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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Principal curvatures of isoparametric hypersurfaces in $\mathbb{C}P^{n}$


Author: Liang Xiao
Journal: Trans. Amer. Math. Soc. 352 (2000), 4487-4499
MSC (2000): Primary 53C40
Published electronically: June 13, 2000
MathSciNet review: 1779484
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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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Additional Information

Liang Xiao
Affiliation: Department of Mathematics, Graduate School, University of Science and Technology of China (Beijing), P.O. Box 3908, Beijing 100039, P.R.China
Email: lxiao@tonghua.com.cn

DOI: https://doi.org/10.1090/S0002-9947-00-02578-2
Received by editor(s): December 20, 1998
Received by editor(s) in revised form: March 25, 1999
Published electronically: June 13, 2000
Article copyright: © Copyright 2000 American Mathematical Society