Normal Scalar Curvature Inequality on the Focal Submanifolds of Isoparametric Hypersurfaces

An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [1], [7], and [25]. This paper determines |CA|⁠, the set of points with Condition A in focal submanifolds. It turns out that the points in |CA| reach an upper bound of the normal scalar curvature |ρ| (sharper than that in DDVV inequality [16], [23]). We also determine the sets |CP| (points with parallel second fundamental form) and |CE| (points with Einstein condition), which achieve two lower bounds of |ρ|⁠.