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

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