of square integrable T M-valued functions: Let π : T M −→ M be the canonical projection. Given any curve σ : I ⊆ IR −→ A, a vector field along σ is a map ζ : For A ⊆ M, the symbol C 1 ([0, 1], A) will denote the set of C 1 -curves defined [0, 1] and with image in A; we also define the Sobolev space H 2 ([0, 1], A) as: :σ is absolutely continuous, and It is not too difficult to prove that the definition of the spaces H i ([0, 1], A) does not indeed depend on the choice of the Riemannian metric g R , nor on the choice of the linear connection ∇ that appears in (2.5). As a matter of fact, the spaces H i ([0, 1], A) can be defined intrinsically for any differentiable manifold A using local charts (see [20]) or, equivalently, using auxiliary structures on A, like for instance a Riemannian metric. In the sequel, we will use the spaces The inner product in T σ H 1 ([0, 1], A) is given by: while the inner product in T σ H 2 ([0, 1], A) is given by: Let k be a fixed positive constant, with −k 2 < sup M Y (q), Y (q) , and U k be the open set: Since Y is Killing, the quantity Y, Y is constant along the integral lines of Y , hence U k is invariant by the flow of Y .
We will denote by p a fixed event of U k and by γ : IR −→ U k a given integral line of Y which does not pass through p. We introduce the spaces p,γ is given by: p,γ is a Hilbert space with respect to the inner products: in the case of T w Ω (1) p,γ and for T w Ω (2) p,γ . Observe that, since ζ(0) = 0 for all ζ ∈ T w Ω (i) p,γ , then the inner products ·, · * and ·, · * * of formulas (2.7) and (2.8) are equivalent in T w Ω (2) p,γ , respectively, to the products ·, · 1 and ·, · 2 of formulas (2.12) and (2.13).
We consider the action functional F on Ω (i) p,γ , given by: It is well known that F is smooth; for σ ∈ Ω (i) p,γ and V ∈ T σ Ω (i) p,γ , the Gateaux derivative dF (σ)[V ] is given by: Finally, for all positive constant k ∈ IR + , we introduce the spaces B (i) p,γ (k), i = 1, 2, by: The main goal of this section is to establish an infinite dimensional differentiable structure on the sets B (2) p,γ (k) and B (1) p,γ (k). The case of B (2) p,γ (k) is easier, and its regularity is proven in the next Proposition. For the set B (1) p,γ (k), we are only able to establish its regularity around some special points; this second case is treated at the end of this section. Proposition 2.1. B (2) p,γ (k) is a smooth submanifold of Ω (2) p,γ . For σ ∈ B (2) p,γ (k), the tangent space T σ B (2) p,γ (k) can be identified with the Hilbert space: T σ B (2) p,γ (k) = ζ ∈ T σ Ω (2) p,γ : ∃ C ζ ∈ IR such that ∇σζ, Y − ζ, ∇σY ≡ C ζ and ∇σζ,σ ≡ T σ C ζ k , endowed with the inner product ·, · 2 of formula (2.13).
Proof. For σ ∈ Ω (2) p,γ , the maps σ, Y , σ, Y 2 and σ,σ are in H 1 ([0, 1], IR). Let k ∈ IR + be a fixed constant. We consider the following map: given by: It is not difficult to prove that F is a smooth map and that, for σ ∈ Ω (2) p,γ and V ∈ T σ Ω (2) p,γ , the Gateaux derivative dF (σ)[V ] is given by: Here we have used the fact that Y is Killing, thus σ, Let C denote the subspace of H 1 ([0, 1], IR) given by the constant functions, and let C − denote the open submanifold of C consisting of negative functions:

(2.22)
It is easy to see that B (2) p, given by the quotient map on the first factor and the identity on the second factor.
To prove the Proposition we use the Inverse Mapping Theorem (see [15]). According to this Theorem, B (2) p,γ (k) is a smooth submanifold of Ω (2) p,γ provided that the map F be transversal over C − × {0}, i.e., if for all σ ∈ B (2) p,γ (k) the composite map: is surjective. This amounts to saying that, for all h 1 , h 2 ∈ H 1 ([0, 1], IR) there exists a constant c ∈ IR such that the system of differential equations: has at least one solution V ∈ T σ Ω (2) p,γ . Using the fact that σ, Y ≡ −kT σ , we can rewrite (2.26) as: To prove the existence of such a vector field Z, consider first the vector field along σ given bẏ σ ⊥ , which is the orthogonal projection ofσ onto the distribution ∆ = Y ⊥ orthogonal to Y . Formally, we have:σ Obviously, we have: Observe thatσ ⊥ ∈ H 1 , and it does not have the required H 2 -regularity. Now, let Z be any section of class H 2 of ∆ which is uniformly close toσ ⊥ , in such a way that Z,σ = 0 as well. 1 Observe in particular that, since Z,σ is continuous, then Z,σ −1 is in L ∞ ([0, 1], IR).
In some parts of the paper (see Section 7) we will need to consider variations of curves in B (2) p,γ (k) by curves σ satisfying the conditions (1.6) and (1.7), but not necessarily with endpoints in p and γ. For this reason, for i = 1, 2 we introduce the sets: where the unions in (2.39) are taken over all γ's that are integral lines of Y having image in U k .
After setting up our variational framework, we are ready to give the following definition: Definition 2.3. A brachistochrone of energy k between p and γ is a stationary point for the travel time functional T on B (2) p,γ (k). A brachistochrone curve σ is said to be minimal if σ is a minimum point for T on B (2) p,γ (k). From Corollary 2.2 and Definition 2.3 we obtain immediately: (2) p,γ (k) is a brachistochrone of energy k between p and γ if and only if for every ζ ∈ T σ B (2) p,γ (k) it is C ζ = ∇σζ, Y − ζ, ∇σY = 0. Remark 2.5. Observe that the definition of brachistochrone of energy k given in Definition 2.3 is different from the one given in ( [9], Definition 1.1) and used in Reference [7]. Namely, in these articles, it was not established a differentiable structure in the set of admissible curves for the variational problem, and the brachistochrones of energy k were defined as curves locally minimizing their travel time. The equivalence of the two definitions will be given in Section 4, where we prove that the two approaches lead to exactly the same solutions.
Remark 2.6. Since T is strictly positive on B (2) p,γ (k), then its critical points coincide with the critical points in B (2) p,γ (k) of the restriction of the action functional F = − 1 2 T 2 . The minimal brachistochrones of energy k are maximum points of F on B (2) p,γ (k). Remark 2.7. The proof of the regularity of the space B (2) p,γ (k) presented in Proposition 2.1 does not apply to the space B (1) p,γ (k); more precisely, the failure of the proof is in the existence of the vector field Z satisfying (2.28). Observe indeed that, in the case of B (1) p,γ (k), the derivativė σ is only defined as an L 2 -function, and in general it is not a continuous curve.
This fact is the reason why we have to introduce here our global variational setup using the space B (2) p,γ (k). However, the same proof of Proposition 2.1 can be adapted to prove that, if σ is C 1 , then a suitable neighborhood of σ in B (1) p,γ (k) has the structure of a smooth Hilbert manifold. The proof is more delicate; we omit the details that can be found in a forthcoming paper (see [8]).
As a matter of facts, we will see that the the solutions to our variational problem as given in Section 4 are indeed smooth curves (see Proposition 4.1). This fact will allow us to to work in the space B (1) p,γ (k) in the second part of the paper (starting from Section 4), when we will be studying the local properties of the brachistochrones, i.e., the properties of objects that are defined only around the brachistochrones, like for instance the second variation of T , the Jacobi fields, conjugate points and the Morse Index Theorem for brachistochrones (Theorem 7.12).
We summarize the main properties of the set B (1) p,γ (k) as follows: Proposition 2.8. B (1) p,γ (k) is a metric space with the metric induced by H 1 ([0, 1], M). The inclusion ι : B (2) p,γ (k) −→ B (1) p,γ (k) is continuous and it has dense image. If σ ∈ B (1) p,γ (k) is a map of class C 1 , then there exists a neighborhood V σ of σ in B (1) p,γ (k) that has the structure of an infinite dimensional Hilbertian manifold. In particular, B (1) p,γ (k) has a dense open subset that is a smooth Hilbert manifold.
