Kaehler submanifolds of the real hyperbolic space

The local classification of Kaehler submanifolds of the hyperbolic space with low codimension under only intrinsic assumptions remains a wide open problem. The situation is quite different for submanifolds in the round sphere, since Florit et al. [7] have shown that the codimension has to be and then that any submanifold is just part of an extrinsic product of two-dimensional umbilical spheres in. The main result of this paper is a version for Kaehler manifolds isometrically immersed into the hyperbolic ambient space of the result in [7] for spherical submanifolds. Besides, we generalize several results obtained by Dajczer and Vlachos [5].