On focal submanifolds of isoparametric hypersurfaces and Simons formula

Abstract

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 (\cite{QTY},\cite{LY},\cite{TY3}). In this paper we study the geometry of the focal submanifolds via Simons formula. We show that all the focal submanifolds with $g\ge 3$ are not normally flat by estimating the normal scalar curvatures. Moreover, we give a complete classification of the semiparallel submanifolds among the focal submanifolds.

12 pages, all comments are welcomed

References in corpus (2)

• New examples of Willmore submanifolds in the unit sphere via isoparametric functions,II
• Willmore submanifolds in the unit sphere via isoparametric functions