If σ ∈ B (1) p,γ (k) is a curve of class C 1 , then, for all σ 1 ∈ V σ , the tangent space T σ1 V σ can be identified with the Hilbert subspace of T σ1 Ω (1) p,γ given by: (2.44) The restriction of the travel time functional T to each neighborhood of the form V σ , for some σ ∈ B (1) p,γ (k) of class C 1 , is smooth, and the same result of Corollary 2.2 holds. Proof. Convergence in the space B (2) p,γ (k) clearly implies the convergence in B (1) p,γ (k), which implies that the inclusion ι : B (2) p,γ (k) −→ B (1) p,γ (k) is continuous. For the second part of the thesis, it suffices to adapt the proof of Proposition 2.1, and the details will be omitted.
We can give the following definition: has the structure of a smooth Hilbert manifold in a neighborhood V σ of σ. By Proposition 2.8, every curve σ ∈ B (1) p,γ (k) of class C 1 is a regular point of To conclude this section, we remark that, in perfect analogy with Proposition 2.8, if σ is a regular point in B (1) p (k) or in B (1) (k), then these two sets have the structure of smooth manifolds around σ. Their tangent spaces are given by: (2.45) and 3. An Abstract Approach to the Lagrange Multiplier Method.
The First Variation of the Travel Time In this Section we use the Lagrange multiplier technique to derive a system of differential equation satisfied by the brachistochrones, and to extend the variational principle proven in [9].
In order to use this technique, we need a global Banach differentiable structure on our set of maps, and for this reason we will work in the space B (2) p,γ (k) rather than B (1) p,γ (k) (see Remark 2.7 and Proposition 2.8). This approach has the unpleasant drawback of making our notations and calculations much heavier then one would expect. This is due to the fact that the duality in the Sobolev spaces H 1 andH 1 , which are the natural images for the map F defined by (2.19), involves also products of the first derivatives of the maps and of the Lagrange multipliers, resulting in very lengthy formulas that make it a complicated task to determine an explicit form of the Euler-Lagrange equation satisfied by the critical points of our functional.
To overcome this difficulty, the authors have decided to use the formalism of generalized functions and distributions on Sobolev spaces, which will make the computations formally similar to the naive calculations made by the classical variationalists of the last century. Unfortunately, our problem does not fit perfectly in the theory of distributions on Sobolev spaces presented in standard textbooks, and we are forced to develop our own theory from scratch. Hopefully, the formalism developed here will be adaptable to the study of other variational problems involving several constraints and requiring a high degree of regularity for the trial maps.
The first part of this section is devoted to this aim, and it is of rather technical nature. A first time reader who wants to avoid technicalities and who is willing to make an act of faith, after reading formula (3.2) can just skip everything that comes before formula (3.10) without seriously jeopardizing his/her general understanding of the subject.
Keeping Remark 2.6 in mind, in the notation of Section 2 (recall in particular formulas (2.20), (2.22) and (2.23), we want to extremize the action functional F (σ) = 1 Then, σ ∈ B (2) p,γ (k) is a solution to our variational problem if and only if there exists an element Λ in the dual space vanishes identically. In this case, Λ is unique, and it is the Lagrange multiplier of σ; from a physical point of view, Λ represents the constraint forces acting on the particle moving along σ.
A Lagrangian multiplier for our variational problem is of the form Λ = (λ, µ), where λ ∈ where λ and µ are seen as generalized functions. Indeed, we know that not all the continuous functionals on the spaces H 1 andH 1 are of the form (3.2) for some function λ and µ.
This argument introduces a substantial simplification in the computations involving dual spaces, but it needs a concrete formalization, which is rather technical and it is done once and for all in the following.
We introduce the following formalism.
We will consider in particular the tangent bundle T M and the cotangent bundle T M * with their canonical projections onto M.
Finally, we denote by D i σ the dual space H i (I, σ, T M * ) * . We remark that there are canonical inclusions D i ⊂ D i+1 and D i σ ⊂ D i+1 σ given by restriction of the linear functionals. By convention, keeping in mind the Riesz representation theorem for Hilbert spaces, we define H 0 = D 0 = L 2 (I, IR) and D 0 σ = L 2 (I, σ, T M). We consider the following operations in the spaces D i , D i σ and H i : Observe that this product is well defined because, H i (I, IR) is an algebra (i.e., closed with respect to products) and the product (f, φ) −→ f φ is continuous in H i (I, IR). The validity of the operations defined in the other items is checked by similar arguments.
is linear and continuous. Note also that λ is a sort of distributional derivative, but keeping in mind that λ is an element of a dual space of functions which do not vanish on the boundary. In particular, even for differentiable functions λ, it is not true that λ = λ ′ . An explicit form ofλ is given in part 10 of Proposition 3.1.
Part 2 is simply the fact that elements in the dual space ofH 1 (I, IR) are characterized by the property of vanishing on constant functions.
For part 3, it suffices to observe that the map For part 4, using (d) and (e), we have: On the other hand, by (e) and (g) we have: which proves the claim. For part 5, observe thatλ is a well defined element in the dual of H i (I, IR). The linearity of the map λ −→λ is trivial; the continuity depends on the fact that convergence in H 1 implies uniform convergence. Finally, the injectivity is simply the Fundamental Theorem of Calculus of Variations. Part 6 is proven analogously. Namely, using an orthonormal frame along σ, one reduces the problem to the case M = IR m . In this case the proof of part 5 can be repeated verbatim for each component of ν.
Using part 5 and 6, we will identify each λ ∈ L 1 (I, IR) withλ ∈ D i and every V ∈ L 1 (I, σ, T M) withV ∈ D i σ . Suppressing the symbolˆ, for all λ ∈ L 1 (I, IR), V ∈ L 1 (I, σ, T M), f ∈ H i (I, IR) and α ∈ H i (I, σ, T M * ) we will write concisely: gives a continuous linear functional on L 2 (I, IR), and the conclusion follows by Riesz Theorem. The second half is proven similarly.
The formulas in part 8 and 9 are the product rules for the˜-derivative. We prove 9 as follows. For α ∈ H i+1 (I, σ, T M * ), we have: On the other hand, we compute: Part 8 is proven similarly. We omit the proof of part 10 and we prove part 11. For α ∈ H i+1 (I, σ, T M * ), using (3.4), we have: This concludes the proof of Proposition 3.1.
The following simple result states the well known fact that the Dirac delta's are not given by any L 1 -function: The proof of the second part of the Lemma is analogous.
We are now ready to determine the Euler-Lagrange equation for the critical points of the travel time functional in B (2) p,γ (k). Recalling (2.21), (3.1) and part 2 of Proposition 3.1, we now fix a curve σ ∈ B (2) p,γ (k). Recall from the definition (2.16) of B (2) p,γ (k) that there exists T σ > 0 such that: We assume that there exist λ, µ ∈ D 1 (see part 2 of Proposition 3.1), with I λ = 0, such that the equation: is satisfied for all V ∈ T σ Ω (2) p,γ . Using the formalism introduced in the first part of the Section, we rewrite equation (3.6) as: In the above formula, the "products" between the dual maps λ and µ with functions or vector fields along σ have to be interpreted in the sense of the operations (a)-(c) above; moreover, the inner product ·, · in (3.7) and (3.8) is meant in the sense of (d).
We use the operation (f) to "integrate by parts" (3.8), and, keeping in mind parts 8 and 9 of Proposition 3.1, we obtain p,γ . Observe that, when using parts 8 and 9 of Proposition 3.1, if the functions involved are only in H 1 (like in this particular case the functionσ) then they must be multiplied by distributions in D 0 = L 2 for the rule to apply. This is where we use Lemma 3.5.
Hence, by Lemma 3.4, the first member of the equality (3.20) is null, and also A 1 = B 1 = 0. Therefore, we obtain the following differential equation for σ: We have proven the following: Proposition 3.6. Let σ ∈ B (2) p,γ (k). Then, σ is a brachistochrone of energy k between p and γ if and only if σ is a curve of class C 2 and there exists T σ > 0 such that σ satisfies the differential equation (3.21).
