$CR$-submanifolds in complex hyperbolic spaces satisfying an equality of Chen

Abstract

Recently, Bang-Yen Chen has introduced new type of Riemannian curvature invariants and obtained sharp inequalitiesinvolving these invariants for arbitrary submanifolds in Riemannian and Kaehlerian space forms. It is natural and interesting to investigate and understand submanifolds which satisfythe equality case of this type of inequalities,and such submanifolds have been investigated by many geometers (cf. for instance, [2-6, 8-10, 12-16]). In this paper, we investigate Ci?-submanifolds of complex hyperbolic spaces which satisfy the equality case of one of Chen's inequalities. Let M be an ^-dimensional Riemannian manifold. Denote by K(n) the sectional curvature of M associated with a plane section n a TpM, p e M. For any orthonormal basis e\,...,en of the tangent space TPM, the scalar curvature r at p is defined to be