On the Kaehler angles of submanifolds

Abstract

We prove that under certain conditions on the mean curvature and on the Kaehler angles, a compact submanifold M of real dimension 2n, immersed into a Kaehler-Einstein manifold N of complex dimension 2n, must be either a complex or a Lagrangian submanifold of N, or have constant Kaehler angle, depending on n=1, n=2, or n \geq 3, and the sign of the scalar curvature of N. These results generalize to non-minimal submanifolds some known results for minimal submanifolds. Our main tool is a Bochner-type technique involving a formula on the Laplacian of a symmetric function on the Kaehler angles and the Weitzenboeck formula for the Kaehler form of N restricted to M.