CR-Warped Products in Complex Projective Spaces with Compact Holomorphic Factor

A submanifold of a Kaehler manifold is called a CR-warped product if it is the warped product NT ×fN⊥ of a complex submanifold NT and a totally real submanifold N⊥. There exist many CR-warped products NT ×fN⊥ in CPh+p, h = dimCNT and p = dimRN⊥ (see [5, 6]). In contrast, we prove in this article that the situation is quite different if the holomorphic factor NT is compact. For such CR-wraped products in CPm (4), we prove the following: (1) The complex dimension m of the ambient space is at least h + p + hp. (2) If m = h + p + hp, then NT is CPh(4). We also obtain two geometric inequalities for CR-warped products in CPm with compact NT.