Topology of Dupin hypersurfaces with six distinct principal curvatures
The classification of isoparametric and, more generally, of compact (proper) Dupin hypersurfaces in spheres with four and six distinct principal curvatures is still an open problem. In the present paper, the author classifies the homeomorphism types of Dupin hypersurfaces with six different principal curvatures and proves that they are homeomorphic to the known examples. There are two cases: either the dimension of the hypersurface is six and it is even diffeomorphic to \(SU(2)\times SU(2)/Q_8\) where \(Q_8=\{\pm 1,\pm i,\pm j,\pm k\}\), or its dimension is twelve and it is homeomorphic to a principal orbit of the adjoint representation of \(G_2\).
scientific article; zbMATH DE number 1336882