On The Constructions Of Finsler Metrics With Certain Attribute

Finsler geometry is Riemannian geometry, and it without the quadratic restriction in measure. Its theory and research method has been widely used in the field of information science and computer tech-nology, become the development direction of differential geometry in the 21st century. How to construct dually flat Finsler metrics is a valu-able problem in Finsler geometry. According to it, we constructed some new class of dually flat Finsler metrics by using dually flat equations.This paper is divided in two parts:The first part: In the Euclidean space of a convex set Ω, we study the dually flat symmetric Finsler metric F(x,y) = |y|(?).Based on the dually flat equation sfts + fss - 2ft = 0, where t =|x|2/2,s=<x,y>/|y|.Then gain two forms of dually flat symmetric Finsler metrics. One form is investigating a solution of polynominal and the other is using a power series on an interval to construct the solutions.The second part: For the general (α,β)-metrics F = αφ(b2, β/α),where α is Riemannian metric, β is a 1-form and b = ||β||α. Then F is dually flat when φ satisfy the PDE: (φ2)2 + φφ22 + 2sφ1φ2 +2sφφ12 - 4φφ1 = 0, and we gain the equivalent dually flat equationψ22 + 2sψ12 -4ψ1 = 0 by introducing variables ψ = φ2. So we gain some new general (α, β)-metrics is dually flat Finsler metrics by investigating a solution of polynominal and separation of variables to the equation.