Trivial doubly warped products

Abstract

The aim of the paper is to provide new obstructions to the existence of doubly warped products. We prove that, if the factor manifolds of a doubly warped product are connected and locally product Riemannian manifolds, then, the almost product structure naturally induced on the doubly warped product is parallel if and only if the manifold is a direct product manifold. We also show that there do not exist doubly warped product K\"{a}hler manifolds (with respect to the naturally induced almost Hermitian structure) with connected K\"{a}hler factors, which are not direct products, neither doubly warped product manifolds which are pointwise slant but not slant submanifolds (with respect to the naturally induced almost Hermitian structure) with pointwise slant factors.