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On doubly warped product of complex Finsler manifolds. (English) Zbl 1374.53108

Summary: Let \((M_1, F_1)\) and \((M_2, F_2)\) be two strongly pseudoconvex complex Finsler manifolds. The doubly warped product complex Finsler manifold \((_{f_2}M_1\times_{f_1}M_2, F)\) of \((M_1, F_1)\) and \((M_2, F_2)\) is the product manifold \(M_1\times M_2\) endowed with the warped product complex Finsler metric \(F^2 = f_2^2F_1^2 + f_1^2F_2^2\), where \(f_1\) and \(f_2\) are positive smooth functions on \(M_1\) and \(M_2\), 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 \((M_1, F_1)\) and \((M_2, F_2)\) are projectively flat, then the DWP-complex Finsler manifold is projectively flat if and only if \(f_1\) and \(f_2\) are positive constants.

MSC:

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53C40 Global submanifolds
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