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Miguel Angel Javaloyes

Publications and source records attributed to Miguel Angel Javaloyes.

At least 19 recordsLinked to original sources
arXiv2026

On the formulations of the Fermat principle in general relativity and beyond

This paper presents a survey of the Fermat principle within the framework of general relativity, tracing its evolution from classical optics to its modern variational formulation in Lorentzian geometry. In particular, we provide its proof in the framework of smooth lightlike curves. We also analyze the mathematical difficulties inherent in the relativistic setting, specifically demonstrating that the space of lightlike curves in the Sobolev topology does not admit a smooth manifold structure due to the cone nature of the null condition. To address these variational obstacles, we discuss alternative frameworks highlighting the role of the quadratic arrival time functional in establishing multiplicity results for light rays. Furthermore, we explore significant extensions of the principle, such as its application to extended sources and receivers, arbitrary arrival curves, timelike geodesics with prescribed proper time, Finsler spacetimes, or settings with a non-continuous interface giving rise to a Snell law.

gr-qc↗
arXiv2024

Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system

We consider a geodesic problem in a manifold endowed with a Randers-Kropina metric. This is a type of singular Finsler metric arising both in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo's navigation problem with a wind represented by a vector field having norm not greater than one. By using Lusternik-Schnirelman theory, we prove existence of infinitely many geodesics between two given points when the manifold is not contractible. Due to the type of nonholonomic constraints that the velocity vectors must satisfy, this is achieved thanks to a recent result about the homotopy type of the set of solutions of an affine control system with a controlled drift and related to a corank one, completely nonholonomic distribution of step 2.

math.DG↗
arXiv2021

Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes

The notion of wind Finslerian structure is developed; this is a generalization of Finsler metrics where the indicatrices at the tangent spaces may not contain the zero vector. In the particular case that these indicatrices are ellipsoids, called here wind Riemannian structures, they admit a double interpretation which provides: (a) a model for classical Zermelo's navigation problem even when the trajectories of the moving objects (planes, ships) are influenced by strong winds or streams, and (b) a natural description of the causal structure of relativistic spacetimes endowed with a non-vanishing Killing vector field (SSTK splittings), in terms of Finslerian elements. These elements can be regarded as conformally invariant Killing initial data on a partial Cauchy hypersurface. The wind Finslerian structure is described in terms of two (conic, pseudo) Finsler metrics, one with a convex indicatrix and the other with a concave one. However, the spacetime viewpoint for the wind Riemannian case gives a useful unified viewpoint. A thorough study of the causal properties of such a spacetime is carried out in Finslerian terms. Randers-Kropina metrics appear as the Finslerian counterpart to the case of an SSTK when the Killing vector field is either timelike or lightlike. Among the applications, we obtain the solution of Zermelo's navigation with arbitrary stationary wind, metric-type properties (distance-type arrival function, completeness, existence of minimizing, maximizing or closed geodesics), as well as description of spacetime elements (Cauchy developments, black hole horizons) in terms of Finslerian elements in Killing initial data. A general Fermat's principle of independent interest for arbitrary spacetimes, as well as its applications to SSTK spacetimes and Zermelo's navigation, are also provided.

math.DG↗
arXiv2021

On equifocal Finsler submanifolds and analytic maps

A relevant property of equifocal submanifolds is that their parallel sets are still immersed submanifolds, which makes them a natural generalization of the so-called isoparametric submanifolds. In this paper, we prove that the regular fibers of an analytic map $π:M^{m+k}\to B^{k}$ are equifocal whenever $M^{m+k}$ is endowed with a complete Finsler metric and there is a restriction of $π$ which is a Finsler submersion for a certain Finsler metric on the image. In addition, we prove that when the fibers provide a singular foliation on $M^{m+k}$, then this foliation is Finsler.

math.DG↗
arXiv2020

The flag curvature of a submanifold of a Randers-Minkowski space in terms of Zermelo data

