A Local Classification Of A Class Of (α,β) Metrics With Constant Flag Curvature

In Finsler manifold,(α,β)metric is a special metric,which is given by the form F αΦ(s),s=β/α,where (?) is a Riemann metric,β=bi(x)yi is a1-form,and Φ=Φ(s)is a positive C∞function on(-b0,b0).In Finsler geometry,one important problem is to classify those metrics of constant flag curvature and a great progress has been made.In this paper,we will study a special(α,β)metric which is in the form Φ(s)=1+Es+2s2-1/3s4,that is,F=α+εβ+2β2/α-β4/3α3.we will discuss the classification of this metrics with constant flag curvature.Firstly,we compute Riemannian curvature and Ricci curvature of(α,β)metric. Then we apply these formulae to discuss a special class(α,β) metric F=α+εβ+2β2/α-β4/3α3,which have constant flag curvature.we obtain the necessary conditions that F have constant flag curvature.Then we prove that such metrics must be locally projectively flat and complete their local classification.This paper consists of three parts:In the first part,we mainly illustrate the backgrounds of our study and the related primary definitions and theorems.In the second part,we show some simple related property of(α,β)metric F=α+εβ+2β2/α-β4/3α3.In the third part,we show the main theorem,and give a simple proof.