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Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 17, No. 3, pp. 171-177 (2001)
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On Ricci curvature of $C$-totally real submanifolds in Sasakian space forms

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Liu Ximin

Dalian University of Technology, China

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

**Keywords:** Ricci curvature, $C$-totally real submanifold, Sasakian space form.

**Classification (MSC2000):** 53C15

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