arXiv subjects

Benigno O. Alves

Publications and source records attributed to Benigno O. Alves.

4 recordsLinked to original sources

arXiv2026

Generalized Semi-Riemannian Submersions and Foliations

In this article, we introduce the concepts of generalized semi-Riemannian submersions and foliations, extending the classical framework to accommodate leaves with varying or degenerate causal characters, such as homogeneous foliations on semi-Riemannian manifolds and codimension-one lightlike foliations on Lorentz manifolds (including pp-waves). We establish that a regular foliation with a basic horizontal distribution is generalized semi-Riemannian if and only if it is transnormal. Furthermore, we introduce stationary foliations and prove that a stationary transnormal foliation induces a classical Riemannian foliation on the spatial rest space of a conformal observer. As a geometric consequence, we obtain a rigidity result ruling out stationary codimension-one lightlike foliations on positively curved Robertson-Walker spacetimes or static spacetimes. Finally, we derive an O'Neill-type sectional curvature formula for submersions with an involutive total distribution.

math.DG↗

arXiv2026

On singular Finsler foliations of $(α,β)$-spaces

We investigate singular Finsler foliations (SFFs) on a manifold equipped with an $(α,β)$-metric. To be precise, we verify that any SFF of an $(α,β)$-space is, under some hypotheses on the metric, a singular Riemannian foliation (SRF). This gives a partial answer to the general question "under which conditions a SFF is a SRF with respect to some Riemannian metric". Moreover, we extend the proof of Molino's conjecture to SFFs whenever they are also a SRFs. Finally, we prove equifocality of the regular leaves for a SFF under the same condition.

math.DG↗

arXiv2018

On Finsler transnormal functions

In this note we discuss a few properties of transnormal Finsler functions, i.e., the natural generalization of distance functions and isoparametric Finsler functions. In particular, we prove that critical level sets of an analytic transnormal function are submanifolds, and the partition of $M$ into level sets is a Finsler partition, when the function is defined on a compact analytic manifold $M$.

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↗