arXiv subjects

Miguel Sánchez

Publications and source records attributed to Miguel Sánchez.

At least 19 recordsLinked to original sources

arXiv2026

Lorentzian Cheeger-Gromov convergence and temporal functions

Uniqueness (up to isometries) and existence of limits are studied in the context of Cheeger-Gromov convergence of spacetimes. To address the non-compactness of the linear isometry group in the semi-Riemannian setting, standard pointed convergence is strengthened to anchored convergence, which in the Lorentzian case requires the convergence of a timelike direction. This allows one to construct a local isometry between the neighborhoods of the basepoints, which can be extended globally when one of the limits is geodesically complete for a single causal type of geodesics and the other one is inextensible; a Misner-type example shows that inextensibility of both limits is not enough. In spacetimes, by using Cauchy temporal functions both as a strengthening of the anchors and as a tool to ``Wick rotate'' metrics, a special notion of convergence for globally hyperbolic spacetimes (including those with timelike boundaries) is introduced. After revisiting the tools related to time functions and studying their connections with the Sormani-Vega null distance, the machinery of Riemannian Cheeger-Gromov theory becomes applicable. In particular, several results of independent interest are obtained, including global and local characterizations of $h$-steep functions, independence of steepness and $h$-steepness for temporal functions, compatibility of both conditions for Cauchy temporal functions, and the stability of the latter.

math.DG↗

arXiv2026

Generalized Fermat's principle and Snell's law for cone structures and applications

Fermat's principle is fully generalized to the case where a smooth interface separates two cone structures -- Lorentz-Finsler lightcones -- representing wave propagation in a potentially inhomogeneous, anisotropic, time-dependent and discontinuous medium. The interface, wave source and receiver are assumed to be a hypersurface, a submanifold and a curve in the spacetime, respectively, of any causal character. For a trajectory to fullfil Fermat's principle -- i.e., to be a critical point of the arrival time functional -- its direction must change at the interface, obeying a precise condition that generalizes Snell's law of refraction when the wave crosses the interface, or the law of reflection when it remains in a single medium. Both laws are analyzed in detail to establish the conditions ensuring the existence and uniqueness of refracted and reflected trajectories, and to determine whether they actually minimize the arrival time. Applications to Zermelo's navigation problem and the determination of geodesics in discretized spacetimes are also emphasized.

math.DG↗

arXiv2026

On the foundations and applications of Lorentz-Finsler Geometry

Finslerian extensions of Special and General Relativity -- commonly referred to as Very Special and Very General Relativity -- necessitate the development of a unified Lorentz-Finsler geometry. However, the scope of this geometric framework extends well beyond relativistic physics. Indeed, it offers powerful tools for modeling wave propagation in classical mechanics, discretizing spacetimes in classical and relativistic settings, and supporting effective theories in fundamental physics. Moreover, Lorentz-Finsler geometry provides a versatile setting that facilitates the resolution of problems within Riemannian, Lorentzian, and Finslerian geometries individually. This work presents a plain introduction to the subject, reviewing foundational concepts, key applications, and future prospects. The reviewed topics include (i) basics on the setting of cones, Finsler and Lorentz-Finsler metrics and their (nonlinear, anisotropic and linear) connections, (ii) the global structure of Lorentz-Finsler manifolds and its space of null geodesics, (iii) links among Riemannian, Finsler and Lorentz geometries, (iv) applications in classical settings as wildfires and seisms propagation, and discretization in classical and relativistic settings with quantum prospects, and (v) Finslerian variational approach to Einstein equations. The new results include the splitting of globally hyperbolic Finsler spacetimes, in addition to the analysis of several extensions of the Lorentz setting, as the case of timelike boundaries.

gr-qc↗

arXiv2026

Cosmological Spacetimes with Sign-Changing Spatial Curvature and Topological Transitions

Observational evidence, together with practical computations and modeling, supports a Euclidean spatial sector in the current cosmological model based on the FLRW metric. This, however, would imply that the total amount of matter and energy immediately after the Big Bang must have been infinite, an implication that could only be avoided through a transition from a closed to an open universe, a process forbidden in standard FLRW models. In this article, we investigate the spacetimes resulting from promoting the spatial curvature $k$ in FLRW spacetimes to a time-dependent function, $k \to k(t)$, allowing it to change sign and thereby allowing changes in the topology of the constant-$t$ slices. Although previously dismissed due to a classical theorem by Geroch, such transitions are shown to be consistent with global hyperbolicity when the comoving time is distinct from a Cauchy time, as recent work by one of the authors demonstrates. We construct three distinct geometries exhibiting this behavior using different representations of constant-curvature spaces. We analyze their global properties and identify mild conditions under which they remain globally hyperbolic. Furthermore, we characterize their Killing vectors, proving a general result for spherically symmetric spacetimes and compare them with known geometries in the literature.

