The Studies On Curvature Properties Of The General (α,β)-Metrics

In this paper,we study and discuss the flag curvature,the Ricci curvature and non-Riemannian geometric quantities Ξ-curvature and H-curvature of the general(α,β)-metrics and the related problems.Firstly,we study the Ricci curvature and Ricci curvature tensor of the general(α,β)-metrics.Under certain conditions on a and β,an equivalent characterization of strong Einstein general(α,β)-metrics is given.Furthermore,by using the Riemann curvature formula of the general(α,β)-metrics given by Xia Qiaoling,a sufficient and necessary condition for a general(α,β)-metric to be a Ricci-quadratic Finsler metric is obtained under the same conditions.Secondly,we study a special kind of the general(α,β)-metrics--Randers metrics and discuss some problems on the flag curvature of Randers metrics.Under a condition on Ξ-curvature,we prove that a Randers metric F=α+β of scalar flag curvature on an n-dimensional manifold M must be of constant flag curvature if β is a Killing 1-form with respect to α.In this case,the structures of Randers metrics with the conditions mentioned above can be determined completely when the dimension n≥3.