On equifocal Finsler submanifolds and analytic maps

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 $π:M^{m+k}\to B^{k}$ 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.

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