On singular Finsler foliation

TL;DR

This paper introduces the concept of singular Finsler foliation, showing its relation to singular Riemannian foliations on Randers manifolds and exploring conditions under which leaves are equifocal submanifolds.

Contribution

It generalizes Finsler foliations, connects them to Riemannian cases, and provides a slice theorem relating local foliations on Finsler and Minkowski spaces.

Findings

Singular Finsler foliation on Randers manifolds induces a singular Riemannian foliation.

Regular leaves are equifocal submanifolds when the wind W is an infinitesimal homothety.

A slice theorem relates local singular Finsler foliations on Finsler and Minkowski spaces.

Abstract

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.