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On conformally flat minimal Legendrian submanifolds in the unit sphere

Part of: Global differential geometry

Published online by Cambridge University Press:  10 May 2024

Cece Li Open the ORCID record for Cece Li [Opens in a new window] ,
Cheng Xing Open the ORCID record for Cheng Xing [Opens in a new window]  and
Jiabin Yin Open the ORCID record for Jiabin Yin [Opens in a new window]
Cece Li
Affiliation:
School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471000, People's Republic of China (ceceli@haust.edu.cn)
Cheng Xing*
Affiliation:
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People's Republic of China (xingcheng@nankai.edu.cn)
Jiabin Yin
Affiliation:
School of Mathematics and Statistics, Guangxi Normal University, Guilin 541001, People's Republic of China (jiabinyin@126.com)
*
*Corresponding author.
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Abstract

This paper is concerned with the study on an open problem of classifying conformally flat minimal Legendrian submanifolds in the $(2n+1)$-dimensional unit sphere $\mathbb {S}^{2n+1}$ admitting a Sasakian structure $(\varphi,\,\xi,\,\eta,\,g)$ for $n\ge 3$, motivated by the classification of minimal Legendrian submanifolds with constant sectional curvature. First of all, we completely classify such Legendrian submanifolds by assuming that the tensor $K:=-\varphi h$ is semi-parallel, which is introduced as a natural extension of $C$-parallel second fundamental form $h$. Secondly, such submanifolds have also been determined under the condition that the Ricci tensor is semi-parallel, generalizing the Einstein condition. Finally, as direct consequences, new characterizations of the Calabi torus are presented.

Keywords

Legendrian submanifoldunit sphereconformally flatsemi-parallelproduct manifoldrigidity theorem

MSC classification

Primary: 53C24: Rigidity results
Secondary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C40: Global submanifolds 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 30
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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