The Constructions Of A Class Of Projectively Flat And Dually Flat Spherically Symmetric Finsler Metrics

It is an important problem in Finsler geometry to construct projectively flat and dually flat Finsler metrics. Based on it, this paper study spherically symmetric Finsler metrics. Analyzing the solution of projectively fiat and dually flat equation, new examples of projectively flat and dually flat Finsler metrics are constructed.The paper is composed of three parts:In the first part, we review the history of Finsler geometry, introduce the research background of projectively flat and dually fiat Finsler metrics, introduce the research significance and situation of spherically symmetric Finsler metrics, introduce some basic knowledge of Finsler geometry, give the main lemmas which we use, summary some of the latest research results at home and abroad, state the main work and the innovation of this thesis.In the second part, we construct a class of projectively flat spherically symmetric Finsler metrics. Let ?=?(t, s) be a function defined by where t = |x|2/2, s=<x,y>/|y|, b is a constant and ?1 is any continuous function,?0 is a polynomial function of degree N where N?n, h0 is a differentiable function which satisfies where C1, C2 are constants. Then the following spherically symmetric Finsler metric on Bm(?), F =| y| ?>(|x|2/2,<x,y>/|y|) is projectively flat. Particularly, when b = 1, n = m = 2, after setting ?0(t) = 0,?1(t) = h0(t)=1/1-2t, we obtain the Funk metric.n the third part, we construct a class of dually fiat spherically symmetric Finsler metrics. Let f= f(t,s) be a function defined by where t = |x|2/2, s=<x,y>/|y|, b is a constant, g(t) is a differentiable function, h(t) is a polynomial function of degree N where N ? n. h(j) (t) denotes the j-order derivative for h(t),?X(t) is any continuous function, where C1, C2 axe constants. Then the following spherically symmetric Finsler metric on Bm(?), F =|y|(?) is dually flat. Particularly, when b = 1, C1 = 0, C2 = 3, r = 2, after setting g(t) = 1/(1-2t)2,h(t) = 0, we obtain It is also obtained by B. Li [5] and C. Yu [7] in other different ways.