Isoparametric and Dupin Hypersurfaces

TL;DR

This survey reviews the classification and properties of isoparametric and Dupin hypersurfaces in real space-forms, highlighting recent advances and open problems in the field of differential geometry.

Contribution

It consolidates recent results and discusses open problems related to isoparametric and Dupin hypersurfaces, emphasizing developments in Lie sphere geometry.

Findings

Complete classification in Euclidean and hyperbolic spaces

Complex classification challenges in spheres

Recent progress and open problems in the field

Abstract

A hypersurface $M^{n-1}$ in a real space-form ${\bf R}^n$, $S^n$ or $H^n$ is isoparametric if it has constant principal curvatures. For ${\bf R}^n$ and $H^n$, the classification of isoparametric hypersurfaces is complete and relatively simple, but as Elie Cartan showed in a series of four papers in 1938-1940, the subject is much deeper and more complex for hypersurfaces in the sphere $S^n$. A hypersurface $M^{n-1}$ in a real space-form is proper Dupin if the number $g$ of distinct principal curvatures is constant on $M^{n-1}$, and each principal curvature function is constant along each leaf of its corresponding principal foliation. This is an important generalization of the isoparametric property that has its roots in nineteenth century differential geometry and has been studied effectively in the context of Lie sphere geometry. This paper is a survey of the known results in these fields with emphasis on results that have been obtained in more recent years and discussion of important open problems in the field.