arXiv subjects
Publications and source records attributed to Miguel Angel Javaloyes.
We give the details of the proof of the equality between the critical groups, with respect the H^1 and C^1 topology, at a non-degenerate critical point of the energy functional of a non-reversible Finsler manifold (M,F), defined on the Hilbert manifold of the H^1 curves connecting two given points on M.
Let $(M,g)$ be a spacetime which admits a complete timelike conformal Killing vector field $K$. We prove that $(M,g)$ splits globally as a standard conformastationary spacetime with respect to $K$ if and only if $(M,g)$ is distinguishing (and, thus causally continuous). Causal but non-distinguishing spacetimes with complete stationary vector fields are also exhibited. For the proof, the recently solved "folk problems" on smoothability of time functions (moreover, the existence of a {\em temporal} function) are used.
For a standard Finsler metric F on a manifold M, its domain is the whole tangent bundle TM and its fundamental tensor g is positive-definite. However, in many cases (for example, the well-known Kropina and Matsumoto metrics), these two conditions are relaxed, obtaining then either a pseudo-Finsler metric (with arbitrary g) or a conic Finsler metric (with domain a "conic" open domain of TM). Our aim is twofold. First, to give an account of quite a few subtleties which appear under such generalizations, say, for conic pseudo-Finsler metrics (including, as a previous step, the case of Minkowski conic pseudo-norms on an affine space). Second, to provide some criteria which determine when a pseudo-Finsler metric F obtained as a general homogeneous combination of Finsler metrics and one-forms is again a Finsler metric ---or, with more accuracy, the conic domain where g remains positive definite. Such a combination generalizes the known (alpha,beta)-metrics in different directions. Remarkably, classical examples of Finsler metrics are reobtained and extended, with explicit computations of their fundamental tensors.
In this review, we collect several results for conformally standard stationary spacetimes (SxR,g) obtained in terms of a Finsler metric of Randers type on the orbit manifold S that we call Fermat metric. This metric is obtained by applying the relativistic Fermat principle and it turns out that it encodes all the causal aspects of the spacetime.
We show that any two non-conjugate points on a forward or backward complete connected Finsler manifold can be joined by infinitely many geodesics which are not covered by finitely many closed ones, provided that the Betti numbers of the based loop space grow unbounded.
In this paper we first study some global properties of the energy functional on a non-reversible Finsler manifold. In particular we present a fully detailed proof of the Palais--Smale condition under the completeness of the Finsler metric. Moreover we define a Finsler metric of Randers type, which we call Fermat metric, associated to a conformally standard stationary spacetime. We shall study the influence of the Fermat metric on the causal properties of the spacetime, mainly the global hyperbolicity. Moreover we study the relations between the energy functional of the Fermat metric and the Fermat principle for the light rays in the spacetime. This allows us to obtain existence and multiplicity results for light rays, using the Finsler theory. Finally the case of timelike geodesics with fixed energy is considered.
We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on $M=\R\times S$ and Randers metrics on $S$. In particular, for stationary spacetimes, we give a simple characterization of when they are causally continuous or globally hyperbolic (including in the latter case, when $S$ is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived. Causality suggests that the role of completeness in many results of Riemannian Geometry (geodesic connectedness by minimizing geodesics, Bonnet-Myers, Synge theorems) is played, in Finslerian Geometry, by the compactness of symmetrized closed balls. Moreover, under this condition we show that for any Randers metric there exists another Randers metric with the same pregeodesics and geodesically complete. Even more, results on the differentiability of Cauchy horizons in spacetimes yield consequences for the differentiability of the Randers distance to a subset, and vice versa.
We obtain a result about the existence of only a finite number of geodesics between two fixed non-conjugate points in a Finsler manifold endowed with a convex function. We apply it to Randers and Zermelo metrics. As a by-product, we also get a result about the finiteness of the number of lightlike and timelike geodesics connecting an event to a line in a standard stationary spacetime.
We study focal points and Maslov index of a horizontal geodesic $γ:I\to M$ in the total space of a semi-Riemannian submersion $π:M\to B$ by determining an explicit relation with the corresponding objects along the projected geodesic $π\circγ:I\to B$ in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary space-time which is orthogonal to a timelike Killing vector field.
In this paper we survey on some recent results on Riemannian orbifolds and singular Riemannian foliations and combine them to conclude the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.f. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.
We prove the semi-Riemannian bumpy metric theorem using equivariant variational genericity. The theorem states that, on a given compact manifold $M$, the set of semi-Riemannian metrics that admit only nondegenerate closed geodesics is generic relatively to the $C^k$-topology, $k=2,...,\infty$, in the set of metrics of a given index on $M$. A higher order genericity Riemannian result of Klingenberg and Takens is extended to semi-Riemannian geometry.
Given a Lorentzian manifold $(M,g)$, a geodesic $γ$ in $M$ and a timelike Jacobi field $\mathcal Y$ along $γ$, we introduce a special class of instants along $γ$ that we call $\mathcal Y$-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the $\mathcal Y$-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field $\mathcal Y$ is obtained as the restriction of a globally defined timelike Killing vector field.
This paper has been withdrawn by the authors because it has been merged with paper arXiv:0903.3501v1 [math.DG]
We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is never vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold $M$ admits a Lorentzian metric with a never vanishing Killing vector field which is timelike somewhere if and only if $M$ admits a smooth circle action without fixed points.
We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Based on an idea of B. White, we prove an abstract genericity result that employs the infinite dimensional Sard--Smale theorem. Applications are given by proving the genericity of metrics without degenerate geodesics between fixed endpoints in general (non compact) semi-Riemannian manifolds, in orthogonally split semi-Riemannian manifolds and in globally hyperbolic Lorentzian manifolds. We discuss the genericity property also in stationary Lorentzian manifolds.
We prove an estimate on the difference of Maslov indices relative to the choice of two distinct reference Lagrangians of a continuous path in the Lagrangian Grassmannian of a symplectic space. We discuss some applications to the study of conjugate and focal points along a geodesic in a semi-Riemannian manifold.
In this paper we prove several multiplicity results of $t$-periodic light rays in conformally stationary spacetimes using the Fermat metric and the extensions of the classical theorems of Gromoll-Meyer and Bangert-Hingston to Finsler manifolds. Moreover, we exhibit some stationary spacetimes with a finite number of $t$-periodic light rays and compute a lower bound for the period of the light rays when the flag curvature of the Fermat metric is $η$-pinched.
We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior.