On Ricci Curvature of 𝐶-totally Real Submanifolds in Sasakian Space Forms

Proceedings Mathematical Sciences

Liu Ximin¹ 

[1] Department of Applied Mathematics, Dalian University of Technology, Dalian 0, China$$

关键词: Ricci curvature;    𝐶-totally real submanifold;    Sasakian space form.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let 𝑀𝑛 be a Riemannian 𝑛-manifold. Denote by $S(p)$ and $overline{Ric}(p)$ the Ricci tensor and the maximum Ricci curvature on 𝑀𝑛, respectively. In this paper we prove that every 𝐶-totally real submanifold of a Sasakian space form $overline{M}^{2m + 1}(c)$ satisfies $S≤ left(frac{(n - 1)(c + 3)}{4} + frac{n^2}{4}H^2ight)g$, where $H^2$ and 𝑔 are the square mean curvature function and metric tensor on 𝑀𝑛, respectively. The equality holds identically if and only if either 𝑀𝑛 is totally geodesic submanifold or 𝑛 = 2 and 𝑀𝑛 is totally umbilical submanifold. Also we show that if a 𝐶-totally real submanifold 𝑀𝑛 of $overline{M}^{2n + 1}(c)$ satisfies $overline{Ric}=frac{(n-1)(c+3)}{4} + frac{n^2}{4}H^2$ identically, then it is minimal.

【 授权许可】

Unknown