Riemannian maps of CR-submanifolds of Kaehler manifolds

Abstract In this paper, Riemannian maps from a CR-submanifold of an almost Hermitian manifold to another almost Hermitian manifold are studied. Such Riemannian maps include the special class of CR-submersions, which are well known in the literature. First, the notion of Riemannian map from CR-submanifold to an almost Hermitian manifold is presented and the invariant of the image spaces of these maps is shown. The harmonicity of these maps is investigated. Also, the relation between the holomorphic sectional curvature of the target manifold and the holomorphic sectional curvature of the CR-submanifold is obtained depending on the second fundamental form of the submanifold and the second fundamental form of the Riemannian map. Finally, the Ricci curvature of the ambient manifold of the CR-submanifold in the direction of the invariant distribution is established in terms of the Ricci curvature of the image space, the Ricci curvature of the anti-invariant distribution, the mean curvature vector fields of the holomorphic distribution and the anti-invariant distribution.