On equifocal Finsler submanifolds and analytic maps

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 π : M^{m+k} → B^k are equifocal whenever M^{m+k} is endowed with a complete Finsler metric and there is a restriction of π 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.