关于对称芬斯勒度量的若干性质

Some Properties of Reversible Finsler Metric

在n(n≥3)维芬斯勒流形(M,F)上,利用芬斯勒几何的基础知识和基本方法得到了对称芬斯勒度量F(reversible Finslermetric)具有若干很好的曲率性质;并进一步证明了对称(α,β)-度量F=αφ(s)具有相对迷向平均Landsberg曲率的充分必要条件是F为黎曼度量或Berwald度量,拓展了沈忠民等人的结果。最后证明了对称芬斯勒度量F具有殆迷向S-曲率时,F必为弱Berwald度量,这时如果F还具有标量旗曲率K(x,y),那么K(x,y)必为常数。

Let （M, F） be an n-dimensional Finsler manifold （n≥3）, using Finsler geometric basic knowledge and methods, it is obtained that reversible Finsler metric F is of many good curvature propertoes. It is proved that reversible （α,β）-metric F=αφ（s） is of relatively isotropic mean Landsberg curvature if and only if the metric F is either Riemannian or Berwaldian. , and it develops the result given by Shen Zhong Ming. Finally, it is obtained that if reversible Finsler metric F is of almost isotropic S-curvature, the metric F must be weak Berwaldian. In this case, if the metric F is of scalar flag curvature K（x,y）, thenK（x,y）must be a constant.