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arXiv · 1610.03912

Normal scalar curvature inequality on the focal submanifolds of isoparametric hypersurfaces

Jianquan Ge·Zizhou Tang·Wenjiao Yan
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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 $ρ^{\bot}$ (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 $ρ^{\bot}$.

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Jianquan Ge, Zizhou Tang, Wenjiao Yan. 2018-02-05. Normal scalar curvature inequality on the focal submanifolds of isoparametric hypersurfaces. https://arxiv.org/abs/1610.03912

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