gr-qc↗

arXiv2025

Finslerian lightconvex boundaries: applications to causal simplicity and the space of cone geodesics $\mathcal{N}$

Our outcome is structured in the following sequence: (1) a general result for indefinite Finslerian manifolds with boundary $(M,L)$ showing the equivalence between local and infinitesimal (time, light or space) convexities for the boundary $\partial M$, (2) for any cone structure $(M,\mathcal{C})$ which is globally hyperbolic with timelike boundary, the equivalence among: (a) the boundary $\partial M$ is lightconvex, (b) the interior $\mathring{M}$ is causally simple and (c) the space of the cone (null) geodesics $\mathcal{N}$ of $(\mathring{M},\mathcal{C})$ is Hausdorff, (3) in this case, the manifold structure of $\mathcal{N}$ is obtained explicitly in terms of elements in $\partial M$ and a smooth Cauchy hypersurface $S$, (4) the known results and examples about Hausdorfness of $\mathcal{N}$ are revisited and extended, leading to the notion of {\em causally simple spacetime with $T_2$-lightspace} as a step in the causal ladder below global hyperbolicity. The results are significant for relativistic (Lorentz) spacetimes and the writing allows one either to be introduced in Finslerian technicalities or to skip them. In particular, asymptotically AdS spacetimes become examples where the $C^{1,1}$ conformal extensions at infinity yield totally lightgeodesic boundaries, and all the results above apply.

math.DG↗

arXiv2025

Cosmological spacetimes with spatially constant sign-changing curvature

Globally hyperbolic spacetimes endowed with a time function $t$ whose spacelike slices $t=t_0$ have constant curvature $k(t_0)$ and where the sign of $k(t_0)$ (as well as the topology of the slice) varies with $t_0$, can be constructed despite some common claims about the implications of the classical Cosmological Principle. Here, we stress the possibilities of these cosmologies and announce the development of new models obtained in collaboration with G. Garc\'ıa-Moreno, B. Janssen, A. Jiménez-Cano, M. Mars and R. Vera

gr-qc↗

arXiv2025

Lorentzian-Euclidean black holes and Lorentzian to Riemannian metric transitions

In recent papers on spacetimes with a signature-changing metric, the concept of a Lorentzian-Euclidean black hole and new elements for Lorentzian-Riemannian signature change have been introduced. A Lorentzian-Euclidean black hole is a signature-changing modification of the Schwarzschild spacetime satisfying the vacuum Einstein equations in a weak sense. Here the event horizon serves as a boundary beyond which time becomes imaginary. We demonstrate that the proper time needed to reach the horizon remains finite, consistently with the classical Schwarzschild solution. About Lorentzian to Riemannian metric transitions, we stress that the hypersurface where the metric signature changes is naturally a spacelike hypersurface which can be identified with the future or past causal boundary of the Lorentzian sector. Moreover, a number of geometric interpretations appear, as the degeneracy of the metric corresponds to the collapse of the causal cones into a line, the degeneracy of the dual metric corresponds to collapsing into a hyperplane, and additional geometric structures on the transition hypersurface (Galilean and dual Galilean) might be explored.

gr-qc↗

arXiv2025

The ladder of Finsler-type objects and their variational problems on spacetimes

The space of anisotropic $r$-contravariant $s$-covariant $α$-homogeneous tensors on a manifold admits a functorial structure where vertical derivatives $\dot{\partial}$ and contractions $\imath_{\mathbb{C}}$ by the Liouville vector field $\mathbb{C}$ are operators which maintain $s+α$ constant. In (semi-)Finsler geometry, this structure is transmitted faithfully to connection-type elements yielding the following ladder: geodesic sprays / nonlinear connections / anisotropic connections / linear (Finslerian) connections. However, it is more loosely transmitted to metric-type ones: Finslerian Lagrangians / Legendre transformations / anisotropic metrics. We will study this structure in depth and apply it to discuss the recent variational proposals (Einstein-Hilbert, Einstein-Palatini, Einstein-Cartan) for generalizing Einstein equations to the Finsler setting.

