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.

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