On Ricci curvature ofC-totally real submanifolds in Sasakian space forms

Abstract

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.