On singular Finsler foliations of $(\alpha,\beta)$-spaces

TL;DR

This paper studies singular Finsler foliations on $(eta,eta)$-spaces, showing under certain conditions they are equivalent to singular Riemannian foliations, extending Molino's conjecture and establishing equifocality of regular leaves.

Contribution

It demonstrates that singular Finsler foliations on $(eta,eta)$-spaces are often SRFs, extends Molino's conjecture to this setting, and proves equifocality of regular leaves.

Findings

SFFs on $(eta,eta)$-spaces are SRFs under certain conditions.

Extended Molino's conjecture to SFFs that are SRFs.

Proved equifocality of regular leaves in SFFs.

Abstract

We investigate singular Finsler foliations (SFFs) on a manifold equipped with an $(\alpha,\beta)$-metric. To be precise, we verify that any SFF of an $(\alpha,\beta)$-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.