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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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