Chen type of some classes of CR-submanifolds in CP^m and CH^m

We survey some of the fundamental classification results on low-type submanifolds of non-at model complex space forms (complex projective and hyperbolic spaces) via the standard embeddings by projection operators. These results include classification of submanifolds of type 1 in these spaces, of CMC and Hopf hypersurfaces of type 2, and investigation of the Chen type of totally real and Kähler submanifolds. Some examples of submanifolds of type 3 are presented. We also give some nonexistence results for certain families of CR-submanifolds of complex space forms of Chen type two. For example, there exist no holomorphic submanifolds of the complex hyperbolic space which are of type 2 via the standard embedding by projectors. This is contrasted with the situation in the complex projective space, where there exist some parallel Einstein Kähler submanifolds of type 2.