On doubly warped product of complex finsler manifolds

Let (M1, F1) and (M2, F2) be two strongly pseudoconvex complex Finsler manifolds. The doubly wraped product complex Finsler manifold (f2M1 × f1M2, F) of (M1, F1) and (M2, F2) is the product manifold M1 × M2 endowed with the warped product complex Finsler metric F2 = f22F12 + f22F22, where f1and f2 are positive smooth functions on M1 and M2, respectively. In this paper, the most often used complex Finsler connections, holomorphic curvature, Ricci scalar curvature, and real geodesics of the DWP-complex Finsler manifold are derived in terms of the corresponding objects of its components. Necessary and sufficient conditions for the DWP-complex Finsler manifold to be Kähler Finsler (resp., weakly Kähler Finsler, complex Berwald, weakly complex Berwald, complex Landsberg) manifold are obtained, respectively. It is proved that if (M1, F1) and (M2, F2) are projectively flat, then the DWP-complex Finsler manifold is projectively flat if and only if f1 and f2 are positive constants.