Research On The Unicorn Problem Of General (?,?)-Metrics

In Finsler geometry,there are two classes of important manifolds,namely Berwald manifolds and Landsberg manifolds.By definition,a Berwald manifold must be a Landsberg one.However,people couldn't find Landsberg manifolds which are not Berwaldian,since L.Berwald introduced these concepts during1920 s.A natural and long-standing problem in Finsler geometry is the following:Is there any Landsberg metric which is not of Berwald type? In 2005,D.Bao suggested to name Landsberg metrics that are not Berwald type as unicorns.Since then this problem is referred to as the unicorn problem.In this paper,we will give a brief review of the unicorn problem,and study Finsler manifolds with general(?,?)-metrics in details.This paper consists of 4 chapters.In chapter 1,we will give a brief introdcution to Finsler geometry,Berwald curvature,Landsberg curvature,and the unicorn problem.In chapter 2,we will study the local properties of some geometric invariants on Finsler manifolds by Chern connection and its curvature.Furthermore,we will give discriptions of some special Finsler manifolds by using parallel transport.In chapter 3,we will introduce the correspondence between the differential geometry of a Minkowski space and the centroaffine geometry of its indicatrix.We will get an equivalence theorem of Minkowski spaces.Thus,we can ascribe the unicorn problem to an equivalence problem of Minkowski spaces and finally to a rigidity problem in affine geometry.Following the strategy in the above two chapters,we will investigate general(?,?)-metrics in chapter 4.Firstly,we will calculate the Cartan tensor of(?,?)-norms in details.Then,we will obtian an equivalence theorem by using the centroaffine differential geometric structure of indicatrices of(?,?)-norms.Finally,by applying the equivalence theorem to the parallel transports of Finsler manifolds,we will prove the main result of this paper: general(?,?)-metrics on smooth manifolds of dimension n ? 4with vanishing Landsberg curvatures must be Berwaldian.