On Doubly Twisted Product of Complex Finsler Manifolds

Let (M1,F1) and (M2,F2) be two strongly pseudoconvex complex Finsler manifolds. The doubly twisted product (abbreviated as DTP) complex Finsler manifold (M1×(λ1,λ2) M2,F) is the product manifold M1×M2 endowed with the twisted product complex Finsler metric F=λ1F 2 1 +λ 2 2F 2 2 , where λ1 and λ2 are positive smooth functions on M1×M2. 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. AMS subject classifications: 53C60, 53C40