Some Properties Of Special (α, β)-Metrics With Flag Curvature And Locally Dually Flat

In this paper,we study the properties of special (α,β)-metrics.In the third part,firstly we study conditions of special (α,β)-metric to be of isotropic S-curvature;Secondly we computethe Ricci curvature of (α,β)-metric F=α+εβ+kβ²/α.Intrigued by C.Robles's work,we finda necessary condition for a class of Ricci-isotropic (α,β)-metrics.Then,we get a result thatthe constant flag curvature K of (α,β)-metric F=α+εβ+kβ²/αis zero.In the forth part,we discuss firstly the special properties of the conformal transformations on (M,F),secondly westudy the Matsumoto metric of locally dually flat.We obtain mainly the following results:Theorem 3.1 F=α²/βis a Kropina metric on a mainfold of dimension n (n≥3),if itis of scaler flag curvature K=K(x,y),then F is of isotropic S-curvature if and only if K is aconstant.In this case,S=0 and K satisfies the following equation:Theorem 3.2 Let F=F=α+εβ+2kβ²/α-k²Î²⁴/3α³ be a Finsler metric on a mainfold of dimension n (n≥3),whereεand k≠0 are constants,if F is of scaler flag curvature K=K(x,y),thenF is of isotropic S-curvature if and only if F is a Berwald metric.In this case,F is a locallyMinkowski metric.Corollary 3.1 Let F=αφ(β/α)be a Finsler metric on a mainfold of dimension n (n≥3),βis a Killing 1-form with respect toα,if F is of sealer flag curvature K=K(x,y),then K=0 if and only ifβis closed. Theorem 3.3 F=(α,β)²/αis a Finsler metric on a mainfold of dimension n (n≥3),if itis of isotropic Ricci curvature,then F is Ricci-flat.Theorem 3.4 Let F=α+εβ+kβ²/αbe a Finsler metric on a mainfold of dimension n(n≥3),if F is Ricci-isotropic,Ric=(n-1)λ(x)F²,whereλ=λ(x)is a scaler function,thenλ=0.In this case,F is Ricci-flat.Corollary 3.2 Let F=α+εβ+kβ²/αbe a Finsler metric on a mainfold of dimension n (n≥3),if F is of isotropic flag curvature K,then K=0.Proposition 4.1 Let F and (?) be two Finsler metrics on an n-dimensional manifold M.If (?)(x,y)=e^c(x)F(x,y),then F is C-reducible if and only if (?) is C-reducible.Proposition 4.2 Let F and (?) be two Finsler metrics on an n-dimensional manifold M.If (?)(x,y)=e^c(x)F(x,y)and F is a Douglas metric,then (?) is a Douglas metric if and only ifF²/2(cⁱy^j-c^jyⁱ)=B_klm^ij(x)y^ky^ly^m.Proposition 4.3 Let F and (?) be two Finsler metrics on an n-dimensional manifold M.If F is conformally flat,the following are equivalent:(a)F is locally dually flat;(b)F is locally projectively flat;(c)c₀F(?)-c_lF=0.where c_l:=(?)c/(?)x^l,F(?)l:=(?)F/(?)y^l,c₀:=c_ky^k.Theorem 4.1 A Matsumoto metric F=α²/α-βon a n-mainfold,αis locally projectivelyflat,if F is locally dually flat,thenαis flat metric andβis parallel.In this case,F is a locallyMinkowski metric.Theorem 4.2 A Matsumoto metric F=α²/α-βon a n-mainfold,αis locally dually flat,if F is locally dually flat with scaler curvature K=K(x,y),then K=0.In this case,F is alocally Minkowski metric.