Isoparametric functions on Finsler space forms

TL;DR

This paper extends the theory of isoparametric functions to Finsler space forms, establishing conditions under which transnormal functions are isoparametric, and explores their geometric relationships and constructions in Finsler geometry.

Contribution

It generalizes a key theorem from Riemannian geometry to Finsler spaces and constructs explicit examples of isoparametric functions on Finsler spheres.

Findings

Transnormal functions are isoparametric under certain conditions.

Relationship between umbilic and isoparametric hypersurfaces established.

Constructed explicit isoparametric functions on Finsler spheres.

Abstract

In this paper, we prove that transnormal functions are isoparametric functions on Finsler space forms (N(c), F) under certain conditions, which generalize Theorem B given by Q.M. Wang in Riemannian case. Next, we discuss the relationship between umbilic hypersurfaces and isoparametric hypersurfaces in (N(c), F). Further more, we construct a locally isoparametric function by a distance function and give a global isoparametric function on a standard Finsler sphere.