On conformally flat minimal Legendrian submanifolds in the unit sphere

2024/05/10 by Cece Li, Cheng Xing, Jiabin Yin
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Mathematics #Mathematics education #Pure mathematics #Unit (ring theory) #Unit sphere

paper · doi:10.1017/prm.2024.57

openalex publication_date 2024/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/26

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 S²ⁿ⁺¹ admitting a Sasakian structure (φ, ξ, η, g) for n≥ 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:=-φ 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.

Citations

• Riemannian Geometry of Contact and Symplectic Manifolds
• Complete locally conformally flat manifolds of negative curvature
• An optimal pinching theorem of minimal Legendrian submanifolds in the unit sphere

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