Observe that any curve σ in H 2 (I, M) that satisfies (3.21) almost everywhere is automatically smooth.
Besides determining the differential equation (3.21), the importance of Proposition 3.6 lies in the fact that, due to the smoothness of the brachistochrones,we will be able to work in the space B (1) p,γ (k) when we are in the vicinity of such a curve (recall Remark 2.7 and Proposition 2.8). This will be done systematically starting from the next Section.
Recalling Definition 2.9, we have the following:

The Brachistochrone Differential Equation and the First Order Variational Principle Revisited
In this section we will take a closer look at the differential equation (3.21) and we will prove that it characterizes the brachistochrones between p and γ among all the curves in Ω (1) p,γ satisfying suitable initial conditions. Proposition 3.6 can be improved as follows: p,γ is a brachistochrone of energy k between p and γ if and only if σ is smooth and there exists a T σ > 0 such that σ satisfies (3.21), with initial velocitẏ σ(0) satisfying: Proof. From Proposition 3.6, all we need to prove is that any smooth curve σ ∈ Ω (1) p,γ that satisfies the differential equation (3.21) and whose initial velocityσ(0) satisfies (4.1) is in B (1) p,γ (k).
We give two more different descriptions of the brachistochrone curves. We first characterize them as curves that minimize locally their travel time.
If q is any point in U k , we denote by γ q the maximal integral line of Y through q. Moreover, if I = [a, b] ⊆ [0, 1] is any interval, and if q 1 , q 2 are any two points in U k , we define B (1) q1,γq 2 (k, I) as the space of curves τ ∈ H 1 (I, U k ) such that τ (a) = q 1 , τ (b) ∈ γ q2 (IR), and satisfying τ , Y ≡ −k T τ , τ ,τ ≡ −T 2 τ for some T τ ∈ IR + . Observe that if σ ∈ B (1) p,γ (k), then, for every I = [a, b] ⊆ [0, 1], the restriction of σ to I is a curve in B (1) σ(a),γ σ(b) (k, I). Note that Definition 4.2 is essentially the definition of brachistochrones of energy k given in [9]. For curves that are local minimizers of the travel time, the differential equation (3.21) was established in [9] by means of a variational principle, that we can now state in a more complete form.
We denote by ∆ the smooth distribution on M given by the orthocomplement of the vector field Y . Observe that, since Y is timelike, the wrong way Schwartz's inequality implies that ∆ is spacelike, i.e., the restriction of the Lorentzian metric g on ∆ is positive definite.
Let ψ : M × IR −→ M be the flow of Y . Recall that, since Y is Killing, then ψ(·, t) is a local isometry for all t ∈ IR; moreover, it is easy to see that the distribution ∆ is ψ-invariant, which means that ψ x (q, t 0 )(∆ q ) = ∆ ψ(q,t0) , where ψ x (q, t 0 ) denotes the differential of the map ψ(·, t 0 ) at the point q. A function φ : M −→ IR is said to be Y -invariant if it is constant along the flow lines of Y ; if φ is C 1 , this amounts to saying that Y, ∇φ ≡ 0.
We define Ω (1) p,γ (∆) to be the subset of Ω (1) p,γ consisting of curves with tangent vector at each point lying in ∆: Using the language of sub-Riemannian geometry, we will call horizontal the curves in Ω (1) p,γ (∆). By the same arguments of Proposition 2.1, one checks immediately that, since Y, Y is never vanishing, Ω (1) p,γ (∆) is a smooth submanifold of Ω (1) p,γ , and that, for w ∈ Ω (1) p,γ (∆), the tangent space T w Ω (1) p,γ (∆) is given by: It will also be useful, as in the case of the spaces B (2) p,γ (k) and B (2) p (k) (see formula (2.39), to introduce the spaces Ω (1) p and Ω (1) p (∆), by: We single out the following simple fact: on Ω (1) p,γ and its restriction to Ω (1) p,γ (∆) have the same critical points. These critical points are geodesics in M with respect to the Riemannian metric φ · g R that join p and γ and that are orthogonal to γ.
Proof. The critical points of E φ in Ω (1) p,γ are precisely the geodesics in M with respect to φ · g R that join p and γ and that are orthogonal to γ, i.e., ẇ(1), Y (w(1)) (R) = 0. Since φ is Yinvariant, then Y is Killing in the metric φ · g R , thus, for every such geodesic w, the quantity ẇ, Y (R) = is constant. Hence ẇ, Y (R) ≡ 0 and w is horizontal. Therefore, the critical points of E φ on Ω (1) p,γ belong to Ω (1) p,γ (∆), and clearly they are critical points of the restriction of E φ to Ω (1) p,γ (∆). Conversely, if w is a critical point of the restriction of E φ to Ω (1) p,γ (∆), then the Gateaux derivative dE φ (w)[V ] vanishes for all V ∈ T w Ω (1) p,γ (∆). Let's define: Since Y is Killing in the metric g R , an easy calculation shows that for all w ∈ Ω (1) p,γ , the Gateaux derivative dE φ (w)[V ] vanishes for all V ∈ T w .
Moreover, for all w ∈ Ω (1) p,γ (∆) it is (see [9]): This concludes the proof. The functional E φ of (4.7) is called the energy functional relative to the metric φ · g R . The critical points of E φ in Ω (1) p,γ (or equivalently in Ω (1) p,γ (∆), see [9]) will be called horizontal geodesics between p and γ with respect to the Riemannian metric φ · g R .
In order to state properly our variational principle, we introduce an operator D that deforms curves in Ω (2) p,γ into horizontal curves using the flow of Y . Let D be the map: (4.10) and r σ is the unique solution on [0, 1] of the Cauchy problem: Using the Killing property of Y it is easily checked that D is well defined, i.e., the maximal solution of (4.11) is defined on the entire interval [0, 1] and the corresponding curve w given by (4.10) is horizontal. Namely, using the fact that the differential d x ψ is an isometry, we compute easily: Observe that, if σ ∈ B (1) p,γ (k), then (4.11) gives: In Section 7 we will need to use the differential dD of D on brachistochrones; the differentiability of D and a formula for dD is established in the next: Proposition 4.4. The map D is smooth around the regular points of B (1) p,γ (k). If σ is a curve of class C 1 in B (1) p,γ (k) and ζ ∈ T σ B (1) p,γ (k), the Gateaux derivative dD(σ)[ζ] is given by: 14) where τ ζ : [0, 1] −→ IR is the function: where C ζ is the constant ∇σζ, Y − ζ, ∇σY . In particular, if σ is a brachistochrone, then τ ζ takes the following form: Proof. The smooth dependence on σ of the solution r σ of (4.11) proves that D is a smooth map. Formulas (4.14), (4.15) and (4.16) are easily obtained by differentiating (4.10) using (2.42), and keeping in mind that d x ψ(σ, r σ )[Y (σ)] = Y (ψ(σ, r σ )). In particular, formula (4.16) follows immediately from (4.15) and Corollary 2.4.
Observe that formula (4.10) allows to extend the definition of the map D to the space B (1) p (k) and with values in Ω (1) p ; these spaces were defined in (2.39) and (4.6). Obviously, Proposition 4.4 remains true for the extension. Now everything is ready to state and prove the following:

Proposition 4.5 (First Variational Principle for Brachistochrones).
Let σ ∈ B (1) p,γ (k) be fixed. The following are equivalent: 1. σ is a brachistochrone of energy k between p and γ; 2. σ is a local minimizer for the travel time; 3. w = D(σ) ∈ Ω (1) p,γ (∆) is a horizontal geodesic between p and γ with respect to the Riemannian metric φ k · g R , where: Moreover, if one of the conditions above is satisfied, then E φ k (w) = 1 2 T σ 2 , where E φ k is the energy functional relative to the metric φ k · g R , given by: The equivalence of condition 2 and 3 is based on the fact that, for σ ∈ B (1) p,γ (k) and w = D(σ), using (3.5), (4.10) and (4.11), one computes easily: Here we have used the facts that Y, Y is constant along the flow lines of Y , that ψ(·, t 0 ) is an isometry for all t 0 ∈ IR and the conservation law of the energy of the Riemannian geodesics. Observe that, since Y is Killing in the metric φ k · g R , then a critical point of E φ k in Ω (1) p,γ (∆) is indeed a geodesic with respect to φ k · g R (see [9]). It follows that the quantity φ k (w) ẇ,ẇ is constant along each horizontal geodesic w.
