Dupin hypersurfaces in Lie sphere geometry

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.