Levi-Civita Ricci-Flat Doubly Warped Product Hermitian Manifolds

Abstract

Let M 1 , g and M 2 , h be two Hermitian manifolds. The doubly warped product (abbreviated as DWP) Hermitian manifold of M 1 , g and M 2 , h is the product manifold M 1 × M 2 endowed with the warped product Hermitian metric G = f 2 2 g + f 1 2 h , where f 1 and f 2 are positive smooth functions on M 1 and M 2 , respectively. In this paper, the formulae of Levi-Civita connection, Levi-Civita curvature, the first Levi-Civita Ricci curvature, and Levi-Civita scalar curvature of the DWP-Hermitian manifold are derived in terms of the corresponding objects of its components. We also prove that if the warped function f 1 and f 2 are holomorphic, then the DWP-Hermitian manifold is Levi-Civita Ricci-flat if and only if M 1 , g and M 2 , h are Levi-Civita Ricci-flat manifolds. Thus, we give an effective way to construct Levi-Civita Ricci-flat DWP-Hermitian manifold.