Levi-Civita Ricci-Flat Doubly Warped Product Hermitian Manifolds

Let M1,g and M2,h be two Hermitian manifolds. The doubly warped product (abbreviated as DWP) Hermitian manifold of M1,g and M2,h is the product manifold M1×M2 endowed with the warped product Hermitian metric G=f22g+f12h, where f1 and f2 are positive smooth functions on M1 and M2, 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 f1 and f2 are holomorphic, then the DWP-Hermitian manifold is Levi-Civita Ricci-flat if and only if M1,g and M2,h are Levi-Civita Ricci-flat manifolds. Thus, we give an effective way to construct Levi-Civita Ricci-flat DWP-Hermitian manifold.