On the topology of Clifford isoparametric hypersurfaces
In [the reviewer, \textit{H. Karcher} and \textit{H.-F. Münzer}, Math. Z. 177, 479-502 (1981; Zbl 0443.53037)] we constructed from each orthogonal representation of the Clifford algebra \(C_{m-1}\) on \({\mathbb{R}}^{\ell}\) an isoparametric function on \(S^{2\ell -1}\), the regular levels of which have 4 distinct principal curvatures. For \(m\equiv 0\) mod 4 there are inequivalent representations in the same dimension \(\ell\). For these the author studies the isoparametric levels in terms of homotopy, homeomorphism, and diffeomorphism types. He obtains a fairly complete classification. As a by-product he answers in the negative a question of Chern, showing that in the sphere (of dimension 199) there are diffeomorphic but incongruent compact minimal hypersurfaces with the same constant scalar curvature.
scientific article; zbMATH DE number 4056484