arXiv subjects

Ivan P. Costa e Silva

Publications and source records attributed to Ivan P. Costa e Silva.

13 recordsLinked to original sources

arXiv2026

Transverse Geometry of Lorentzian foliations with applications to Lorentzian orbifolds

We prove a transverse diameter theorem in the context of Lorentzian foliations, which can be interpreted as a Hawking--Penrose-type singularity theorem for timelike geodesics transverse to the foliation. In order to develop the necessary machinery we introduce and study a novel causality structure on the leaf space via the transverse Lorentzian geometry on the foliated manifold. We describe the initial rungs of a transverse causal ladder and relate them to their standard counterparts on an underlying foliated spacetime. We show how these results can be interpreted as doing Lorentzian (and more generally semi-Riemannian) geometry on low-regularity spaces that can be realized as leaf spaces of foliations. Accordingly, we discuss how all of these concepts and results apply to Lorentzian orbifolds, insofar as these can be seen as leaf spaces of a specific class of Lorentzian foliations. In particular, we derive an associated Lorentzian timelike diameter theorem on orbifolds.

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↗

arXiv2025

A Hadamard theorem in transversely affine geometry with applications to affine orbifolds

We introduce and investigate a novel notion of transversely affine foliation, comparing and contrasting it to the previous ones in the literature. We then use it to give an extension of the classic Hadamard's theorem from Riemannian geometry to this setting. Our main result is a transversely affine version of a well-known "Hadamard-like" theorem by J. Hebda for Riemannian foliations. Alternatively, our result can be viewed as a foliation-theoretic analogue of the Hadamard's theorem for affine manifolds proven by Beem and Parker. Namely, we show that under the transverse analogs of pseudoconvexity and disprisonment for the family of geodesics in the transverse affine geometry, together with an absence of transverse conjugate points, the universal cover of a manifold endowed with a transversely affine foliation whose leaves are compact and with finite holonomy is diffeomorphic to the product of a contractible manifold with the universal cover of a leaf. This also leads to a Beem-Parker-type Hadamard-like theorem for affine orbifolds.

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

Omniscient foliations and the geometry of cosmological spacetimes

We identify certain general geometric conditions on a foliation of a spacetime (M,g) by timelike curves that will impede the existence of null geodesic lines, especially if (M,g) possesses a compact Cauchy hypersurface. The absence of such lines, in turn, yields well-known restrictions on the geometry of cosmological spacetimes, in the context of Bartnik's splitting conjecture. Since the (non)existence of null lines is actually a conformally invariant property, such conditions only need to apply for some suitable conformal rescaling of g.

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

A note on causality conditions on covering spacetimes

A number of techniques in Lorentzian geometry, such as those used in the proofs of singularity theorems, depend on certain smooth coverings retaining interesting global geometric properties, including causal ones. In this note we give explicit examples showing that, unlike some of the more commonly adopted rungs of the causal ladder such as strong causality or global hyperbolicity, less-utilized conditions such as causal continuity or causal simplicity do not in general pass to coverings, as already speculated by one of the authors (EM). As a consequence, any result which relies on these causality conditions transferring to coverings must be revised accordingly. In particular, some amendments in the statement and proof of a version of the Gannon-Lee singularity theorem previously given by one of us (IPCS) are also presented here that address a gap in its original proof.

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

Some remarks on conformal symmetries and Bartnik's splitting conjecture

Inspired by the results in a recent paper by G. Galloway and C. Vega (see arXiv:1712.00785), we investigate a number of geometric consequences of the existence of a timelike conformal Killing vector field on a globally hyperbolic spacetime with compact Cauchy hypersurfaces, especially in connection with the so-called Bartnik's splitting conjecture. In particular we give a complementary result to the main theorem in Galloway and Vega's paper.

math.DG↗

arXiv2018

Hausdorff closed limits and the causal boundary of globally hyperbolic spacetimes with timelike boundary

We show that when a spacetime $\mathcal{M}(=M \cup \partial M)$ is globally hyperbolic with (possibly empty) smooth timelike boundary $\partial M$, a metrizable topology, the closed limit topology (CLT) introduced by F. Hausdorff himself in the 1950's in set theory, can be advantageously adopted on the Geroch-Kronheimer-Penrose causal completion of M, retaining essentially all the good properties of previous topologies in this ambient. In particular, we show that if the globally hyperbolic spacetime $M$ admits a conformal boundary, defined in such broad terms as to include all the standard examples in the literature, then the latter is homeomorphic to the causal boundary endowed with the CLT. We also discuss how our recent proposal arXiv:1807.00152 for a definition of null infinity using only causal boundaries can be translated when using the CLT, simplifying a number of technical aspects in the pertinent definitions. In a more technical vein, in the appendix we discuss the relationship of the CLT with the more generally applicable (but not always Hausdorff) chronological topology, and show that they coincide exactly in those cases when the latter happens to be Hausdorff.

math-ph↗

arXiv2016

Rigidity of geodesic completeness in the Brinkmann class of gravitational wave spacetimes

We consider restrictions placed by geodesic completeness on spacetimes possessing a null parallel vector field, the so-called Brinkmann spacetimes. This class of spacetimes includes important idealized gravitational wave models in General Relativity, namely the plane-fronted waves with parallel rays, or pp-waves, which in turn have been intensely and fruitfully studied in the mathematical and physical literatures for over half a century. More concretely, we prove a restricted version of a conjectural analogue for Brinkmann spacetimes of a rigidity result obtained by M.T. Anderson for stationary spacetimes. We also highlight its relation with a long-standing 1962 conjecture by Ehlers and Kundt. Indeed, it turns out that the subclass of Brinkmann spacetimes we consider in our main theorem is enough to settle an important special case of the Ehlers-Kundt conjecture in terms of the well known class of Cahen-Wallach spaces.

math-ph↗

arXiv2016

On the splitting problem for Lorentzian manifolds with an $\mathbb{R}$-action with causal orbits

We study the interplay between the global causal and geometric structures of a spacetime $(M,g)$ and the features of a given smooth $\mathbb{R}$-action $ρ$ on $M$ whose orbits are all causal curves, building on classic results about Lie group actions on manifolds described by Palais. In the first part of this paper, we prove that $ρ$ is free and proper (so that $M$ splits topologically) provided that $(M,g)$ is strongly causal and $ρ$ does not have what we call weakly ancestral pairs, a notion which admits a natural interpretation in terms of "cosmic censorship". Accordingly, such condition holds automatically if $(M,g)$ is globally hyperbolic. We also prove that $M$ splits topologically if $(M,g)$ is strongly causal and $ρ$ is the flow of a complete conformal Killing causal vector field. In the second part, we investigate the class of Brinkmann spacetimes, which can be regarded as null analogues of stationary spacetimes in which $ρ$ is the flow of a complete parallel null vector field. Inspired by the geometric characterization of stationary spacetimes in terms of standard stationary ones [24], we obtain an analogous geometric characterization of when a Brinkmann spacetime is isometric to a standard Brinkmann spacetime. This result naturally leads us to discuss a conjectural null analogue for Ricci-flat $4$-dimensional Brinkmann spacetimes of a celebrated rigidity theorem by Anderson [1], and highlight its relation with a long-standing 1962 conjecture by Ehlers and Kundt [13]. If true, our conjecture provides strong mathematical support to the idea that gravitational plane waves are the most natural "boundary conditions" at infinity for vacuum solutions of the Einstein equation modeling regions outside gravitationally radiating sources.

math-ph↗