On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms

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 at a point $p\in M^n$, respectively. First we show that every isotropic submanifold of a complex space form $\widetilde M^m(4\,c)$ satisfies $S\leq ((n-1)c+ {n^2 \over 4} H^2)g$, where H ² and g are the squared mean curvature function and the metric tensor on M ⁿ, respectively. The equality case of the above inequality holds identically if and only if either M ⁿ is totally geodesic submanifold or n = 2 and M ⁿ is a totally umbilical submanifold. Then we prove that if a Lagrangian submanifold of a complex space form $\widetilde M^m(4\,c)$ satisfies $\overline {Ric}= (n-1)c+ {n^2 \over 4} H^2$ identically, then it is a minimal submanifold. Finally, we describe the geometry of Lagrangian submanifolds which satisfy the equality under the condition that the dimension of the kernel of second fundamental form is constant.