On Focal Submanifolds of Isoparametric Hypersurfaces and Simons Formula.

The focal submanifolds of isoparametric hypersurfaces in spheres are all minimal Willmore submanifolds, mostly being $${\mathcal{A}}$$ -manifolds in the sense of A.Gray but rarely Ricci-parallel, see Li et al. (Sci China Math 58, 2015), Qian et al. (Ann Glob Anal Geom 43:47-62, 2013), Tang and Yan (Isoparametric foliation and a problem of Besse on generalizations of Einstein condition , 2013). In this paper we study the geometry of the focal submanifolds via Simons formula. We show that all the focal submanifolds with g ≥ 3 are not normally flat by estimating the normal scalar curvatures. Moreover, we give a complete classification of the semiparallel submanifolds among the focal submanifolds. [ABSTRACT FROM AUTHOR]