Levi-Civita Ricci-Flat Doubly Warped Product Hermitian Manifolds

Abstract

Let $\left(M_1,g\right)$ and $\left(M_2,h\right)$ be two Hermitian manifolds. The doubly warped product (abbreviated as DWP) Hermitian manifold of $\left(M_1,g\right)$ and $\left(M_2,h\right)$ is the product manifold $M_1\times M_2$ 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 $\left(M_1,g\right)$ and $\left(M_2,h\right)$ are Levi-Civita Ricci-flat manifolds. Thus, we give an effective way to construct Levi-Civita Ricci-flat DWP-Hermitian manifold.