On The Berwald Scalar Curvature And Related Problems

In this paper,we study the Berwald scalar curvature of Finsler manifolds,the Unicorn problem and the Finsler metric with vanishing mean Chern curvature.We get some interesting results.Firstly,we study the relationship between the E-curvature and the Berwald scalar curvature.Furthermore,we prove that the Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds.We proved that there is no generalized Unicorn for(α,β)metric with vanishing Berwald scalar curvature.Secondly,we study the Finsler metric with vanishing mean Chern curvature,and we proved that the Finsler metric with vanishing mean Chern curvature has closed Cartan type 1-form.Based on the above result,the Finsler metrics with scalar flag curvature and vanishing mean Chern curvature are completely classified.