Locally Dual Flat Finsler Metrics

The metrics of dual flat is one class of important metrics in geometry, which has significant applications in large scale artificial neural networks, information geometry, and superstring theory. Professor Shen has done a lot research on information geometry from the perspective of Finsler geometry. however, it is a must to study the Finsler metrics of dual flat or locally dual flat. Among the known Finsler metrics, Funk metrics and locally Minkowsian metrics are the only two which are locally dual flat. We should study the Randers metrics and (α,β) metrics before the research of locally dual flat Finsler metrics. In this paper, we have studied the locally dual flat Randers metrics and square Randers metrics and mainly obtained the following results.Theorem 3.3 Let (M,F) be a Finsler space of dimension n(≥3). Ifαis locally projectively flat, then the following are equivalent:(1) F is locally dual flat;(2)P=c(x)F;(3)F_x^l=2PF_y^l;(4)F_x^l=2c(x)FF_y^l.Where P=(F_x^ky^k)/(2F) is the projective factor of F.Theorem 3.4 Let (M,F) be a Finsler space of dimension n(≥3). F=(?)(s) (s=β/α) is an (α,β)-metric, whereα:=(a_ijyⁱy^j)^1/2 is a Riemannian metrics andβ:=b_iyⁱ is a 1-form, and(?)≠k₁(1+k₂s²)^1/2+k₃s. Then F is locally dual flat and locally projectively flat if and only if F is locally Minkowskian. Theorem 4.1 Let (M, F) be a Finsler space of dimension n(≥3). F =α+βis a Randers metric, whereα:=(a_ijyⁱy^j)/(1/2) is a locally projectively flat Riemannian metric andβ:=b_iyⁱ is a 1-form. Then F is locally dual flat if and only if one of the following holds(1) F is locally Minkowskian.(2) F is locally isometric to the following metricWhereμ=-4c², c is a constant. When c = 1/2, (F|^) is a Fuck metric.Theorem 4.2 Let (M, F) be a Finsler space of dimension n(≥3). F=α+βis a Randers metric, whereα:= (a_ijyⁱy^j)/(1/2) is a Riemannian metric andβ:=b_iyⁱ is a closed 1-form. Then F is locally dual flat if and only if the following conditions holds(1)b_i|j=(?)/2(a_ij-b_ib_j)+2(?)(b_ib_j-b²a_ij);(2)G_α^m=θ_y^m+(?)α²b^m-(?)β_y^mWhere∈=∈(x) , (?)=(?)(x) are scalar functions, andθ=(6(?)+(?))/2β.When∈=0, F is locally isometric to the followingWhereμ=-4c², and c is a constant. If c=1/2, (F|^) is a Fuck metric.Theorem 5.1 Let (M, F) be a Finsler space of dimension n(≥3). F=α(?)(s) (s=β/α) is a square Randers metric, whereα:=(a_ijyⁱy^j)/(1/2) is a locally projectively flat Riemannian metric andβ:=b_iy^j is a 1-form. Then F is locally dual flat if and only if F is locally Minkowskian.Theorem 5.2 Let (M, F) be a Finsler space of dimension n(≥3). F=α(?)(s) (s=β/α) is a square Randers metric, whereα:=(a_ijyⁱy^j)/(1/2) is a Riemannian metric andβ:=b_iyⁱ is a closed 1-form. Then F is locally dual flat if and only if F is locally Minkowskian.