On the classification of isoparametric hypersurfaces with four distinct curvatures in spheres

Abstract

In this paper we give a new proof for the classification result in [3]. We show that isoparametric hypersurfaces with four distinct principal curvatures in spheres are of Clifford type provided that the multiplicities m 1 , m 2 of the principal curvatures satisfy m 2 > 2m 1 - 1. This inequality is satisfied for all but five possible pairs (m 1 , m 2 ) with m 1 ≤ m 2 . Our proof implies that for (m 1 ,m 2 ) ≠ (1,1) the Clifford system may be chosen in such a way that the associated quadratic forms vanish on the higher-dimensional of the two focal manifolds. For the remaining five possible pairs (m 1 , m 2 ) with m 1 ≤ m 2 (see [13], [1], and [15]) this stronger form of our result is incorrect: for the three pairs (3,4), (6,9), and (7,8) there are examples of Clifford type such that the associated quadratic forms necessarily vanish on the lower-dimensional of the two focal manifolds, and for the two pairs (2,2) and (4,5) there exist homogeneous examples that are not of Clifford type; cf. [5, 4.3, 4.4].