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

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*Liu Ximin*

**Address.** Department of Applied Mathematics, Dalian University of
Technology, Dalian 116024, China

**E-mail:** xmliu@dlut.edu.cn
**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.