Normal scalar curvature inequality on the focal submanifolds of isoparametric hypersurfaces

TL;DR

This paper investigates the set of points satisfying Condition A on focal submanifolds of isoparametric hypersurfaces, revealing bounds on normal scalar curvature and characterizing special point sets with geometric significance.

Contribution

It precisely characterizes the set of points with Condition A and establishes sharper bounds on normal scalar curvature than previous inequalities, also identifying points with parallel second fundamental form and Einstein condition.

Findings

Points in $C_A$ reach an upper bound of $ ho^{ot}$, sharper than DDVV inequality.

Sets $C_P$ and $C_E$ achieve lower bounds of $ ho^{ot}$.

Provides a detailed geometric classification of focal submanifold points.

Abstract

An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], [Chi16] and [Miy13]. This paper determines $C_A$, the set of points with Condition A in focal submanifolds. It turns out that the points in $C_A$ reach an upper bound of the normal scalar curvature $\rho^{\bot}$ (sharper than that in DDVV inequality [GT08], [Lu11]). We also determine the sets $C_P$ (points with parallel second fundamental form) and $C_E$ (points with Einstein condition), which achieve two lower bounds of $\rho^{\bot}$.