Ricci tensor of slant submanifolds in complex space forms

Abstract

B.-Y. Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. The Lagrangian version of this inequality was proved by the same author.\newline In this article, we obtain a sharp estimate of the Ricci tensor of a slant submanifold $M$ in a complex space form $\widetilde M(4c)$, in terms of the main extrinsic invariant, namely the squared mean curvature. If, in particular, $M$ is a Kaehlerian slant submanifold which satisfies the equality case identically, then it is minimal.