Abstract. Let $M^n$ be a Riemannian $n$-manifold. Denote by $S(p)$
and
$\overline{\Ric}(p)$ 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 $S\leq ((n-1)c+\frac{n^2}{4}H^2)g$,
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 $\overline{\Ric}=(n-1)c+\frac{n^2}{4}H^2$
identically, then it is minimal.
AMSclassification. 53C40, 53C42
Keywords. Ricci curvature, totally real submanifolds, quaternion projective space.