Finsler等参函数与等参超曲面

Isoparametric functions and hypersurfaces in Finsler geometry

本文对Riemann-Finsler几何中等参理论研究进展和若干尚未解决的问题作概要综述.首先,本文简要回顾Riemann情形下等参问题的研究历史以及Riemann空间形式中等参超曲面相关问题的重要成果.随后,本文着重介绍Finsler流形中等参超曲面的相关概念和一些研究进展,特别是Finsler空间形式中等参问题的研究进展.内容主要包括超常函数、等参函数及其焦点集的性质,特殊Finsler空间中等参超曲面的分类,以及研究Finsler等参问题的一种重要方法—导航术.

In this paper,we provide an overview of the progress and some open problems on isoparametric theory in Riemann-Finsler geometry.Firstly,we briefy recall the history of isoparametric problems in the Riemannian case and introduce some important results on isoparametric hypersurfaces in Riemannian space forms.Then we elaborate on the isoparametric problems in the Finsler setting.We introduce some concepts and development of isoparametric hypersurfaces in Finsler geometry,especially some progress in Finsler space forms,containing some properties of the transnormal functions,the isoparametric functions,and the focal varieties,and some classifications of isoparametric hypersurfaces in some special Finsler spaces.In particular,we present an important method-navigation technology-to study isoparametric problems in Finsler geometry.