Recalling (2.43), integrating formula (4.19) yields: From (4.19) it follows that σ is a local minimizer for the travel time if and only if w is a local minimizer for the energy functional E φ k in Ω (1) p,γ (∆), i.e., if and only if w is a horizontal geodesic between p and γ with respect to φ k · g R .
The last statement of the thesis follows easily by integrating (4.19) over [0, 1].
The result of Proposition 4.5 remains true for brachistochrones and horizontal geodesics with free endpoints in U k . The correct statement of this fact is obtained by replacing the spaces B (1) p,γ (k) and Ω (1) p,γ (∆) respectively with B (1) p (k) and Ω (1) p (∆), which were defined in formulas (2.39) and (4.6).

The Second Variation of the Travel Time
In this section we want to investigate the problem of whether a given stationary point σ in B (1) p,γ (k) for the travel time functional is a local minimum, maximum or a saddle point. To this aim, we need a second order variational formula for our variational problem.
In the first part of the Section we will discuss the abstract problem of relating the Hessians of smooth functions on Banach manifolds that are intertwined by a Banach manifold morphism; then we use the first part to determine the relation between the Hessian of the travel time T and the Hessian of the Riemannian action E φ k .
Let M be a Banach manifold and f : M −→ IR be a smooth map. If x 0 ∈ M is a critical point for f , i.e., df (x 0 ) = 0, then it makes sense to define the Hessian of f at x 0 , denoted by H f (x 0 ), which is a continuous symmetric bilinear form on T x0 M , in the following way. Choose a coordinate system around x 0 , φ : U ⊂ M −→ U 0 ⊂ E, where E is some Banach space. Define: for v, w ∈ T x0 M . Using the fact that x 0 is critical for f , it is easy to see that this definition will not depend on the chart (U, φ). Indeed, is is easily seen that for every smooth curve s −→ y s ∈ M such that y 0 = x 0 and y ′ 0 = v ∈ T x0 M , we have: Proof. Since both sides of (5.3) are symmetric, it suffices to prove the equality in the case v = w. Let y(s), s ∈ ] − ε, ε [ be a smooth curve in M such that y(0) = x 0 and y ′ (0) = v. Then, clearly,ỹ = D • y is a smooth curve in N such thatỹ(0) = D(x 0 ) andỹ ′ (0) = dD(x 0 )[v]. Using (5.2), we have: Let σ ∈ B (1) p,γ (k) be a brachistochrone and w = D(σ). Then, for all ζ 1 , ζ 2 ∈ T σ B (1) p,γ (k), we have: From (2.43) and (5.2) we obtain easily: for all brachistochrone σ ∈ B (1) p,γ (k). From (5.4) and (5.5) we obtain also: for all brachistochrone σ ∈ B (1) p,γ (k) and all ζ 1 , ζ 2 ∈ T σ B (1) p,γ (k).

The Riemannian Morse Index Theorem
For Riemannian geodesics, the classical Morse Index Theorem (see References [1,3,12,18] for the different versions of this Theorem) relates the index of the action functional with some geometrical properties of the geodesic. The main ingredients for the theory are given by the curvature tensor of the metric and the concepts of Jacobi fields and conjugate or focal points along a geodesic.
In view to applications to the brachistochrone problem, in this section we quickly review some known results about the Morse Index Theorem for Riemannian geodesics joining a curve with a point, as presented, for instance, in [12]. Then, we prove a different version of this theorem in the case of an orthogonal geodesic between the integral line of a Killing vector field and a point.
In order to simplify the formulas, in this section we interchange the role of p and γ, that is, we consider curves starting at the curve γ and ending at the point p. Clearly, the final results (Theorems 6.8 and 6.9) will not be affected by this change. Moreover, all the results and the formulas of the previous sections remain true after changing the variable t with 1 − t in the interval [0, 1], and, in particular, the role of the endpoints t = 0 and t = 1 will be interchanged. To avoid confusion, in this section we will use the symbols Ω (1) γ,p and Ω (1) γ,p (∆) to indicate the spaces of curves in U k of class H 1 from γ to p. If we denote by O the direction reversing map for curves w :  ) is smooth, and its differential is formally given by: Observe also that the energy functional E φ k can be defined in Ω (1) γ,p by the same formula (4.18); obviously, a curve w is a critical point for E φ k in Ω (1) p,γ if and only if O(w) is a critical point for E φ k in Ω (1) γ,p . In this case, we have: By Lemma 4.3, we know that the critical points of the Riemannian energy functional E φ k corresponding to the metric φ k · g R on the spaces Ω (1) γ,p and Ω (1) γ,p (∆) are the same. However, given a horizontal geodesic w between p and γ, the Morse index of E φ k at w (see Definition 6.5) in the Hilbert manifold Ω (1) γ,p (∆) may be strictly less then the Morse index of E φ k at w in the manifold Ω (1) γ,p . The purpose of this section is to prove that the two indices are indeed equal; we accomplish this result by proving an index theorem for the Morse index m(w, E φ k ) restricted the space T w Ω (1) γ,p (∆) ⊥ , defined by: Observe that T w Ω (1) γ,p (∆) ⊥ is a (closed) Hilbert subspace of T w Ω (1) γ,p (∆); moreover, if w is a horizontal geodesic with respect to the metric φ k · g R in Ω (1) γ,p , then, for a vector field V ∈ T w Ω (1) γ,p (∆) we have: where ∇ {k} is the covariant derivative of the Levi-Civita connection of the Riemannian metric On the other hand, if 0 , the quantity φ k (w) · V,ẇ (R) is constant and (6.4) is proven. In particular, since V (1) = 0 (recall that we are considering curves ending at the fixed point p), if w is a horizontal geodesic and V ∈ T w Ω (1) γ,p (∆) ⊥ , then C V = 0. Hence, a vector field V ∈ T w Ω (1) γ,p (∆) belongs to T w Ω (1) γ,p (∆) ⊥ if and only if it is everywhere perpendicular to w, which is the reason for the notation.
Remark 6.1. From (6.4) it is easy to see that, if we think of the elements in T w Ω (1) γ,p as variational vector fields relative to variations w s of the horizontal geodesic w, then the condition V ∈ T w Ω (1) γ,p (∆) ⊥ means that, up to infinitesimals of order larger than 1, the curves w s are horizontal, and they are parameterized by a constant multiple of arclength: We can easily write (6.4) in terms of the Lorentzian structure, by differentiating the above expression using the Lorentzian covariant derivative. Given a horizontal geodesic w and V ∈ T w Ω (1) γ,p (∆), we have that V ∈ T w Ω (1) γ,p (∆) ⊥ if and only if the following equation holds: ∇φ k (w), V · ẇ,ẇ + 2 φ k (w) · ∇ẇV,ẇ = 0. (6.5) We recall the basic facts concerning the Morse Index Theorem for Riemannian geodesics between a point and a curve, as it is presented, for instance, in Ref. [12].
Given a horizontal geodesic w in between p and γ with respect to the Riemannian metric φ k · g R , let ∇ {k} and R {k} denote respectively the Levi-Civita connection and the curvature tensor (chosen with the same sign convention as in (2.1)) of the metric φ k · g R , and let J {k} w be the finite dimensional vector space of all the Jacobi fields J along w with respect to φ k · g R , i.e., all smooth vector fields satisfying the second order differential equation: We recall that, in analogy with the Riemannian case, given a submanifold Σ of M whose tangent bundle T Σ is non degenerate, i.e., the restriction of g to the tangent space T q Σ is non degenerate for all q ∈ Σ, one can define the second fundamental form S Σ (also known as the shape tensor of Σ) as follows. For each q ∈ Σ and each vector n ∈ T q Σ ⊥ , the second fundamental form of Σ in the direction of n is the bilinear form S Σ n : T q Σ × T q Σ −→ IR defined by: where V 2 is any smooth vector field on Σ that takes value v 2 at q. One can show that S Σ n is well defined (i.e., formula (6.7) does not indeed depend on the choice of the extension V 2 of v 2 ), and it is symmetric (see for instance [2] and [19]).
