| The conformal property of metrics is meaningful in Riemann and Finsler geometry.It is one hot issue that how to characterize conformally flat metrics in Finsler geometry.There are many important results in conformal Finsler geometry.(α,β)-metric is a class of significant Finsler metric,and scholars have done a lot of research in recent years.Chen-He-Shen[6]proved that conformally flat(α,β)-metrics with constant flag curvature must be locally Minkowskian or Riemannian.Yu-Youf33]showed that the spray coefficients Gi and Ricci curvature Ric of m th root metric must be rational functions in y.Cubic metrics are useful in physics and biology.Tayebi-Razgordani-Najafi[25]showed that the conformally flat weak Einstein cubic metric and the conformally flat cubic metric with almost vanishing x curvature are either a locally Minkowskian metric or a Riemannian metric.Further,Chen-Xia[4]gave the explicit expressions of Ricci curvature and scalar curvature of(α,β)-metrics which are conformally flat.They studied the conformally flat(α,β)-metrics with weakly isotropic scalar curvature and showed that ifφ(s)in F is a polynomial of degree m(≥2)and F is of weakly isotropic scalar curvature r,then r≡0.There are many non-Riemannian quantities(i.e.it is identically vanishing in Riemann geometry,but uncertainly equal to zero in Finsler geometry)in Finsler geometry,for example:S curvature,Landsberg curvature,χ curvature,H curvature,etc.Studying those special curvature properties could be meaningful to global results.And the study of non-Riemannian quantity x curvature properties is one of basic problems in Finsler geometry.Shen[29]considered the relationship between flag curvature and χ curvature,and proved that S curvature of Randers metric is isotropic if and only if x is almost vanishing.Later,Chen-Liu[7]gave the equivalent condition of Kropina metric with almost vanishing x curvature.Cheng-Yuan[11]showed that the χ curvature vanished for m(≥2)th polynomial(α,β)-metrics with almost vanishing χ curvature.And if F is conformally flat,then F must be locally Minkowskian.There are a lot of research of(α,β)-metrics for conformal and χ curvature properties.We mainly studied the conformally flat cubic metric with weakly isotropic scalar curvature or vanishing scalar curvature,and a class of(α,β)-metrics with vanishing χ curvature.The main findings are summarized as below:1.Firstly,in the second section of this paper,we studied cubic metric F=(?)on an n(≥3)dimensional manifold M,it not only belongs to(α,β)metrics,but also belongs to m-th root metrics.By the properties of(α,β)-metrics and m-th root metrics,with the condition of conformally flat,we found that the scalar curvature must vanish if it is of weakly isotropic scalar curvature.And the cubic metric would be reduced to locally Minkowskian metric if it is of weakly isotropic scalar curvature.2.Secondly,in the third section of this paper,we gave the expression of χ curvature for a class of(α,β)-metrics,and considered a class of non-Riemannian(α,β)-metrics that is of vanishing χ curvature on an n(≥3)dimensional manifold M.To simplify computation,we took an orthonormal basis on the tangent space TxM at x with respect to α and the suitable coordinate transformation.Then we obtained the characterized equations.Based on it,we constructed a class of(α,β)-metrics with vanishing χ curvature. |
