单位球面中具有共形Maslov形式和常截面曲率的Legendrian子流形
Legendrian submanifolds in the unit sphere with conformal Maslov form and constant sectional curvature
- Chongqing University of Technology（重庆理工大学）
- Henan Normal University（河南师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文分类了单位球面中具有共形Maslov形式和常截面曲率的Legendrian子流形，并将Dillen-Vrancken夹逼定理从极小情形推广到共形Maslov类，证明在曲率界0≤sec≤1下仅有两类闭子流形。
AI中文摘要：
本文研究单位球面$\mathbb{S}^{2n+1}$中具有共形Maslov形式的Legendrian子流形，其中$n\ge2$，该球面具有Sasakian结构$(\varphi,\xi,\eta,g)$。作为主要结果，受具有常截面曲率的极小Legendrian子流形分类结果的启发，我们对这类具有常截面曲率的子流形进行了分类。此外，我们证明：对于$\mathbb{S}^{2n+1}$中具有共形Maslov形式的闭Legendrian子流形$M^n$，若其截面曲率满足夹逼条件$0\leq\sec_g\leq1$，则要么$M^n$是满足$\sec_g=1$的全测地Legendrian球面，要么$M^n$是满足$\sec_g=0$的闭嵌入加权Clifford环面。这将在相同曲率界下Dillen--Vrancken（J Math Pures Appl 69:85--93 1990）的相应夹逼定理从极小Legendrian子流形推广到共形Maslov类。
英文摘要：
This paper is concerned with the study on Legendrian submanifolds with conformal Maslov form in the unit sphere $\mathbb{S}^{2n+1}$, which admits a Sasakian structure $(φ,ξ,η,g)$ for $n\ge2$. As the main result, we classify such submanifolds with constant sectional curvature, motivated by the classification result of the minimal Legendrian submanifolds with constant sectional curvature. Moreover, we prove that, for a closed Legendrian submanifold $M^n$ in $\mathbb{S}^{2n+1}$ with conformal Maslov form, if its sectional curvature satisfies the pinching $0\leq\sec_g\leq1$, then either $M^n$ is the totally geodesic Legendrian sphere with $\sec_g=1$, or $M^n$ is a closed embedded weighted Clifford torus with $\sec_g=0$. This extends the corresponding pinching theorem of Dillen--Vrancken (J Math Pures Appl 69:85--93 1990) from minimal Legendrian submanifolds to the conformal Maslov class under the same curvature bounds.
