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

Abstract.

A submanifold of a Kaehler manifold is called a CR-warped product if it is the warped product N _T × _f N _⊥ of a complex submanifold N _T and a totally real submanifold N _⊥. There exist many CR-warped products N _T × _f N _⊥ in CP ^h+p, h = dim_C N _T and p = dim_R N _⊥ (see [5, 6]). In contrast, we prove in this article that the situation is quite different if the holomorphic factor N _T is compact. For such CR-wraped products in CP ^m (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 N _T is CP ^h(4). We also obtain two geometric inequalities for CR-warped products in CP ^m with compact N _T .