Finsler metrics of weakly isotropic flag curvature

الأصل · EN

Finsler metrics of scalar flag curvature play an important role to show the complexity and richness of general Finsler metrics. In this paper, on an n-dimensional manifold M we study the Finsler metric F=F(x,y) of scalar flag curvature K = K(x,y) and discover some equations K should be satisfied. As an application, we mainly study the metric F of weakly isotropic flag curvature K = 3 θ/F + σ, where θ=θᵢ(x) yⁱ ≠ 0 is a 1-form and σ=σ(x) is a scalar function. We prove that in this case, F must be a Randers metric when dim(M) ≥ 3. Further, without the restriction on the dimension we prove that projectively flat Finsler metrics of such weakly isotropic flag curvature are Randers metrics too.