On Ricci curvature of totally real submanifolds in a quaternion projective space

summary:Let $M^n$ be a Riemannian $n$-manifold. Denote by $S(p)$ and $\overline{\operatorname{Ric}}(p)$ the Ricci tensor and the maximum Ricci curvature on $M^n$, respectively. In this paper we prove that every totally real submanifolds of a quaternion projective space $QP^m(c)$ satisfies $S\le ((n-1)c+\frac{n^2}{4}H^2)g$, 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 Lagrangian submanifold of $QP^m(c)$ satisfies $\overline{\operatorname{Ric}}=(n-1)c+\frac{n^2}{4}H^2$ identically, then it is minimal