On Ricci curvature ofC-totally real submanifolds in Sasakian space forms
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LetM^{n} be a Riemanniann-manifold. Denote byS(p) and Ric(p) the Ricci tensor and the maximum Ricci curvature onM_{n}, respectively. In this paper we prove that everyC-totally real submanifold of a Sasakian space formM^{2m+1}(c) satisfies$$S \leqslant (\frac{{(n - 1)(c + 3)}}{4}) + \frac{{n^2 }}{4}H^2 )g$$, whereH^{2} andg are the square mean curvature function and metric tensor onM^{n}, respectively. The equality holds identically if and only if eitherM^{n} is totally geodesic submanifold or n = 2 andM^{n} is totally umbilical submanifold. Also we show that if aC-totally real submanifoldM^{n} ofM^{2n+1} (c) satisfies$$\overline {Ric} = \frac{{(n - 1)(c + 3)}}{4} + \frac{{n^2 }}{4}H^2 $$ identically, then it is minimal.
Liu Ximin^{1} ^{2} ^{}
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November 2019
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