math.DG↗

arXiv2024

Complete translating solitons in Lorentzian products

Obstructions to the existence of spacelike solitons depending on the growth of the mean curvature $H$ are proved for Lorentzian products $(M\times \mathbb{R}, \bar g=g_M-dt^2)$ with lowerly bounded curvature. The role of these bounds for both the completeness of the soliton $Σ$ and the applicability of the Omori-Yau principle for the drift Laplacian, is underlined. The differences between bounds in terms of the intrinsic distance $g$ of the soliton and the distance $g_M$ in the ambiance are analyzed, and lead to a revision of classic results on completeness for spacelike submanifolds. In particular, primary bounds, including affine $g$-bounds and logarithmic $g_M$-bounds, become enough to ensure both completeness and Omori-Yau's. Therefore, they become an obstruction to the existence of solitons when the ambiance Ricci is non-negative. These results, illustrated with a detailed example, deepen the uniqueness of solutions of elliptic equations which are not uniformly elliptic, providing insights into the interplay between mean curvature growth and the global properties of this geometric flow.

math.DG↗

arXiv2024

Lie groups with all left-invariant semi-Riemannian metrics complete

For each left-invariant semi-Riemannian metric $g$ on a Lie group $G$, we introduce the class of bi-Lipschitz Riemannian Clairaut metrics, whose completeness implies the completeness of $g$. When the adjoint representation of $G$ satisfies an at most linear growth bound, then all the Clairaut metrics are complete for any $g$. We prove that this bound is satisfied by compact and 2-step nilpotent groups, as well as by semidirect products $K \ltimes_ρ\mathbb{R}^n$ , where $K$ is the direct product of a compact and an abelian Lie group and $ρ(K)$ is pre-compact; they include all the known examples of Lie groups with all left-invariant metrics complete. The affine group of the real line is considered to illustrate how our techniques work even in the the absence of linear growth and suggest new questions.

math.DG↗

arXiv2024

Towards certification: A complete statistical validation pipeline for supervised learning in industry

Methods of Machine and Deep Learning are gradually being integrated into industrial operations, albeit at different speeds for different types of industries. The aerospace and aeronautical industries have recently developed a roadmap for concepts of design assurance and integration of neural network-related technologies in the aeronautical sector. This paper aims to contribute to this paradigm of AI-based certification in the context of supervised learning, by outlining a complete validation pipeline that integrates deep learning, optimization and statistical methods. This pipeline is composed by a directed graphical model of ten steps. Each of these steps is addressed by a merging key concepts from different contributing disciplines (from machine learning or optimization to statistics) and adapting them to an industrial scenario, as well as by developing computationally efficient algorithmic solutions. We illustrate the application of this pipeline in a realistic supervised problem arising in aerostructural design: predicting the likelikood of different stress-related failure modes during different airflight maneuvers based on a (large) set of features characterising the aircraft internal loads and geometric parameters.

cs.LG↗

arXiv2024

A general model for wildfire propagation with wind and slope

A geometric model for the computation of the firefront of a forest wildfire which takes into account several effects (possibly time-dependent wind, anisotropies and slope of the ground) is introduced. It relies on a general theoretical framework, which reduces the hyperbolic PDE system of any wave to an ODE in a Lorentz-Finsler framework. The wind induces a sort of double semi-elliptical fire growth, while the influence of the slope is modeled by means of a term which comes from the Matsumoto metric (i.e., the standard non-reversible Finsler metric that measures the time when going up and down a hill). These contributions make a significant difference from previous models because, now, the infinitesimal wavefronts are not restricted to be elliptical. Even though this is a technical complication, the wavefronts remain computable in real time. Some simulations of evolution are shown, paying special attention to possible crossovers of the fire.

math.DG↗

arXiv2024

A Gauss-Bonnet-Chern type obstruction for Killing vector fields on Lorentzian manifolds

A new curvature obstruction to the existence of a timelike (resp. causal) Killing or homothetic vector field $X$ on an even-dimensional (odd-dimensional) Lorentzian manifold, in terms of its timelike (resp. null) sectional curvature is given. As a consequence for the compact case, the well-known Gauss-Bonnet-Chern obstruction to the existence of semi-Riemannian metrics is extended from non-zero constant sectional curvature to non-zero timelike sectional curvature on $X$.

math.DG↗

arXiv2024

Gielis superformula and wildfire models

Experimental data on the propagation of wildfires show that its short-time spread has a double semi-elliptical shape. Our main goal is to show that this shape can be accurately approximated in polar coordinates by choosing suitable parameters in Gielis superformula and, then, implemented in a geometrical model for wildfire propagation. In this model, the firefront can be determined by computing the lightlike geodesics of a specific Finsler spacetime, and we derive a concise and efficient expression of the geodesic equations in these coordinates.

