On Some Special Douglas Finsler Metics

As a differential geometry disciplines of wide application prospect, Finsler geometry isdeveloping forward vigorously. In recent years, due to the Finsler geometric application inbiological physical, mental, etc, Finsler metrics receive more and more attention. As a researchdirection of Finsler metrics , Douglas metrics get more the favour of geometric scholars. Thispaper is based on this to research and discuss on some special Douglas Finsler metrics .This article is divided into six parts:In the first chapter, introduce the research history of Finsler geometry and related content .Inthe second chapter, introduce the basical knowledge . In the third chapter, we discuss a class ofimportant Finsler metrics—m ?th root Berwald metrics, and find the equivalent condition ofBerwaldian m ?th root Finsler metrics. With the result, we can construct some special Berwaldianm ?th root Finsler metrics. In the fourth chapter, we study another class of important Finslermetrics—m ?th root Douglas metrics. We prove that if a Finsler metric is an m ?th root Finslermetric, then it must be Berwald metric. In the fifth chapter , we discuss S ? curvature and the flagcurvature,and study m ?th root metrics and generalized m ?th root metrics with scalar flagcurvature . We prove that if the metrics of this class with scalar flag curvature and isotropicS ? curvature, then the flag curvature must be constant. The final chapter of this paper issummarized in this paper. The main part summarizes the research process of this paper and makespositive outlook to the future development situation of this direction.