Using Lie Sphere Geometry to Study Dupin Hypersurfaces in ${\bf R}^n$

TL;DR

This paper introduces Lie sphere geometry as a powerful framework for analyzing Dupin hypersurfaces in Euclidean and spherical spaces, providing new insights and classifications of these geometric structures.

Contribution

It presents a detailed introduction to using Lie sphere geometry to study Dupin hypersurfaces, including proofs of key results and invariance properties.

Findings

Lie sphere geometry effectively classifies proper Dupin hypersurfaces.

Dupin properties are invariant under Lie sphere transformations.

The paper provides fundamental proofs related to Dupin hypersurfaces in this setting.

Abstract

A hypersurface $M$ in ${\bf R}^n$ or $S^n$ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on $M$, i.e., the number of distinct principal curvatures is constant on $M$. The notions of Dupin and proper Dupin hypersurfaces in ${\bf R}^n$ or $S^n$ can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective setting for the study of Dupin hypersurfaces, and many classifications of proper Dupin hypersurfaces have been obtained up to Lie sphere transformations. In these notes, we give a detailed introduction to this method for studying Dupin hypersurfaces in ${\bf R}^n$ or $S^n$, including proofs of several fundamental results.