Some Explicit Constructions Of Spherically Symmetric Finsler Metrics

Finsler geometry is just Riemannian geometry without quadratic restriction.The various canonical Finsler metrics on domains of Euclidean spaces,such as spherically Symmetric metrics,projectively flat metrics,dually flat metrics,are important models in Finsler geometry which reveal different geometric properties with Riemannian geometry.The equations of dually flat Finsler metric and projectively flat Finsler metric have some linear structures.Motivated by well-known Finsler metrics and the analytic method,mathematicians have explored many non-Riemannian Finsler metrics which contribute to the study of Finsler geometry.This present theis follows this idea to find some explicit dually flat Finsler metrics and projectively flat Finsler metrics.Our thesis is organized as follows:In Chapter 1,we will briefly recall the history of Finsler geometry and introduce our motivations as well as main results.In Chapter 2,we will introduce some basic notions of Finsler geometry,such as the geodesic coefficient,curvatures,sphericalty symmetric metrics,dually fglat Finsler metrics and projectively flat Finsler metrics.Some basic properties of Finsler metrics on open domains of Euclidean spaces will be present.In Chapter 3,inspired by the decomposition of Funk metric and the linear structure of square of dually flat equations,we apply the analytical method of Huang-Mo[13]to explore more sphericalty symmetric dually fat Finsler metrics with the follwing form:(?) where(?) and p ∈R\{0,1},s=<x,y>/|y|.t=|x|2/2,fl(t)fl(t)≠0,g(t),h(t)are differentiable functions and h(n)(t)is constant.Moreover,the corresponding necessary and sufficient conditions for the dually flat Finsler metrics with the above form is derived.As an application,the classification of dually flat spherically symmetric metrics with the above form under l=6 is established.A similar result will be given for the case l=8.Note that Huang and Mo[13]have discussed the case p=1/2,l=4.At the end of this chapter,such analytic method is generalized to study the following dually flat sphericalty symmetric metrics:(?) where q∈R,gj(t)is a differentiable function,gl(t)≠0,and I(t,s)is as above metrics.It is a variation of the metrics in Chen’s thesis[6].In Chapter 4,we explore some projectively flat(α.β)-metrics formed by polynomials with degree 8:F=α(1+a1s+a2s2+a4s4+a6s6+a8s8),where s=β/α,ai(i=1,2,4,6,8)are constants.By the unique factorization properties of polynomials,the relationship between the coefficients and explicit representations of projectively flat(α,β)-metrics will be derived.