A New Quantity In Finsler Geometry

Recently, a great progress has been made in studying Finsler metrics of weakly isotropicflag curvature. These Finsler metrics are of scalar curvature whose flag curvature is in a specialform K=θ/F+σ, where θ is a1f orm and σ is a scalar function on M. Shen proposed tostudy Finsler metrics of this form. Finsler metrics of weakly isotropic flag curvature not onlyinclude Finsler metrics of constant flag curvature, but also include Finsler metrics of almostisotropic S-curvature and of scalar flag curvature.In this paper, we find the desired quantity to characterise Finsler metrics of weakly isotropicflag curvature and call it the C curvature. The new quantity is defined by Riemannian curvatureand is a non-Riemannian quantity for a Finsler manifold with dimension n=3. We find that theC curvature is closely related to the flag curvature and the H curvature. In this paper, we givethe relation between the C curvature and the flag curvature K, as well as the H curvature whenthe metric F is of scalar curvature. We show the main theorem that, for an n(≥3) dimensionalFinsler manifold of scalar curvature with Flag curvature K(x,y), the flag curvature K is weaklyisotropic if and only if the C curvature vanishes. The theorem tells us that C curvature givesa better description of Finsler metrics of weakly isotropic flag curvature than H curvature. Inthe end, we give a simple proof of the Najafi-Shen-Tayebi’ theorem by using the main result.This paper consists of three parts: In the first part, we mainly illustrate the backgroundsof our study and the related primary definitions and theorems. In the second part, we give theconstruction of C curvature, and also show some simple related properties of C curvature. Inthe third part, we prove the main theorem, and give a simple proof of the Najafi-Shen-Tayebi’theorem.