Dupin hypersurfaces in Lie sphere geometry

TL;DR

This paper demonstrates how the method of moving frames effectively classifies Dupin hypersurfaces in Lie sphere geometry, revealing their congruence and simplifying their understanding.

Contribution

It provides a unified classification approach for Dupin hypersurfaces across different geometries using the method of moving frames.

Findings

All nonumbilic Dupin immersions are Lie sphere congruent.

The method simplifies classification of cyclides of Dupin.

Elementary proof of classification in space form geometries.

Abstract

The method of moving frames in Lie sphere geometry has produced significant results in the classification of Dupin hypersurfaces in spheres. What is the secret of its effectiveness? The answer emerges in the classification of nonumbilic isoparametric surfaces in the space form geometries. Using the method of moving frames, the proof of this classification is an elementary exercise. The same proof classifies the cyclides of Dupin in M\"obius geometry and finally in Lie sphere geometry, where all nonumbilic Dupin immersions are Lie sphere congruent to each other.