Clifford systems, algebraically constant second fundamental form and isoparametric hypersurfaces (Q1275225)

The main result of the paper is that a submanifold with parallel mean curvature in a real space form, whose second fundamental form has the same algebraic type as the one of a symmetric submanifold, is locally symmetric. As an application, the authors give a new proof of \textit{É. Cartan}'s classification of isoparamametric hypersurfaces in spheres with three different principal curvatures [Math. Z. 45, 335-367 (1939; Zbl 0021.15603)]. It should be pointed out that \textit{N. H. Kuiper} already associated Clifford systems to the more general situation of tight \((n-1)\)-connected \(2n\)-dimensional submanifolds in Section 5B of [Invent. Math. 10, 209-238 (1970; Zbl 0187.18902)].