In the following, we will denote by S γ the second fundamental form of the timelike submanifold γ(IR) of M.
A point w(t 0 ) along w is said to be a γ-focal point if dim(J {k} w (t 0 )) > 0; the multiplicity of the a γ-focal point w(t 0 ) is the dimension of J {k} w (t 0 ) (which is clearly finite, because the Jacobi fields are solutions of a second order linear system of differential equations).
Remark 6.2. It is well known that the set of γ-focal points along every Riemannian geodesic is discrete, hence there is only a finite number of γ-focal points along each compact portion of a geodesic. For the reader's convenience, we sketch a simple proof of this fact based on [19,Ex. 8,p. 299]). The set of γ-Jacobi field along a given geodesic w has dimension equal to m = dim(M ). If J 1 , J 2 , . . . , J m is a family of linearly independent γ-Jacobi fields and E 1 , E 2 , . . . , E m is a parallely transported orthonormal basis along w, then one considers the smooth function g(t) = det( J i (t), E j (t) ). Using elementary arguments, one proves that t 0 is a zero of order d for g, i.e., g(t 0 ) = g ′ (t 0 ) = . . . g (d−1) (t 0 ) = 0 and g (d) (t 0 ) = 0, if and only if w(t 0 ) is a γ-focal point of multiplicity d. In particular, the set of γ-focal points is discrete, as is the set of simple zeroes of a smooth function. Equation (6.6) is obtained by linearizing the geodesic equation in the metric φ k ·g R ; hence, it is satisfied by vector fields along w that correspond to variations w s , s ∈ ]−ε, ε [ for some ε > 0, of w consisting of geodesics. Loosely speaking, the arrow-head of J traces out infinitesimally close neighboring geodesics to w.
The condition 1 means that, in a first order approximation, these geodesics start on γ; condition 3 means that they pass through w(t 0 ). Condition 2 means that these geodesics start orthogonally at γ; observe that orthogonality to the vector field Y is equivalent in the three metrics g, g R and φ k · g R , and for this reason it is possible to write this condition using the Lorentzian Levi-Civita connection ∇ and the Lorentzian second fundamental form S γ of γ. Using the Riemannian metric φ k · g R , condition 2 can also be written as: Observe that, since Y is Killing, we obtain easily that, if J satisfies the differential equation 6.6, then the condition ∇ẇJ, Y + ẇ, ∇ J Y = 0 is satisfied identically on [0, 1] provided that it is satisfied at one single point t 0 ∈ [0, 1]. Indeed, using the fact that Killing vector fields satisfy the Jacobi equation (see [19,Lemma 26,p. 252]), it is easy to see that the quantity ∇ẇJ, Y + ẇ, where the last equality follows easily from well known symmetry properties of the curvature tensor R.
From Remark 6.3 and formula (4.5), we obtain immediately the following characterization of the γ-Jacobi fields along a horizontal geodesic w: Lemma 6.4. Let w be a horizontal geodesic in Ω (1) γ,p (∆) and W a Jacobi field along w. Then, W is a γ-Jacobi field if and only if W ∈ T w Ω (1) γ,p (∆).
Given a horizontal geodesic w, we denote by I {k} the index form on T w Ω (1) γ,p , or more in general on T w H 1 ([0, 1], M), given by the symmetric bilinear form: The symmetry of I {k} follows easily from the symmetry properties of the curvature tensor R {k} ; moreover, from the fundamental Lemma of Calculus of Variations, a simple integration by parts in (6.9) shows that a vector field W along w is a Jacobi field if and only if On the other hand, suppose that X is a closed subspace of T x0 M and that the restriction of H f (x 0 ) to X is nondegenerate. Let X 1 be the orthogonal space to X relatively to the bilinear form H f (x 0 ), which is the closed subspace of T x0 M defined by: If w is a horizontal geodesic between p and γ with respect to the Riemannian metric φ k · g R , or equivalently, w is a critical point for E φ k in Ω (1) γ,p , then the Hessian H E φ k (w) is computed easily in terms of the metric φ k · g R as: Since V (0) is tangent to the curve γ andẇ(0) is orthogonal to γ, then the term is tensorial in V , i.e., it only depends on the value V (0). This is precisely the second fundamental form of the curve γ in the direction of the normal vectorẇ(0) with respect to the metric φ k · g R .
We can give a different expression of the Hessian H E φ k (w) in terms of the Lorentzian metric g. This is done by direct computation in the following: Proposition 6.7. Let w ∈ Ω (1) p,γ (∆) be a horizontal geodesic between p and γ with respect to the Riemannian metric φ k ·g R . Then, the Hessian H E φ k (w) is given by the following symmetric bilinear map on T w Ω (1) p,γ (∆): Proof. The geodesic equation for the metric φ k · g R is easily computed as the Euler-Lagrange equation for the functional E φ k , and it is given by: ∇ẇ φ k (w)ẇ = 1 2 ∇φ k (w) ẇ,ẇ . (6.14) In analogy with the proof of Proposition A.3, let V be a fixed vector field in T w Ω (1) p,γ (∆) and let w s denote a variation of w in Ω (1) p,γ (∆) such that V = d ds s=0 w s .
Then, we compute as follows: Using (6.14) and the commutation relations (A.5), we have: Keeping in mind that w s (0) ≡ p and arguing as in the proof of Proposition A.3 (see formula A.15), the boundary term in (6.16) can be computed as: Finally, we have: Let's now go back to the study of the second variation of E φ k in terms of the Riemannian metric φ k · g R . Using integration by parts in the Index formula (6.9), it is easy to see that the set of γ-Jacobi fields J {k} w (t 0 ) can be also described as the kernel of the Hessian H E φ k (w) restricted to the interval [0, t 0 ]; in particular: The geometric index µ {k} (w) of the horizontal geodesic w is defined as the natural number: Recall from Remark 6.2 that the number of γ-focal points along w is finite, hence the sum in (6.20) is finite.
The Morse Index Theorem says that, if p is not a γ-focal point along w, the Morse index m(w, E φ k ) of E φ k in the space T w Ω (1) γ,p is given by the number of γ-focal points along w, counted with multiplicity: Theorem 6.8. Let w be a critical point of E φ k in Ω (1) γ,p , i.e., a geodesic from γ to p in the metric φ k · g R that starts orthogonally to γ. Then, the Morse index m(w, E φ k ) is finite; moreover, if p is not a γ-focal point along w, we have: m(w, E φ k ) = µ {k} (w). (6.21) Theorem 6.8 is obtained as a special case of [12, The Index Theorem, p. 342]. Observe that Theorem 6.8 holds without any assumption that γ be the integral line of a Killing vector field.
LetẼ denote the energy functional of the metricg in the space Ω (1) γ,p ; let w be a critical point ofẼ in Ω (1) γ,p (or, equivalently, in Ω (1) γ,p (∆)), and let HẼ(w) be the Hessian ofẼ at w. Then, if p is not a γ-focal point along w, the three indices are equal: m(w, HẼ) = m(w, HẼ, T w Ω (1) γ,p (∆)) = m(w, HẼ, T w Ω (1) γ,p (∆) ⊥ ). (6.22) Proof. The condition that the Killing vector field Y is never vanishing is needed to prove that the space Ω (1) p,γ (∆) is a smooth submanifold of Ω (1) p,γ (see for instance Ref. [9]). We start proving the second equality in (6.22); we denote by∇ andR respectively the covariant derivative and the curvature tensor of the Levi-Civita connection ofg; moreover, let I denote the index form in T w Ω (1) γ,p with respect to the metricg, defined as in (6.9). Moreover, letμ(w) be the geometric index of the geodesic w in the metricg, defined as in (6.20).
