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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.
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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