On Doubly Twisted Product of Complex Finsler Manifolds

Abstract

Let $(M_1,F_1)$ and $(M_2,F_2)$ be two strongly pseudoconvex complex Finsler manifolds. The doubly twisted product (abbreviated as DTP) complex Finsler manifold $(M_1\times_{(\lambda_1,\lambda_2)}M_2,F)$ is the product manifold $M_1\times M_2$ endowed with the twisted product complex Finsler metric $F^2=\lambda_1^2F_1^2+\lambda_2^2F_2^2$, where $\lambda_1$ and $\lambda_2$ are positive smooth functions on $M_1\times M_2$. In this paper, the relationships between the geometric objects (e.g. complex Finsler connections, holomorphic and Ricci scalar curvatures, and real geodesic) of a DTP-complex Finsler manifold and its components are derived. The necessary and sufficient conditions under which the DTP-complex Finsler manifold is a Kähler Finsler (respctively weakly Kähler Finsler, complex Berwald, weakly complex Berwald, complex Landsberg) manifold are obtained. By means of these, we provide a possible way to construct a weakly complex Berwald manifold, and then give a characterization for a complex Landsberg metric that is not a Berwald metric.