A Class Of Special Projectively Flat(α, β)-Metric And The Dual Flat Finsler Space

The purpose of this paper is to study a special projectively flat (α,β)metric,and dual flat Finsler space. The chapter three obtained necessary and sufficient conditions of the Mastumoto metric F =α²/(α-β) which is projectively flat, whereα= (α_ijyⁱy^j)^1/2 is Riemannian metric andβ= b_iyⁱ is a 1-form, and completely determine the local structure of the projectively flat Mastumoto metric F =α²/(α-β) with constant curvature. The chapter four disscussed the dual flat Finsler space, and obtained the equivalent proposition, of dual flat Finsler space.The paper make up some examples which is dual flat metric . The primary results follows:Theoery 3.1 F =α²/(α-β)is projectively flat if and only ifαis projectively flat ,andβis parallelize with respect toα.Lema 3.1 F =α²/(α-β)is projectively flat with constant curvature k=â…¤= constant,thenâ…¤= 0.Theoery 3.2 If F =α²/(α-β)is projectively flat with constant curvature K = 0 ,then F is local Minkowskian metric.Proposition 4.1 Let (M, F) is a Finsler space, F is dual flat if only if there exit a scalar function Q on TM , Q(â…¤y) =â…¤Q(y),Ⅴ＞0 , satisfies there Q =(F²)_x^ky^k/(2F²).Proposition 4.2 Let(M, F) is a Finsler space ,F is dual flat if and only ifLema 4.1 LetΘbe the Funk metric on strongly convex domainΩ∈Rⁿ and Finsler metric L = F² induces (?) if and only if L satisfiesTheorem 4.1 LetΘ=Θ(x, y) be the Funk metric On a strongly convex domainΩ(?)Rⁿ.For an arbitrary pointα= (αⁱ)∈Ω,define a Function F = F(x, y) on TΩ=Ω×Rⁿ by Then L is dual flat.The concrete expression isTheorem 4.2 Letφ=φ(y) be a Minkowski norm. Let (?) given by denote the spray defined inΩ= {y∈Rⁿ|φ(y)＜1} . If L = L(x, y) is a Finsler metric on a neighborhood of the origin that induces spray (?) ,then L is given by whereψ(y) = L(0, y).Conversely,for any Minkowski normφ=φ(y)on Rⁿ, the function L defined in the obove expression induces (?), hence L is a dual flat Finsler metric.