Pogorelov Integral Representation And Dually Flat Finsler Metrics

Hilbert's Fourth problem in the smooth case is to characterize and study projectively flat Finsler metrics on an open subset U in Rn,it is a key point of studying Finsler geometry.Dually flat Finsler metrics is from the geometric information,they also have important research value.We find that the solution of projective flat equation and the solution of dually flat equation have one-to-one correspondence,through the analysis of projective flat euqation and dual flat e-quations.We use the correspondence and the projectively flat Finsler metrics we have known to get new dually flat Finsler metrics.In[19],Using the integral representation of two-dimensional reversible projective Finsler metrics due to A.V.Pogorelov.In[11],Mo Xiaohuan and Huang Libing also used the corresponding relationship and A.V.Pogorelov.integral representation to construct two new families of dually flat(not necessarily reversible)Finsler metrics with or-thogonal invariance of arbitrary dimensional.Applying the idea of this method,we continue to produce new dually flat Finsler metrics.This paper consists of three parts:In the first part,we mainly illustrate the backgrounds of our research and the related definitions,theorems and conclusions,prepared for the following discussion.In the second part,we introduce some relevant knowledge of Hamel's differen-tial equation and Pogorelov integral function.At the beginning,using the Pogorelov function(?)(x1,x2,y1,y2)=(?)(y1 cos ?+y2 sin ?)?(x1 cos ?+x2 sin ?,?)d? we manufacture d-ually flat Finsler metrics;In the third part,we respectively to produce new dually flat Finsler metrics with orthogonal invariance by the two variables in this function.