Warped product Kaehler manifolds and Bochner-Kaehler metrics

Abstract

Using as an underlying manifold an alpha-Sasakian manifold we introduce warped product Kaehler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kaehler manifold is of quasi-constant holomorphic sectional curvatures with special distribution. Conversely, we prove that any Kaehler manifold of quasi-constant holomorphic sectional curvatures with special distribution locally has the structure of a warped product Kaehler manifold whose base is an alpha-Sasakian space form. Considering the scalar distribution generated by the scalar curvature of a Kaehler manifold, we give a new approach to the local theory of Bochner-Kaehler manifolds. We study the class of Bochner-Kaehler manifolds whose scalar distribution is of special type. Taking into account that any manifold of this class locally is a warped product Kaehler manifold, we describe all warped product Bochner-Kaehler metrics. We find four families of complete metrics of this type.