?????- Metric And The Important Conformal Properties

A Finsler metrjc which can be regarded as generalized Riemann metric is a function defined on the tangent bundle F:TM→[0,∞)satisfying the follos-ing properties(1)F(x,y) is smooth on the slit tangent bundle;(2)F(x,y)is positively homogeneous function of degree one on y.;(3)the fundamental tensor (gih(x,y):=1/2[F2]yiyj)is positive definite. If gij is independent of x,then F is called a Minkowski metric.If gij is independent of y, then F is called a Rimann metric.Given a Ricmann metric α=(?)aijyiy3and a1-form β=biyi,1et F=αφ(s),s=β/α, where φ(s)is a positive smooth funciton defined on(-bo,bo).If a Finsler metric F satisfies the following properties:(1)‖βx‖α:=(?)aijbi(x)bj(x)<bo,(2)(?)(s)-s(?)(s)+(b2-s2)(?)">0,|S|≤b<bo, then F is called a regular (α,β)-metric.In this paper we discusss two problems on(α,β)-metrics.The first is the general unicorn problem,and the second is the conformal problem.The canonical parallel translation of a tangent vector U∈TxM on Finsler manifold(M,F)is defined by the differential function Ui(t)+Ujгjki(σ(t)); U(t)) σk=O,where гjki=(Gi)yjyk. Berwald metrics are Finsler metrics which canon-ical parallel translations are all linear processes, which means гijk=Ti(x). Define the diffeomorphism (?)t: TxM\{O}→Tσ(t)M\{O} by the canonical parallel translation as (?)t(x, U):=(σ(t),U(t)). That diffeomorphism keeps F invariant.which means (?)*tF=F.While whether (?)t keeps the Riemann metric gx:=gij(x,y)dyi(?)dyi which induced by F defined on punctured tangent space TxM\{O}invariant may not be true.In Finsler geometry it is well known that a Berwald metric must be a Landsberg metric. Then finding a Landsberg metric which is not of Berwald type is a long existing open problem.D.Bao called it as the unicorn problem.Landsberg metrics also can be defined by Landsberg curvature L:=Lijkdxi(?) dxj(?)dxk where Lijk(x,y):=Cijk;mym. Ji:=gjkLijk is induced by contracting Lijk by9jk.The mean Landsberg curvature J is defined by J:=Jidxi.A Finsler metric F is called a weak Landsberg metric if and only if J=0. Clearly s Landsberg metric is also a weak Landsberg metrjc. We call non-Berwald-typing Landsberg metrics as the generalized unicorns, and we discuss the problem of the existence of the generalized unicorn on regular (α,β)-metrics. We prove that on an n (n>2) dimensional space, if a regular (α,β)-metric F=α(?)(s), s=β/α satisfies that (?)(s) is a polynomial in s, then F is a weak Landsberg metric if and only if F is a Berwald metric. The conclusion means that there is no generalized unicorn on (α,β)-metrics which are having polynomial typing.If a Finsler metric F satisfies the equation J+c(x)FI=0, then F is called of relative isotropic mean Landsberg curvature, where I is the mean Cartan cur-vature. A square metric is a special (α,β) metric satisfying We prove that on an n (n>2) dimensional space, if a square metric is of relative isotropic mean Landsberg curvature, then it must be a Berwald metric. Clearly weak metrics must be of relative isotropic mean Landsberg curvature, thus this conclusion is the follow-up discuss of the generalized unicorn problem.The two Finsler metircs F and F are called conformally related if and only if there is a function k(x) on manifold satisfying F=ek(x)F. If a Finsler metric F conformally related to a Minkowski metric, then F is called a conformally flat Finsler metric. We study conformally flat weak Landsberg (α,β)-metrics and ob-tain that conformally flat weak Landsberg (α,β)-metrics must be either Riemann metrics or locally Minkowski metrics. Further we study conformally flat (α,β)-metrics of relative isotropic mean Landsberg curvature and prove that if (?)(s) is a polynomial in s, then a conformally flat (α,β)-metric F=α(?)(s), s=β/α which is of relative isotropic mean Landsberg curvature must be either Rieman-nian metrics or locally Minkowski metrics.The geodesic coefficients Gi of a Finsler metirc F are defined by Gi:=1/4gil(x, y){[F2]xkyl (x, y)yk-[F2]xl (x, y)}. F is called a Douglas metric if and only if geodesic coefficients Gi satisfying Gi=1/2гjki(x)yiyk+P(x,y)yi, where гjki(x) is a scalar function on M and P(x, y) is a positively homogeneous function of degree one. We study the conformal transformations between Douglas (α,β)-metrics, and we prove that on an n (n>2) dimensional space, let F and F be two non Riemannian regular (α,β)-metrics. If F is a non-Randers-typing Douglas metric, then F is also a Douglas metric if and only if the conformal transforamation is a homothety.The distortion (?) is difined by The rate of change of the distortion r along geodesics of F can be used to de-fine the S-curvature which means S:=T;mym. F is of isotropic S-curvature if S=c(x)(n+1)F. In this paper we study the conformal transformation between two (α,β)-metrics of isotropic S-curvature and prove that on an n (n>2) di-mensional space, let F and F be two non Riemannian regular (α,β)-metrics. If F is of isotropic S-curvature, then F is also of isotropic S-curvature if and only if the conformal transforamation is a homothety.