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

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.