On singular Finsler foliation

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.

22 pages, we add Corollary 1.2, correct several misprints and improve a few arguments

References in corpus (1)

• Closure of singular foliations: the proof of Molino's conjecture

Cited by in corpus (3)

• Huygens' envelope principle in Finsler spaces and analogue gravity
• On Finsler transnormal functions
• Homogeneous nonlinear splittings and Finsler submersions