Isoparametric Hypersurfaces in Minkowski Spaces

TL;DR

This paper extends the concept of isoparametric hypersurfaces to Minkowski and Finsler spaces, classifies them in certain cases, and challenges existing theorems in Finsler geometry with counterexamples.

Contribution

It introduces isoparametric functions in Finsler manifolds, classifies hypersurfaces in Minkowski spaces, and provides a counterexample to a previous theorem.

Findings

Hyperplanes, Minkowski hyperspheres, and cylinders are isoparametric hypersurfaces.

Complete classification of isoparametric hypersurfaces in Randers-Minkowski spaces.

Counterexample disproves Wang's Theorem B in Finsler geometry.

Abstract

In this paper, we introduce isoparametric functions and isoparametric hypersurfaces in Finsler manifolds and give the necessary and sufficient conditions for a transnormal function to be isoparametric. We then prove that hyperplanes, Minkowski hyperspheres and $F^*$-Minkowski cylinders in a Minkowski space with $BH$-volume (resp. $HT$-volume) form are all isoparametric hypersurfaces with one and two distinct constant principal curvatures respectively. Moreover, we give a complete classification of isoparametric hypersurfaces in Randers-Minkowski spaces and construct a counter example, which shows that Wang's Theorem B in \cite{WQ} does not hold in Finsler geometry.