Transnormal Functions and Focal Varieties on Finsler Manifolds

On Finsler transnormal functions

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 $M$.

Some Rigidity Results on Complete Finsler Manifolds

We provide an extension of Obata's theorem to Finsler geometry and establish some rigidity results based on a second order differential equation. Mainly, we prove that every complete connected Finsler manifold of positive constant flag curvature is isometrically homeomorphic to an Euclidean sphere endowed with a certain Finsler metric and vice versa. Based on these results, we present a classification of Finsler manifolds which admit a transnormal function. Specifically, we show that if a complete Finsler manifold admits a transnormal function with exactly two critical points, then it is homeo…

Finsler空间形式上的等参函数

Isoparametric functions on Finsler space forms

本文在Finsler空间形式$(N(c),F)$上证明在一定条件下超正则函数是等参函数，推广了Riemann情形的定理B；并在前向Finsler空间形式$(\overrightarrow{N}(c),F)$上讨论脐超曲面与等参超曲面的关系，利用距离函数构造全局和局部等参函数。

Isoparametric Hypersurfaces in Minkowski Spaces

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 sh…

Isoparametric hypersurfaces in Finsler space forms

In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain the Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.

The isoparametric functions on a class of Finsler spheres

In this paper, we give global expressions of geodesics and isoparametric functions on a Randers sphere by navigation. We obtain isoparametric families and focal submanifolds in (S^{n}; F; dμ_{BH}) by Cartan-Münzner polynomials. Further more, we construct some examples of closed and non-closed geodesics, isoparametric functions, isoparametric families and focal submanifolds.