math.DG↗

arXiv2023

An account on links between Finsler and Lorentz Geometries for Riemannian Geometers

Some links between Lorentz and Finsler geometries have been developed in the last years, with applications even to the Riemannian case. Our purpose is to give a brief description of them, which may serve as an introduction to recent references. As a motivating example, we start with Zermelo navigation problem, where its known Finslerian description permits a Lorentzian picture which allows for a full geometric understanding of the original problem. Then, we develop some issues including: (a) the accurate description of the Lorentzian causality using Finsler elements, (b) the non-singular description of some Finsler elements (such as Kropina metrics or complete extensions of Randers ones with constant flag curvature), (c) the natural relation between the Lorentzian causal boundary and the Gromov and Busemann ones in the Finsler setting, and (d) practical applications to the propagation of waves and firefronts.

math.DG↗

arXiv2023

The Einstein-Hilbert-Palatini formalism in Pseudo-Finsler Geometry

A systematic development of the so-called Palatini formalism is carried out for pseudo-Finsler metrics $L$ of any signature. Substituting in the classical Einstein-Hilbert-Palatini functional the scalar curvature by the Finslerian Ricci scalar constructed with an independent nonlinear connection $\mathrm{N}$, the affine and metric equations for $(\mathrm{N},L)$ are obtained. In Lorentzian signature with vanishing mean Landsberg tensor Lan$_i$, both the Finslerian Hilbert metric equation and the classical Palatini conclusions are recovered by means of a combination of techniques involving the (Riemannian) maximum principle and an original argument about divisibility and fiberwise analyticity. Some of these findings are also extended to (positive definite) Riemannian metrics by using the eigenvalues of the Laplacian. When Lan$_i\neq 0$, the Palatini conclusions fail necessarily, however, a good number of properties of the solutions remain. The framework and proofs are built up in detail.

math.DG↗

arXiv2023

A class of cosmological models with spatially constant sign-changing curvature

We construct globally hyperbolic spacetimes such that each slice $\{t=t_0\}$ of the universal time $t$ is a model space of constant curvature $k(t_0)$ which may not only vary with $t_0\in\mathbb{R}$ but also change its sign. The metric is smooth and slightly different to FLRW spacetimes, namely, $g=-dt^2+dr^2+ S_{k(t)}^2(r) g_{\mathbb{S}^{n-1}}$, where $g_{\mathbb{S}^{n-1}}$ is the metric of the standard sphere, $S_{k(t)}(r)=\sin(\sqrt{k(t)}\, r)/\sqrt{k(t)}$ when $k(t)\geq 0$ and $S_{k(t)}(r)=\sinh(\sqrt{-k(t)}\, r)/\sqrt{-k(t)}$ when $k(t)\leq 0$. In the open case, the $t$-slices are (non-compact) Cauchy hypersurfaces of curvature $k(t)\leq 0$, thus homeomorphic to $\mathbb{R}^n$; a typical example is $k(t)=-t^2$ (i.e., $S_{k(t)}(r)=\sinh(tr)/t$). In the closed case, $k(t)>0$ somewhere, a slight extension of the class shows how the topology of the $t$-slices changes. This makes at least one comoving observer to disappear in finite time $t$ showing some similarities with an inflationary expansion. Anyway, the spacetime is foliated by Cauchy hypersurfaces homeomorphic to spheres, not all of them $t$-slices.

gr-qc↗

arXiv2022

Anisotropic connections and parallel transport in Finsler spacetimes

The general notion of anisotropic connections $\nabla$ is revisited, including its precise relations with the standard setting of pseudo-Finsler metrics, i.e., the canonic nonlinear connection and the (linear) Finslerian connections. In particular, the vertically trivial Finsler connections are identified canonically with anisotropic connections. So, these connections provide a simple intrinsic interpretation of a part of any Finsler connection closer to the Koszul formulation in $M$. Moreover, a new covariant derivative and parallel transport along curves is introduced, taking first a self-propagated vector ({\em instantaneous observer}) so that it serves as a reference for the propagation of the others. The covariant derivative of any anisotropic tensor is given by the natural derivative of a curve of tensors obtained by parallel transport along a curve and, in the case of pseudo-Finsler metrics, this is used to characterize the Levi-Civita--Chern anisotropic connection as the one that preserves the length of parallely propagated vectors.

math.DG↗