Normal scalar curvature inequality on the focal submanifolds of isoparametric hypersurfaces

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 ${\rho }^{\perp }$ (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 ${\rho }^{\perp }$.

37 pages, to appear in International Mathematics Research Notices, Dedicated to Professor Chiakuei Peng on the Occasion of His 75th Birthday

References in corpus (2)

• New examples of Willmore submanifolds in the unit sphere via isoparametric functions,II
• Moebius geometry of three dimensional Wintgen ideal submanifolds in S^5

Cited by in corpus (1)

• DDVV-type inequality for Hermitian matrices