The main result of this paper is an expression of the flag curvature of a submanifold of a Randers-Minkowski space $({\mathscr V},F)$ in terms of invariants related to its Zermelo data $(h,W)$. More precisely, these invariants are the sectional curvature and the second fundamental form of the positive definite scalar product $h$ and some projections of the wind $W$. This expression allows for a promising characterization of submanifolds with scalar flag curvature in terms of Riemannian quantities, which, when a hypersurface is considered, seems quite approachable. As a consequence, we prove that any $h$-flat hypersurface $S$ has scalar $F$-flag curvature and the metric of its Zermelo data is conformally flat. As a tool for making the computation, we previously reobtain the Gauss-Codazzi equations of a pseudo-Finsler submanifold using anisotropic calculus.

math.DG↗
arXiv2020

Anisotropic conformal invariance of lightlike geodesics in pseudo-Finsler manifolds

In this paper, we prove that lightlike geodesics of a pseudo-Finsler manifold and its focal points are preserved up to reparametrization by anisotropic conformal changes, using the Chern connection and the anisotropic calculus and the fact that geodesics are critical points of the energy functional and Jacobi fields, the kernel of its index form. This result has applications to the study of Finsler spacetimes.

math.DG↗
arXiv2019

On the definition and examples of cones and Finsler spacetimes

A systematic study of (smooth, strong) cone structures $\C$ and Lorentz-Finsler metrics $L$ is carried out. As a link between both notions, cone triples $(Ω,T, F)$, where $Ω$ (resp. $T$) is a 1-form (resp. vector field) with $Ω(T)\equiv 1$ and $F$, a Finsler metric on $\ker (Ω)$, are introduced. Explicit descriptions of all the Finsler spacetimes are given, paying special attention to stationary and static ones, as well as to issues related to differentiability. In particular, cone structures $\C$ are bijectively associated with classes of anisotropically conformal metrics $L$, and the notion of {\em cone geodesic} is introduced consistently with both structures. As a non-relativistic application, the {\em time-dependent} Zermelo navigation problem is posed rigorously, and its general solution is provided.

math.DG↗
arXiv2018

On singular Finsler foliation

In this paper we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and (regular) Finsler foliations. We show that if $\mathcal{F}$ is a singular Finsler foliation on a Randers manifold $(M,Z)$ with Zermelo data $(\mathtt{h},W),$ then $\mathcal{F}$ is a singular Riemannian foliation on the Riemannian manifold $(M,\mathtt{h} )$. As a direct consequence we infer that the regular leaves are equifocal submanifolds (a generalization of isoparametric submanifolds) when the wind $W$ is an infinitesimal homothety of $\mathtt{h}$ (e.,g when $W$ is killing vector field or $M$ has constant Finsler curvature). We also present a slice theorem that relates local singular Finsler foliations on Finsler manifolds with singular Finsler foliations on Minkowski spaces.

math.DG↗
arXiv2017

Wind Riemannian spaceforms and Randers metrics of constant flag curvature

Recently, wind Riemannian structures (WRS) have been introduced as a generalization of Randers and Kropina metrics. They are constructed from the natural data for Zermelo navigation problem, namely, a Riemannian metric $g_R$ and a vector field $W$ (the wind), where, now, the restriction of mild wind $g_R(W,W)<1$ is dropped. Here, the models of WRS spaceforms of constant flag curvature are determined. Indeed, the celebrated classification of Randers metrics of constant flag curvature by Bao, Robles and Shen, extended to the Kropina case in the works by Yoshikawa, Okubo and Sabau, can be used to obtain the local classification. For the global one, a suitable result on completeness for WRS yields the complete simply connected models. In particular, any of the local models in the Randers classification does admit an extension to a unique model of wind Riemannian structure, even if it cannot be extended as a complete Finslerian manifold. Thus, WRS's emerge as the natural framework for the analysis of Randers spaceforms and, prospectively, wind Finslerian structures would become important for other global problems too. For the sake of completeness, a brief overview about WRS (including a useful link with the conformal geometry of a class of relativistic spacetimes) is also provided.

