Doubly Twisted Product Complex Finsler Manifolds

Doubly twisted product is a important method used to construct manifolds with special geometric properties in Riemann geometry or real Finsler geometry.We extend the notion of doubly twisted product to complex Finsler geometry,and systematically study the geometric objects of a doubly twisted product complex Finsler manifold.We also construct weakly complex Berwald manifold and complex locally Minkowski manifold by doubly twisted product.In the first part,we deduce the Chern-Finsler connection(or the complex Rund connection,the complex Berwald connection,and the complex Hashiguchi connection)of the doubly twisted product complex Finsler manifold,which are expressed by the corresponding connections of its components,respectively.Then,we derive the formulae of the holomorphic curvature,Ricci curvature and real geodesic of the doubly twisted product complex Finsler manifold,which are expressed by the geometric objectives of its components,respectively.We also give a characterization for a doubly twisted product complex Finsler manifold with constant holomorphic curvature.In the second part,the necessary and sufficient conditions under which the doubly twisted product complex Finsler manifold to be K¨ahler Finsler(resp.weakly K¨ahler Finsler,complex Berwald,weakly complex Berwald,complex Landsberg)manifold are obtained.We provide a possible way to construct weakly complex Berwald(resp.Herimitian,complex locally Minkowski)manifold,and give a characterization for a complex Landsberg manifold that is not a K¨ahler Berwald manifold.In the third part,we give a necessary and sufficient condition for a doubly twisted product of complex Finsler manifolds to be complex Einstein-Finsler manifold,and get the conclusion that a doubly twisted product complex Finsler manifold to be complex EinsteinFinsler manifold if and only if its components manifolds are weakly complex EinsteinFinsler manifolds when logarithm functions of twisted functions both are pluriharmonic functions.We also prove that a generalized complex Einstein-Finsler doubly twisted product manifold has vanishing holomorphic curvature.In the last part,we give a characterization for a doubly twisted product complex Finsler manifold to be locally dually flat manifold and locally conformally flat manifold,respectively.And we obtain the relationships between the flatness of locally dually flat of a doubly twisted product complex Finsler manifold and its components.