On A Class Of Projectively Flat (α, β)-metrics

In this paper, We find a sufficient and necessary condition for an important class of (α,β)-metrics in the form F =αφ(α/β) to be projectively flat. we prove the followingTheorem 1.2 Let F =αφ(α/β) be a Finsler metric on a manifold M, where a is a Riemann metric,β= b_iyⁱ a 1-form with (?)x∈M, b := ||β_x|| < b₀,φ=φ(s) a C^∞function on (-b₀, b₀) satisfyingandwhere p≠0. The solutionsφof the above condition are analyic near the origin and the power series of the solutions are of the formwhere C₁ are arbitrary constants. If C₁≠0 and r≠1 or C₁ = 0 but r =-1/2k, where k is any positive integer, then F =αφ(α/β) is projectively flat if and only ifand the spray cefficients G_αⁱ ofαsatisfy...