On equifocal Finsler submanifolds and analytic maps

TL;DR

This paper establishes conditions under which the fibers of an analytic map in a Finsler manifold are equifocal submanifolds, extending the concept of isoparametric submanifolds to Finsler geometry and analyzing their foliations.

Contribution

It proves that regular fibers of certain analytic maps are equifocal in Finsler manifolds and shows that singular foliations formed by these fibers are Finsler, generalizing known Riemannian results.

Findings

Regular fibers are equifocal under Finsler submersion conditions

Singular foliations by fibers are Finsler foliations

Extends isoparametric submanifold theory to Finsler geometry

Abstract

A relevant property of equifocal submanifolds is that their parallel sets are still immersed submanifolds, which makes them a natural generalization of the so-called isoparametric submanifolds. In this paper, we prove that the regular fibers of an analytic map $\pi:M^{m+k}\to B^{k}$ are equifocal whenever $M^{m+k}$ is endowed with a complete Finsler metric and there is a restriction of $\pi$ which is a Finsler submersion for a certain Finsler metric on the image. In addition, we prove that when the fibers provide a singular foliation on $M^{m+k}$, then this foliation is Finsler.