Multiplicities of Dupin hypersurfaces
There are two types of examples of isoparametric hypersurfaces in spheres: (1) the homogeneous ones which are principal orbits of isotropy representations of symmetric spaces of rank two; and (2) the Clifford examples of Ferus, Karcher and Münzner. All examples of compact (proper) Dupin hypersurfaces are related to these two classes of isoparametric hypersurfaces. The author proves in this remarkable paper that the multiplicities of the principal curvatures of an isoparametric hypersurface, and, more generally, of a Dupin hypersurface are as in the known examples. \textit{H. F. Münzner} [Math. Ann. 256, 215-232 (1981; Zbl 0451.53043)] had previously proved that the number of different prinicipal curvatures of isoparametric hypersurfaces is \(1\), \(2\), \(3\), \(4\) or \(6\), i.e., as in the examples, and the reviewer extended this to compact (proper) Dupin hypersurfaces in [Bull. London Math. Soc. 15, 493-498 (1983; Zbl 0529.53044)]. Previous work on the multiplicity question was done by \textit{U. Abresch} [Math. Ann. 264, 283-302 (1983; Zbl 0514.53050)], \textit{K. Grove} and \textit{S. Halperin} [J. Differ. Geom. 26, 479-502 (1987; Zbl 0637.53075)] and \textit{F. Fang} in [``Multiplicities of principal curvatures of isoparametric hypersurfaces'', preprint MPI 1996-80 (1996)]. The proof of the main result is homotopy theoretic and uses the Hopf invariant and the EHP-sequence.
scientific article; zbMATH DE number 1383526