Some Non-Riemannian Curvature Properties Of The Douglas Metrics And The Finsler Metrics Of Randers Type

In this paper, we first study the complete Douglas spaces ( M , F ). It is proved that, if the Cartan torsion of the complete Douglas space is bounded and F satisfies that H =0 and E jk ?l m= 0,then F is a Berwald metric. Here E denotes the mean Berwald curvature of F and H is the geometric quantity which characterizes the rate of the change of E along geodesics,"|"and"."denote the horizontal and vertical covariant derivatives of F with giving connection, respectively. Moreover, this paper considers some properties of S-curvature of the Finsler metric F = eτ( x)(α+β). It is proved that this kind of metrics is of isotropic S-curvature if and only if it is of isotropic mean Berwald curvature. Finally, We study the Finsler metrics of Randers type in the form of and obtain a sufficient and necessary condition for this metrics to be of isotropic mean Berwald curvature, where is a Riemannian metric andβ= bi ( x ) yi is a 1-form with on an n-dimensional manifold M and k1 > 0, k 3≠0.