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On Ricci curvature of $$C$$-totally real submanifolds in Sasakian space forms. (English) Zbl 1012.53039
Summary: Let $$M^n$$ be a Riemannian $$n$$-manifold. Denote by $$S(p)$$ and $$\overline{\text{Ric}}(p)$$ the Ricci tensor and the maximum Ricci curvature on $$M^n$$, respectively. In this paper we prove that every $$C$$-totally real submanifold of a Sasakian space form $$\overline M^{2m+1}(c)$$ 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 $$\overline M^{2n+1}(c)$$ satisfies $$\overline{\text{Ric}}=\frac{(n-1)(c+3)}4+\frac{n^2} 4H^2$$ identically, then it is minimal.
##### MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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