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On hypersurfaces of $\mathbb{S}^2\times \mathbb{S}^2$ and $\mathbb{H}^2\times \mathbb{H}^2$ with recurrent Ricci tensor

Part of: Global differential geometry Local differential geometry

Published online by Cambridge University Press:  30 April 2025

Qianshun Cui Open the ORCID record for Qianshun Cui [Opens in a new window] ,
Zejun Hu Open the ORCID record for Zejun Hu [Opens in a new window] ,
Xiaoge Lu Open the ORCID record for Xiaoge Lu [Opens in a new window]  and
Zeke Yao Open the ORCID record for Zeke Yao [Opens in a new window]
Qianshun Cui*
Affiliation:
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, People’s Republic of China
Zejun Hu
Affiliation:
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, People’s Republic of China
Xiaoge Lu
Affiliation:
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, People’s Republic of China
Zeke Yao
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou, People’s Republic of China
*
Corresponding author: Qianshun Cui; Email: cuiqszzu@163.com
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Abstract

In this paper, we investigate hypersurfaces of $\mathbb{S}^2\times \mathbb{S}^2$ and $\mathbb{H}^2\times \mathbb{H}^2$ with recurrent Ricci tensor. As the main result, we prove that a hypersurface in $\mathbb{S}^2\times \mathbb{S}^2$ (resp. $\mathbb{H}^2\times \mathbb{H}^2$) with recurrent Ricci tensor is either an open part of $\Gamma \times \mathbb{S}^2$ (resp. $\Gamma \times \mathbb{H}^2$) for a curve $\Gamma$ in $\mathbb{S}^2$ (resp. $\mathbb{H}^2$), or a hypersurface with constant sectional curvature. The latter has been classified by H. Li, L. Vrancken, X. Wang, and Z. Yao very recently.

Keywords

Recurrent Ricci tensorRicci eigenvaluesconstant sectional curvaturenon-Hopf hypersurfacessemi-parallel Ricci tensor

MSC classification

Primary: 53B25: Local submanifolds
Secondary: 53C30: Homogeneous manifolds 53C42: Immersions (minimal, prescribed curvature, tight, etc.)

Information

Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 21
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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