Classifications of Dupin Hypersurfaces in Lie Sphere Geometry

TL;DR

This survey reviews classification results of Dupin hypersurfaces within Lie sphere geometry, emphasizing their relation to isoparametric hypersurfaces and detailing key concepts and constructions in the field.

Contribution

It provides a comprehensive overview of classification results and key concepts in Lie sphere geometry related to Dupin hypersurfaces, including new constructions.

Findings

Classification of Dupin hypersurfaces in spheres and Euclidean spaces

Connections between Dupin and isoparametric hypersurfaces

Construction methods for special Dupin hypersurfaces

Abstract

This is a survey of local and global classification results concerning Dupin hypersurfaces in $S^n$ (or ${\bf R}^n$) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres. Along with these classification results, many important concepts from Lie sphere geometry, such as curvature spheres, Lie curvatures, and Legendre lifts of submanifolds of $S^n$ (or ${\bf R}^n$), are described in detail. The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.