To this goal, we will use also some abstract arguments in functional analysis on Hilbert spaces, and we introduce the following notation.
For all t ∈ ]0, 1], let (T t , ·, · t ) be a real Hilbert space with relative inner product, defined by: (∇ẇζ 1 ,∇ẇζ 2 ) dr. (6.25) Observe that ·, · t is non degenerate on T t , because of the condition ζ(t) = 0. Let · t = ·, · 1 2 t be the relative norm. We also define a continuous symmetric bilinear form H t on T t , by: observe that for t = 1, the Hilbert space (T t , ·, · t ) coincide with T w Ω (1) γ,p (∆) and the bilinear form H t is precisely the Hessian HẼ(w). The symmetry of H t is easily obtained using the symmetry of the curvature tensor and of the second fundamental form of γ. Observe also that, since ζ 1 (0) and ζ 2 (0) are multiples of the Killing field Y , then we have: g(ẇ(0),∇ ζ1(0) ζ 2 ) = −g (ζ 1 (0), Y (w(0))) ·g(ζ 2 (0), Y (w(0))) g(Y (w(0)), Y (w(0))) 2g (Y (w(0)),∇ẇ (0) Y ). (6.27) Using the Riesz representation theorem, we can write H t as: where L t is a self-adjoint linear operator on T t . Comparing (6.25) and (6.26), we see that we can write: where I t is the identity on T t and K t is the self-adjoint operator on T t defined by: Since the inclusions of H 1 ([0, t], IR m ) into L 2 ([0, t], IR m ) and into C 0 ([0, t], IR m ) are compact (see [4]) and keeping in mind (6.27), formula (6.30) tells us that K t is a compact operator for every t ∈]0, 1]. For all t, let {λ l (t)} k∈I N be the sequence of all the eigenvalues of K t ; they can be characterized by the following minimax property: where the first maximum is taken over all possible subspaces V of T t having dimension equal to l.
By standard arguments (see for instance [16]) using the above characterization of the λ l 's one proves that the map is continuous.
We now prove the following claims: 1. for t small enough, H t is positive definite in T t ; 2. for all t, the kernel of H t consists precisely of all γ-Jacobi fields along w [0,t] that vanish at t; 3. for all k ∈ IN , the map t −→ λ l (t) is increasing on ]0, 1]; moreover, if for some t 0 ∈ ]0, 1[ it is λ l (t 0 ) = 1, then λ l (t) > 1 for all t ∈ ]t 0 , 1]. Observe that the proof will be concluded once the above claims are proven. Indeed, by definition, a point w(t 0 ) is a γ-focal point along w with multiplicity d if and only if there exists k > 0 such that λ l (t 0 ) = λ l+1 (t 0 ) = . . . = λ l+d−1 (t 0 ) = 1. From (6.29), the Morse index of HẼ(w) on T w Ω (1) γ,p (∆) is given by the sum of the dimensions of the eigenspaces of K 1 corresponding to eigenvalues λ l (1) which are strictly larger than one. By the claims 1, 2 and 3 above, such number is given by the sum of the dimensions of the kernels of H t , the sum being taken over all t ∈ ]0, 1]. By definition, this number is equal to the geometric indexμ(w) of w.
Let's prove the claim 1; observe that another way of stating this claim is that, for all k ∈ IN and for t > 0 small enough, we have: For ζ ∈ T t , since ζ(t) = 0, we have: hence, using Schwartz's inequality we have: Moreover, another application of Schwartz's inequality gives us: , from which we obtain the inequality: The proof of claim 1 follows immediately from (6.26), (6.27), (6.32) and (6.33).
Let us set: (6.35) and From the definition (6.37) of µ it is easily checked that ζ 1 ∈ T t ; we compute as follows: We now show that H t (ζ, µ · Y ) = 0. Since Y is Killing, then its restriction to w is a Jacobi field (see [19,Lemma 26,p. 252]), and so it satisfies: Integration by parts and (6.39) yield: (6.40) where in the last equality we have used the anti-symmetry of the map (a, b) →g(a,∇ b Y ).
By the symmetry of the curvature tensor, we have: hence, we have: because ζ ∈ T t (see formula (6.24)).
If we use the equality H t (ζ, ζ 1 ) = 0 we get: Since (6.42) holds for all smooth vector field V along w vanishing at the endpoints, the fundamental lemma of Calculus of Variations tells us that: which is the first condition in (6.34). The other three conditions of (6.34) are satisfied by any vector field in T t , hence claim 2 is proven.
Let's go now to the proof of claim 3. Let's fix 0 < t 1 < t 2 in [0, 1]; we prove first that, for all l, we have: To this goal, let l be fixed and let V 1 be a l-dimensional subspace of T t1 such that: We define a linear and continuous map I t1,t2 : T t1 −→ T t2 given by: We observe that, with the above definition, I t1,t2 (ξ) does indeed belong to T t1 (see formula 6.24)); observe also that I t1,t2 is an isometry, and in particular injective. Moreover, the following equality holds trivially: Let V 2 be the l-dimensional subspace of T t2 defined by: Then, by (6.44), we have: (6.45) which proves (6.43).
To prove the second part of claim 3, it suffices to observe that if λ l (t 0 ) = 1 then w(t 0 ) is a γ-focal point along w. Since the set of γ-focal points along w is discrete (see Remark 6.2), it follows that, if λ l (t 0 ) = 1, then λ l (t) = 1 in a neighborhood of t 0 . Finally, by the monotonicity of λ l , we conclude that λ l (t) > 1 in ]t 0 , 1], and we are done. Theorem 6.9 can be applied to the Riemannian metricg = φ k · g R defined in U k . Recalling thatm(w, E φ k ) denotes the Morse Index of the restriction of the Hessian H E φ k on the space T w Ω (1) γ,p (∆) ⊥ , we have thus proven the equality:

The Index Theorem for Brachistochrones
We want to study now the Morse index of the travel time functional at a given brachistochrone σ, which is defined as the index of the symmetric bilinear form H T (σ) (see Definition 6.5). In this section we extend the classical the Morse Theory for Riemannian geodesics, in order to obtain a weak version of the Morse Index Theorem for brachistochrones (Theorem 7.12), by introducing the concepts of b-Jacobi fields and b-focal points along a brachistochrone σ (see Definitions 7.1 and 7.6 below).
We now begin with the study of the Hessian of the travel time functional. Let σ ∈ B (1) p,γ (k) be a brachistochrone, since T σ > 0, formula (5.5) tells us that: m(σ, T ) = m(σ, −F ), and Ker H T (σ) = Ker H F (σ) . (7.1) We emphasize that from now on we will consider brachistochrone curves whose endpoints may vary in the open set U k , whereas the value of their energy constant k is a fixed positive number. For the sake of shortness, when speaking of brachistochrones we will omit to specify the value of their energy constant without danger of confusion.
In this section and in the rest of the paper we will be speaking of variations of a given curve in some fixed space, which will be a family of curves of the same type, in a sense that will be clarified in the different situations, parameterized by a suitable variable, denoted by s. Whenever not specified, we will tacitly assume that s varies in an interval of the form ] − ε, ε [ for some ε > 0. A formal definition of smooth variation of a given curve z ∈ Ω (1) p,γ is given in Appendix A (Definition A.2).
We also warn the reader that, in the course of the section, we will switch back and forth among the three Hessians H T , H F and H E φ k , keeping in mind the basic relations among them given by formulas (5.4) and (5.6).
We mimic the classical Morse theory and we proceed as follows. Let σ ∈ B (1) p,γ (k) be a fixed brachistochrone, and, recalling the definition of the space B (1) (k) given in (2.39), we consider a variation σ s ∈ B (1) (k) of σ, depending smoothly on the parameter s ∈ ] − ε, ε [ and such that σ 0 = σ. Suppose that each curve σ s is a brachistochrone of energy k between σ s (0) and γ σs(1) , where γ σs(1) is the integral line of Y passing through σ s (1).
This means that each σ s satisfies the differential equation (3.21) and with initial tangent vectorσ s (0) satisfying the two conditions: (1) p,γ (k) is called a b-Jacobi field if there exists a variation σ s ∈ B (1) (k) of σ as above such that V = d ds s=0 σ s .
