arXiv subjects

Francisco C. Caramello Jr

Publications and source records attributed to Francisco C. Caramello Jr.

5 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↗

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.

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arXiv2023

Equivariant basic cohomology of singular Riemannian foliations

We introduce the notion of equivariant basic cohomology for singular Riemannian foliations with transverse infinitesimal actions, and prove some elementary properties such as its invariance under homotopies. For the particular case of singular Killing foliations, there is the natural transverse action of its structural algebra. We prove that, modulo torsion, its equivariant basic cohomology localizes to the set of closed leaves of the foliation, in the spirit of the classical localization theorem of Borel. As applications, we obtain that the basic Euler characteristic also localizes to this set, and that the dimension of the basic cohomology of the localized foliation is less than or equal to that of the whole foliation, with equality occurring precisely in the equivariantly formal case.

math.DG↗

arXiv2022

Leaf closures of Riemannian foliations: a survey on topological and geometric aspects of Killing foliations

A smooth foliation is Riemannian when its leaves are locally equidistant. The closures of the leaves of a Riemannian foliation on a simply connected manifold, or more generally of a Killing foliation, are described by flows of transverse Killing vector fields. This offers significant technical advantages in the study of this class of foliations, which nonetheless includes other important classes, such as those given by the orbits of isometric Lie group actions. Aiming at a broad audience, in this survey we introduce Killing foliations from the very basics, starting with a brief revision of the main objects appearing in this theory, such as pseudogroups, sheaves, holonomy and basic cohomology. We then review Molino's structural theory for Riemannian foliations and present its transverse counterpart in the theory of complete pseudogroups of isometries, emphasizing the connections between these topics. We also survey some classical results and recent developments in the theory of Killing foliations. Finally, we review some topics in the theory of singular Riemannian foliations and discuss singular Killing foliations.

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arXiv2022

Introduction to orbifolds

We introduce orbifolds from the classical point of view, using charts, and present orbifold versions of elementary objects from Algebraic Topology, such as the fundamental group, coverings and Euler characteristic; Differential Topology/Geometry, including orbibundles, differential forms, integration and (equivariant) De Rham cohomology; and Riemannian Geometry, surveying generalizations of classical theorems to this setting.

math.DG↗