具有特殊旗曲率性质的芬斯勒度量的若干定理

Some Theorems of Finsler Metrics with Special Flag Curvature Properties

首先研究了n(≥3)维流形上具有弱迷向旗曲率K=3θ/F+σ的芬斯勒度量F,得到了θ和σ所满足的一个偏微分方程组,其中θ=θi(x)yi是一个1-形式,σ=σ(x)是流形上的一个标量函数.其次,证明了具有常数平均Berwald曲率的芬斯勒度量的H-曲率必然为零.进一步地,讨论了具有标量旗曲率且具有常数平均Berwald曲率的芬斯勒度量,得到了旗曲率K所满足的一个恒等式,并在维数n大于2的条件下,证明了此时芬斯勒度量具有常数旗曲率.

In this paper,we first study the Finsler metric Fof weakly isotropic flag curvature with K=3θ/F+σon a manifold M of dimension n(≥3),whereθ=θi(x)yi is a 1-form andσ=σ(x)is a scalar function on manifold M.We obtain a system of partial differential equations thatθandσsatisfy.Next,we prove that the H-curvature vanishes when Fis of constant mean Berwald curvature.Finally,we discuss Finsler metrics of scalar flag curvature and of constant mean Berwald curvature.In this case,we find an identity that the flag curvature Ksatisfies and prove that Kis actually a constant when nis greater than 2.