Mathematical Sciences: RUI: Dupin Submanifolds

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9504535 Cecil A hypersurface in Euclidean n-space is said to be isoparametric if it has constant principal curvatures; it can be shown that such a hypersurface must be a plane, a sphere or a spherical cylinder. Isoparametric hypersurfaces in the unit n-sphere, however, are a rich source of interesting examples; its complete classification is still lacking. Dupin hypersurfaces generalize isoparametric hypersurfaces in that the principal curvatures are assumed to be constant only along certain directions. The proposed research will attempt to classify Dupin hypersurfaces with 4 or 6 distinct principal curvatures. Cartan showed that such a hypersurface can have 1, 2, 3, 4, or 6 distinct principal curvatures; when there are 1 or 2 distinct principal curvatures the classification problem iq routine, and the case of 3 distinct principal curvatures was recently done by the proposer and G. Jensen. Isoparametric and Dupin hypersurfaces form a particularly beautiful subarea of classical differential geometry; these surfaces are highly symmetric and nontrivial at the same time.