math.DG↗
arXiv2016

Penrose's singularity theorem in a Finsler spacetime

We translate Penrose's singularity theorem to a Finsler spacetime. To that end, causal concepts in Lorentzian geometry are extended, including definitions and properties of focal points and trapped surfaces, with careful attention paid to the differences that arise in the Finslerian setting.

math.DG↗
arXiv2015

On a monodromy theorem for sheaves of local fields and applications

We prove a monodromy theorem for local vector fields belonging to a sheaf satisfying the unique continuation property. In particular, in the case of admissible regular sheaves of local fields defined on a simply connected manifold, we obtain a global extension result for every local field of the sheaf. This generalizes previous works of Nomizu for semi-Riemannian Killing fields, of Ledger--Obata for conformal fields, and of Amores for fields preserving a $G$-structure of finite type. The result applies to Finsler or pseudo-Finsler Killing fields and, more generally, to affine fields of a spray. Some applications are discussed.

math.DG↗
arXiv2015

Geodesics and Jacobi fields of pseudo-Finsler manifolds

In this paper, we derive the first and the second variation of the energy functional for a pseudo-Finsler metric using the family of affine connections associated to the Chern connection. This opens the possibility to accomplish computations with coordinate-free methods. Using the second variation formula, we introduce the index form and present some properties of Jacobi fields.

math.DG↗
arXiv2014

Zermelo navigation in pseudo-Finsler metrics

We generalize the notion of Zermelo navigation to arbitrary pseudo-Finsler metrics possibly defined in conic subsets. The translation of a pseudo-Finsler metric $F$ is a new pseudo-Finsler metric whose indicatrix is the translation of the indicatrix of $F$ by a vector field $W$ at each point, where $W$ is an arbitrary vector field. Then we show that the Matsumoto tensor of a pseudo-Finsler metric is equal to zero if and only if it is the translation of a semi-Riemannian metric, and when $W$ is homothetic, the flag curvature of the translation coincides with the one of the original one up to the addition of a non-positive constant. In this case, we also give a description of the geodesic flow of the translation.

math.DG↗
arXiv2014

Chern connection of a pseudo-Finsler metric as a family of affine connections

We consider the Chern connection of a (conic) pseudo-Finsler manifold $(M,L)$ as a linear connection $\nabla^V$ on any open subset $Ω\subset M$ associated to any vector field $V$ on $Ω$ which is non-zero everywhere. This connection is torsion-free and almost metric compatible with respect to the fundamental tensor $g$. Then we show some properties of the curvature tensor $R^V$ associated to $\nabla^V$ and in particular we prove that the Jacobi operator of $R^V$ along a geodesic coincides with the one given by the Chern curvature.

math.DG↗
arXiv2013

Almost isometries of non-reversible metrics with applications to stationary spacetimes

We develop the basics of a theory of almost isometries for spaces endowed with a quasi-metric. The case of non-reversible Finsler (more specifically, Randers) metrics is of particular interest, and it is studied in more detail. The main motivation arises from General Relativity, and more specifically in spacetimes endowed with a timelike conformal field K, in which case \emph{conformal diffeomorphisms} correspond to almost isometries of the Fermat metrics defined in the spatial part. A series of results on the topology and the Lie group structure of conformal maps are discussed.

math.DG↗
arXiv2012

Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime is equal to the index of its spatial projection as a geodesic of a Finsler metric associated to the spacetime. Moreover we obtain the Morse relations of lightlike geodesics connecting a point to an integral line of the standard timelike Killing vector field by using Morse theory on the associated Finsler manifold. To this end, we prove a splitting lemma for the energy functional of a Finsler metric. Finally, we show that the reduction to Morse theory of a Finsler manifold can be done also for timelike geodesics.

math.DG↗