(α, β)-metrics with Special Curvature Properties

(α,β)-metrics form an important and computable class of Finsler metrics. In this paper, we obtain firstly a formula of mean Cartan torsion for (α,β)-metrics and characterize Riemann metrics among (α,β)- metrics. Further, we study the (α,β)- metrics with relatively isotropic mean Landsberg curvature and obtain a sufficient and necessary condition for the (α,β)-metrics to be of relatively isotropic mean Landsberg curvature, which generalizes the classification theorem on the weakly Landsberg (α,β)-metrics given by Chinese-American mathematician Z. Shen and B. Li released on“Science in China Series A”.