Abstract
This chapter is an outline of the method for studying submanifolds of Euclidean space R n or the sphere S n in the context of Lie sphere geometry. For Dupin hypersurfaces this has proven to be a valuable approach, since Dupin hypersurfaces occur naturally as envelopes of families of spheres, which can be handled well in Lie sphere geometry. Since the Dupin property is invariant under Lie sphere transformations, this is also a natural setting for classification theorems.
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