On Finsler transnormal functions

TL;DR

This paper explores properties of transnormal Finsler functions, showing that their critical level sets are submanifolds and that the level set partition forms a Finsler partition on compact analytic manifolds.

Contribution

It establishes new geometric properties of transnormal Finsler functions, including submanifold structure of critical sets and Finsler partitions on compact manifolds.

Findings

Critical level sets are submanifolds.

Partition of the manifold into level sets is a Finsler partition.

Results apply to analytic functions on compact manifolds.

Abstract

In this note we discuss a few properties of transnormal Finsler functions, i.e., the natural generalization of distance functions and isoparametric Finsler functions. In particular, we prove that critical level sets of an analytic transnormal function are submanifolds, and the partition of $M$ into level sets is a Finsler partition, when the function is defined on a compact analytic manifold $M$.