In this chapter, we concentrate on local results that have been obtained using Lie sphere geometry. The main results presented here are the classification of proper Dupin submanifolds with two principal curvatures (cyclides of Dupin) in Section 5.4 and the classification of proper Dupin hypersurfaces with three principal curvatures in R4 in Section 5.7. To obtain these classifications, we develop the method of moving Lie frames which can be used in the further study of Dupin submanifolds, or more generally, Legendre submanifolds.
KeywordsPrincipal Curvature Isoparametric Hypersurface Cartan Form Distinct Principal Curvature Curvature Sphere
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