On focal submanifolds of isoparametric hypersurfaces and Simons formula

TL;DR

This paper investigates the geometry of focal submanifolds of isoparametric hypersurfaces in spheres, using Simons formula to analyze their curvature properties and classify semiparallel cases.

Contribution

It provides a detailed analysis of focal submanifolds' curvature, proving they are not normally flat for g≥3 and classifying semiparallel submanifolds.

Findings

Focal submanifolds with g≥3 are not normally flat.

Complete classification of semiparallel focal submanifolds.

Focal submanifolds are minimal Willmore and mostly -manifolds.

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.