arXiv subjects

José L. Flores

Publications and source records attributed to José L. Flores.

8 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

A Lifting principle of curves under exponential-type maps

We develop a lifting theory for the exponential map of semi-Riemannian manifolds that overcomes the classical obstruction caused by its singularities. We show that every smooth path in the manifold admits, up to a nondecreasing reparametrization, a partial lift through the exponential map which is inextensible in its domain of definition. If the exponential map satisfies the path-continuation property-a natural topological condition-these lifts extend globally, yielding a general path-lifting theorem. This lifting approach yields new, alternative proofs of (generalizations of) a number of foundational results in semi-Riemannian geometry: the Hopf-Rinow theorem and Serre's classic theorem about multiplicity of connecting geodesics in the Riemannian case, as well as the Avez-Seifert theorem for globally hyperbolic spacetimes in Lorentzian geometry. More broadly, our results reveal the central role of the continuation property in obtaining geodesic connectivity across a wide range of semi-Riemannian geometries. This offers a unifying geometric principle that is complementary to the more traditional analytic, variational methods used in to investigate geodesic connectedness, and provides new insight into the structure of geodesics, both on geodesically complete and non-complete manifolds. We also briefly point out how the lifting theory developed here can etend to more general flow-inducing maps on the tangent bundle other than the geodesic flow, suggesting broader geometric applicability beyond the exponential map.

math.DG↗

arXiv2023

On the existence of sections with constant surface gravity on null hypersurfaces

We identify, in spacetimes satisfying the null convergence condition, a certain natural class of null hypersurfaces that admit null sections with constant surface gravity. Our work is meant to offer complementary results to previous work on null hypersurfaces of zero expansion arising from the study of Cauchy and black hole horizons in general relativity.

gr-qc↗

arXiv2022

Locally Extremal Timelike Geodesic Loops on Lorentzian Manifolds

Conditions for the existence of closed geodesics is a classic, much-studied subject in Riemannian geometry, with many beautiful results and powerful techniques. However, many of the techniques that work so well in that context are far less effective in Lorentzian geometry. In revisiting this problem here, we introduce the notion of timelike geodesic homotopy, a restriction to geodesics of the more standard timelike homotopy (also known as $t$-homotopy) of timelike loops on Lorentzian manifolds. This tool is combined with a local shortening/stretching of length argument to provide a number of new results on the existence of closed timelike geodesics on compact Lorentz manifolds.

math.DG↗

arXiv2021

Path-lifting properties of the exponential map with applications to geodesics

We revisit certain path-lifting and path-continuation properties of abstract maps as described in the work of F. Browder and R. Rheindboldt in 1950-1960s, and apply their elegant theory to exponential maps. We obtain thereby a number of novel results of existence and multiplicity of geodesics joining any two points of a connected affine manifold, as well as ${\it causal}$ geodesics connecting any two causally related points on a Lorentzian manifold. These results include a generalization of the well-known Hadamard-Cartan theorem of Riemannian geometry to the affine manifold context, as well as a new version of the so-called Lorentzian Hadamard-Cartan theorem using weaker assumptions than global hyperbolicity and timelike 1-connectedness required in the extant version. We also include a general discription of ${\it pseudoconvexity}$ and ${\it disprisonment}$ of broad classes of geodesics in terms of suitable restrictions of the exponential map. The latter description sheds further light on the the relation between pseudoconvexity and disprisonment of a given such class on the one hand, and geodesic connectedness by members of that class on the other.

math.DG↗

arXiv2020

Geodesic Connectedness of Affine Manifolds

We discuss new sufficient conditions under which an affine manifold $(M,\nabla)$ is geodesically connected. These conditions are shown to be essentially weaker than those discussed in groundbreaking work by Beem and Parker and in recent work by Alexander and Karr, with the added advantage that they yield an elementary proof of the main result.

math.DG↗

arXiv2019

Ehlers-Kundt Conjecture about Gravitational Waves and Dynamical Systems

Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane ${\mathbb R}^2$ with a very simple formulation in Classical Mechanics: given a non-necessarily autonomous potential $V(z,u)$, $(z,u)\in {\mathbb R}^2\times {\mathbb R}$, harmonic in $z$ (i.e. source-free), the trajectories of its associated dynamical system $\ddot{z}(s)=-\nabla_z V(z(s),s)$ are complete (they live eternally) if and only if $V(z,u)$ is a polynomial in $z$ of degree at most $2$ (so that $V$ is a standard mathematical idealization of vacuum). Here, the conjecture is solved in the significative case that $V$ is bounded polynomially in $z$ for finite values of $u\in {\mathbb R}$. The mathematical and physical implications of this {\em polynomial EK conjecture}, as well as the non-polynomial one, are discussed beyond their original scope.

gr-qc↗

arXiv2019

Structure of globally hyperbolic spacetimes with timelike boundary

Globally hyperbolic spacetimes with timelike boundary $(\overline{M} = M \cup \partial M, g)$ are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if $\overline{M}$ is obtained by means of a conformal embedding) can be posed. $\partial M$ represents the naked singularities and can be identified with a part of the intrinsic causal boundary. Apart from general properties of $\partial M$, the splitting of any globally hyperbolic $(\overline{M},g)$ as an orthogonal product ${\mathbb R}\times \barΣ$ with Cauchy slices with boundary $\{t\}\times \barΣ$ is proved. This is obtained by constructing a Cauchy temporal function $τ$ with gradient $\nabla τ$ tangent to $\partial M$ on the boundary. To construct such a $τ$, results on stability of both, global hyperbolicity and Cauchy temporal functions are obtained. Apart from having their own interest, these results allow us to circumvent technical difficulties introduced by $\partial M$. As a consequence, the interior $M$ both, splits orthogonally and can be embedded isometrically in ${\mathbb L}^N$, extending so properties of globally spacetimes without boundary to a class of causally continuous ones.

gr-qc↗