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 $m_2 \geq 2m_1 – 1$. This inequality is satisfied for all but five possible pairs $(m_1, m_2)$ with $m_1 \leq m_2$. Our proof implies that for $(m_1, m_2) \neq (1,1)$ 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 $m_1 \leq m_2$ (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].