In other words, a b-Jacobi field along σ is a variational vector field corresponding to variations made of brachistochrones with the same energy constant and, possibly, with different endpoints. By definition, the b-Jacobi fields are characterized by the property of satisfying the linearized brachistochrone equation; this second order differential equation has a rather ugly aspect and it is presented only for the sake of completeness in the following Proposition.
Proposition 7.2. Let σ ∈ B (1) p,γ (k) be a brachistochrone of travel time T σ and let V ∈ T σ B (1) (k) be a variational vector field along σ, with constant C V = ∇σV, Y − V, ∇σY . If V is a b-Jacobi field then V satisfies the second order linear differential equation: and the initial condition: −T σ C V + k ∇σV (0),σ(0) = 0. (7.4) Proof. The equation (7.3) is obtained by patiently linearizing the brachistochrone differential equation (3.21), using the following dictionary: The formulas above are obtained by considering the basic properties of the Levi-Civita connection and the curvature tensor of g. In particular, in the sixth formula we have used the fact that R(σ, Y ) Y, Y = 0, by the anti-symmetry in the last two variables. The initial condition (7.4) is obtained by linearizing the first equation of formula (7.2).
A partial converse to Proposition 7.2 is provided by the following Proposition: be a brachistochrone and suppose that V is a smooth vector field along σ satisfying the differential equation (7.3), the initial condition (7.4) and with V (0) = 0. Then, V is a b-Jacobi field along σ, i.e., there exists a variation σ s of σ consisting of brachistochrones between p and γ s , Proof. We use a sort of brachistochrone exponential map, as follows. Given a vector v 0 ∈ T p M such that v 0 , Y (p) 2 + k 2 v 0 , v 0 = 0, and v 0 , v 0 < 0, (7.5) then there exists a unique brachistochrone σ v0 ∈ B (1) p (k) and such thatσ v0 (0) = v 0 . This is obtained by solving the differential equation (3.21) with initial conditions σ(0) = p anḋ σ(0) = v 0 .
Moreover, the map v 0 −→ σ v0 ∈ B (1) p (k) is C 1 , due to the regular dependence on the data of the solution of the differential equation (3.21).
Let S ⊂ T p M be the set of vectors v 0 satisfying the conditions (7.5); S is a submanifold of T p M. Indeed, the condition v 0 , v 0 < 0 is open; moreover, the gradient of the smooth map Multiplying by Y (p) we obtain: where the last inequality depends on the fact that both v 0 and Y (p) are timelike, hence v 0 , Y (p) = 0, and Y (p), Y (p) + k 2 > 0 in U k . This implies that G ′ = 0, hence G −1 (0) is a smooth submanifold of T p M. Clearly,σ(0) ∈ S.
In order to conclude the proof, we need to show that the variational fieldṼ = d ds s=0 σ s coincides with V . By Proposition 7.2,Ṽ satisfies the second order differential equation (7.3), while V satisfies (7.3) by assumption, andṼ (0) = V (0) = 0, because we are fixing the initial point p. By uniqueness, in order to prove thatṼ = V along σ it suffices to show that ∇σ (0)Ṽ = ∇σ (0) V . This is easily established by the following calculation, that concludes the proof: Corollary 7.4. If σ is a brachistochrone and V is a b-Jacobi field along σ such that V (0) = 0, then V ∈ T σ B (1) p (k). Proof. Following the proof of Proposition 7.3, V is the variational vector field corresponding to a variation σ s ∈ B (1) p (k) of σ.
In general, it may not be true that a b-Jacobi field V along a brachistochrone σ satisfying V (0) = 0 and V (1) ∈ IR · Y (σ(1)) is the variational vector field corresponding to a family of brachistochrones in B (1) p,γ (k). However, such vector fields belong to the tangent space T σ B (1) p,γ (k), and they are in the kernel of the Hessian H F (σ): Corollary 7.5. If σ ∈ B (1) p,γ (k) is a brachistochrone and V is a b-Jacobi field along σ such that V (0) = 0 and V (1) is parallel to Y (σ(1)), then V ∈ T σ B (1) p,γ (k), and V ∈ Ker H F (σ) .
To prove the second part of the thesis, we need to show that H F (σ)[V, W ] = 0 for all W ∈ T σ B (1) p,γ (k). By Corollary 5.2, we have: hence, to conclude the proof it suffices to show that dD(σ)[V ] is in the kernel of the Hessian H E φ k (D(σ)). By (6.19), this amounts to proving that X = dD(σ)[V ] is the variational vector field corresponding to a smooth variation w s of w = D(σ) consisting of horizontal geodesics in the metric φ k · g R between p and some integral curve γ s of Y lying in U k (recall that a vector field along a geodesic is Jacobi if and only if it is the variational vector field corresponding to a variation by geodesics).
To see this, let σ s be a smooth variation of σ consisting of brachistochrones in B (1) p (k) between p and some curve γ s in U k , and with variational vector field V . Such a variation exists by Proposition 7.3.
Then, if we consider the curves w s = D(σ s ), by part 3 of Proposition 4.5, each w s is a horizontal geodesic between p and γ s ; by Proposition 4.4, w s is a smooth variation of w. Finally, we have: which concludes the proof.
We will see later (Proposition 7.11) that the kernel of the Hessian H F (σ) consists precisely of the b-Jacobi fields along σ; this fact can also be checked directly using the explicit formula for the Hessian H F (σ) given in Appendix A and the Lagrange multipliers technique.
We are now ready to define the notion of a b-focal point along a brachistochrone.
Definition 7.6. Let σ ∈ B (1) p,γ (k) be a brachistochrone. A point σ(t 0 ) of σ is said to be a b-focal point if there exists a non zero b-Jacobi field V along σ [t0,1] that vanish at t 0 , that is, a non zero vector field V along σ for which the quantity C V = ∇σV, Y − V, ∇σY is constant along σ, such that V (t 0 ) = 0, satisfying the differential equation (7.3) and the condition: In the above situation, we will also say that σ(t 0 ) is b-conjugate to σ(1) = p along σ.
For every t 0 ∈ [0, 1], the set J σ (t 0 ) of vector fields V satisfying the above conditions in the interval [t 0 , 1] is a vector field; if σ(t 0 ) is a b-focal point along σ, then multiplicity µ σ (t 0 ) of σ(t 0 ) is the dimension of J σ (t 0 ). The geometric index µ(σ) of the brachistochrone σ is defined to be the (possibly infinite) number: Observe that every vector field along σ [t0,1] which is solution of the linear differential equation We now want to relate the b-focal points along a brachistochrone σ with the γ-focal points along the corresponding Riemannian geodesic w = D(σ). This is done in Theorem 7.12 below, which is preceded by some preliminary results, aimed to determine the relation of the notions of Jacobi fields along σ and w.
More precisely, we will show that the linear map dO • dD(σ) gives an isomorphism of the spaces J σ (t 0 ) and J {k} w (γ, t 0 ) (recall that the map O is the direction reversing map defined in (6.1)).
Given a horizontal geodesic w, a Jacobi field along w is a (smooth) vector field J along w satisfying the differential equation (6.6). From (2.11) and (4.5), such a vector field J belongs to the tangent space T w Ω (1) γ,p (∆) if and only if J(1) = 0, J(0) ∈ IR · Y (w(0)) (recall that we are considering curves w starting on γ and arriving at p), and ∇ẇJ, Y + ẇ, ∇ J Y ≡ 0. Recalling Remark 6.3, this last equality is satisfied identically on [0, 1] provided that it is satisfied at some point t 0 ∈ [0, 1].
Hence, recalling the definitions 1, 2 and 3 of page 25 and Remark 6.3, we have that the set of Jacobi fields in T w Ω (1) γ,p (∆) coincides with the finite dimensional vector space J {k} w (γ, 0): We introduce the following map: G : Ω (1) p,γ −→ Ω (1) p,γ , (7.11) given by: As in the case of the map D, it is easy to see that G is smooth; moreover, using (4.19) one checks that it is a left-inverse for D in B (1) p,γ (k), i.e., for all σ ∈ B (1) p,γ (k), we have: G(D(σ)) = σ. It is easily computed: which concludes the proof.
