Principal curvatures of isoparametric hypersurfaces in ℂℙⁿ

Xiao, Liang

doi:10.1090/S0002-9947-00-02578-2

Principal curvatures of isoparametric hypersurfaces in $\mathbb {C}P^{n}$
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Trans. Amer. Math. Soc. 352 (2000), 4487-4499 Request permission

Abstract:

Let $M$ be an isoparametric hypersurface in $\mathbb {C}P^{n}$, and $\overline {M}$ the inverse image of $M$ under the Hopf map. By using the relationship between the eigenvalues of the shape operators of $M$ and $\overline {M}$, we prove that $M$ is homogeneous if and only if either $g$ or $l$ is constant, where $g$ is the number of distinct principal curvatures of $M$ and $l$ is the number of non-horizontal eigenspaces of the shape operator on $\overline {M}$.

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