Isoparametric foliation and a problem of Besse on generalizations of Einstein condition

Abstract

The focal sets of isoparametric hypersurfaces in spheres with $g=4$ are all Willmore submanifolds, being minimal but mostly non-Einstein [25], [22]. Inspired by A. Gray's view, the present paper shows that these focal sets are all $\mathcal{A}$-manifolds but rarely Ricci parallel, except possibly for the only unclassified case. As a byproduct, it gives infinitely many simply-connected examples to the problem 16.56(i) of Besse concerning generalizations of the Einstein condition.