On Ricci curvature of \(C\)-totally real submanifolds in Sasakian space forms (Q1348753)

In this paper the following results are given. A \(C\)-totally real submanifold \((M^{n},g)\) of a Sasakian space form \(M^{2m+1}(c)\) satisfies \(S\leq \{\frac{(n-1)(c+3)}{4} + \frac{n^2}{4}H^2\}g\), where \(S\) and \(H\) denote the Ricci tensor and the mean curvature on \(M^n\), and the equality holds identically if and only if \(M^n\) is totally geodesic or \(n=2\) and \(M^n\) is totally umbilical. Moreover, if a \(C\)-totally real submanifold \((M^{n},g)\) of a Sasakian space form \(M^{2n+1}(c)\) satisfies \(\overline{\text{Ric}}=\frac{(n-1)(c+3)}{4} + \frac{n^2}{4}H^2\) identically, where \(\overline{\text{Ric}}\) denotes the maximum Ricci curvature function on \(M^n\), then it is minimal.