Kaehler submanifolds of the real hyperbolic space

The local classification of Kaehler submanifolds M2n of the hyperbolic space H2n+p with low codimension 2≤p≤n−1 under only intrinsic assumptions remains a wide open problem. The situation is quite different for submanifolds in the round sphere S2n+p, 2≤p≤n−1, since Florit et al. [7] have shown that the codimension has to be p=n−1 and then that any submanifold is just part of an extrinsic product of two-dimensional umbilical spheres in S3n−1⊂R3n. 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].