Doubly Twisted Product Finsler Metrics

Warped product and twisted product are important geometric models in Riemann geometry and Finsler geometry.Using them to construct special metrics in Finsler geometry has become a hot issue.In this paper,we mainly extend the twisted product of Finsler metric to doubly twisted product,and systematically study the connection induced by doubly twisted product Finsler metrics.On this basis,we construct several special Finsler metrics through doubly twisted product,and obtain the relationship between these special Finsler metrics.At the same time,we study the locally projectively flat doubly twisted product Finsler metric,and find the relationship between it and Minkowski metric.Firstly,we give the definition of doubly twisted product Finsler metric.The spray coefficient,nonlinear connection coefficient,Chern connection coefficient and Cartan tensor coefficient of doubly twisted product Finsler metric are derived.The connection expressions induced by doubly twisted product Finsler metric are given,such as Berwald connection,Cartan connection,Chern connection and Hashiguchi connection.Secondly,we derive the coefficients of Berwald curvature and mean Berwald curvature of the doubly twisted product Finsler metric.Then the necessary and sufficient conditions for the doubly twisted product Finsler metric to be Berwald metric or weakly Berwald metric respectively are gived.And we prove that doubly twisted product Finsler metric with isotropic Berwald curvature is Berwald metric.It is proved that the doubly twisted product Finsler metric with isotropic mean Berwald curvature is a weakly Berwald metric under certain conditions.Finally,we study the locally projectively flat doubly twisted product Finsler metric.we give the differential equation characterization of locally projectively flat doubly twisted product Finsler metric.On this basic,we prove that a locally projectively flat doubly twisted product Finsler metric is a locally Minkowski metric.