In analogy with formula (4.14), for all a ∈ [0, 1[ we can define a linear map L a on the space of vector fields along σ [a,1] satisfying the two conditions appearing in (2.44) on the interval [a, 1], and taking values in the space of vector fields along w [a,1] .
The map L a is given by: In particular, L 0 = dD(σ); observe also that, of ζ(a) = 0, then L a [ζ](a) = 0. The result of Propositions 7.7 and 7.8 can be extended immediately to the maps dO • L t0 : Proof. The proofs of Propositions 7.7 and 7.8 can be repeated verbatim, by replacing the initial point p with the point σ(t 0 ). The only technical subtlety to worry about is that, when replacing the initial point, it will not hold, in general, that σ(t 0 ) = w(t 0 ). Nevertheless, this fact is not essential, because one can always reduce to this case by considering a suitable isometry of U k given by x −→ ψ(x, t).
We now prove that the kernel of the Hessian H F (σ) in T σ B (1) p,γ (k) consists precisely of b-Jacobi fields. This gives an analytical characterization of the b-Jacobi fields along a brachistochrone.
Proof. Corollary 7.5 proves that any b-Jacobi field along σ is in the kernel of H F (σ). Conversely, let σ be a fixed brachistochrone and ζ ∈ Ker H F (σ) . From Corollary 7.10, it suffices to prove that the vector field J = dO • dD(σ)[ζ] is a γ-Jacobi field with respect to the Riemannian metric φ k · g R along the geodesic w = O(D(σ)). Moreover, since J ∈ T w Ω (1) γ,p (∆), from Lemma 6.4 it suffices to show that J is a Jacobi field along w, i.e., that it satisfies equation (6.6). Observe that, by Proposition 7.9, J is in T w Ω (1) γ,p (∆) ⊥ , hence it satisfies the two equations: 2 the point here is that the variational fields in TwΩ (1) p,γ are given by variations ws of w that not necessarily have endpoints on γ(IR). The only thing that can be said about such variations ws is that ws(1) is infinitesimally close to γ as s → 0 with an order of infinitesimal bigger than 1.
To prove that J is Jacobi, let V ∈ C ∞ o ([0, 1], T M) be any smooth vector field along w vanishing at the endpoints. We set: (7.18) where λ and µ are functions in H 1 ([0, 1], IR) to be determined in such a way that the resulting vector field W belongs to T w Ω (1) γ,p (∆). Straightforward computations show this condition is satisfied by setting: dr, dr. (7.19) Observe that, with the definitions above, since w is a geodesic with respect to φ k · g R one has: λ(0) = λ(1) = µ(1) = 0. (7.20) Arguing as in the proof of Theorem 6.9 since Y is Killing in the metric φ k ·g R , then its restriction to w is a Jacobi field (see also (6.39)):  From (6.9), integrating by parts and using formulas (7.17), (7.21) and the symmetry of the curvature tensor R {k} , we have: Since W ∈ T w Ω (1) γ,p (∆), then W is in the image of dO • dD, say W = dO • dD(σ)[ζ 1 ] for some ζ 1 ∈ T σ B (1) p,γ (k). Since ζ ∈ Ker H F (σ) and J = dO • dD(σ)[ζ], then, by Corollary 5.2 and formula (6.2), it is H E φ k (w)[J, W ] = −H F (σ)[ζ, ζ 1 ] = 0, and, in particular, I {k} (J, V ) = 0. Hence, we have that I {k} (J, V ) = 0 for all smooth vector field along w vanishing at the endpoints, and by (6.10) this implies that J is a Jacobi field, concluding the proof.
We are finally ready to state and prove the Morse Index Theorem for the travel time brachistochrones: Theorem 7.12 (Morse Index Theorem for Relativistic Brachistochrones). Let σ ∈ B (1) p,γ (k) be a brachistochrone and w = O(D(σ)) ∈ Ω (1) γ,p the corresponding horizontal geodesic. Then, a point σ(t 0 ) is a b-focal point along σ if and only if w(t 0 ) is a γ-focal point along w, in which case the two focal points have the same multiplicity. In particular, we have µ(σ) = µ {k} (w). (7.25) Moreover, if p is not a b-focal point along σ, then the Morse index m(σ, T ) is equal to the geometric index µ(σ) of σ: m(σ, T ) = µ(σ). If p is not a γ-focal point along w, or equivalently if p is not a b-focal point along σ, then, Theorem 6.8 implies: m(w, E φ k ) = µ {k} (w); (7.30) the equality (7.26) follows at once from (7.25) and (7.27)-(7.30). This concludes the proof.
From finiteness of the index m(σ, T ) we get the following: Corollary 7.13. Let σ ∈ B (1) p,γ (k) be a brachistochrone. Then, σ is never a local maximum for T .
From the equality (7.26) we get that, if µ(σ) = 0, then the Morse index of the travel time vanishes at σ, hence σ is a local minimum for T . Therefore, we have: Corollary 7.14. Let σ ∈ B (1) p,γ (k) be a brachistochrone and w = O(D(σ)). Suppose that there are no γ-focal points along w. Then, σ is a local minimum for the arrival time functional T .

The Global Morse Relations
In this section we will use the infinite dimensional Morse theory to prove some equalities relating the differential structure of the travel time brachistochrone problem and the topological structure carried by the set of continuous paths joining p and γ in U k .
Most of the technical results needed are obtained using the same ideas and techniques employed in Reference [5], where the authors prove the Morse relations for geodesics in a convex subset of a stationary Lorentzian manifold. In order to keep our exposition short, we will omit some of the proofs that can be deduced easily from analogous proofs presented in details in [5].
Throughout the section, we will make the following assumptions: 1. the vector field Y is complete in U k , i.e., its integral lines are defined over the entire real line; 2. γ : IR −→ U k is an integral line of Y without self-intersection; 3. p is an event in U k ; 4. k 2 is a regular value for the function − Y, Y ; 5. U k = U k ∂U k is complete with respect to the Riemannian metric (2.2); 6. the function − Y, Y is bounded away from 0 in U k , i.e., there exists a positive constant ν such that − Y, Y ≥ ν > 0 in U k ; 7. p and γ are not b-conjugate, i.e., for any brachistochrone σ of energy k in B (1) p,γ (k), the points σ(0) = p and σ(1) are not b-conjugate along σ.
The following is the main result of the Section: • E ε satisfies the Palais-Smale condition at every level c ∈ IR, i.e., every sequence {w n } in E c ε such that 3 dE ε (w n ) tends to 0 as n → ∞, has a convergent subsequence in E c ε ; • for all c ∈ IR there exists δ(c) > 0 and ε(c) ∈ ]0, 1] such that, for all ε ∈ ]0, ε(c)] and for all critical point w ε of E ε in Ω (1) p,γ (U k ) with E ε (w ε ) ≤ c, then w ε is also a critical point for E φ k , and the following inequality holds: Ψ k (w ε (t)) ≥ δ(c), ∀ t ∈ [0, 1].
In particular, it follows that if c is a regular value for E φ k , i.e., if there are no critical point for E φ k in E −1 φ k (c), using (8.4) and (8.5) we obtain the existence of a number ε ′ (c) ∈ ]0, ε(c)] such that, for all ε ∈ ]0, ε ′ (c)], c is a regular value also for the functional E ε , and a curve w ∈ Ω (1) p,γ is a critical point for E ε if and only if it is a critical point for E φ k (with E ε (w) = E φ k (w)) and: where m(z, G) denotes the Morse Index of the functional G at the critical point z.
Then, using assumption 7, for all ε ∈ ]0, ε ′ (c)], every critical point w of E ε in E c ε is nondegenerate, which allows to obtain the Morse Relations in E c ε (see Ref. [17]): of g in the directions of ∂η ∂s and ∂η ∂t for vector fields along η; the usual symbols d ds and d dt will denote the differentials with respect to s and t of functions along η.
Since where R is the curvature tensor of the Lorentzian metric g, defined in (2.1).
We have the following: