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On focal submanifolds of isoparametric hypersurfaces and Simons formula (1501.07043v1)
Published 28 Jan 2015 in math.DG
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 $g\geq3$ are not normally flat by estimating the normal scalar curvatures. Moreover, we give a complete classification of the semiparallel submanifolds among the focal submanifolds.