On The Einstein (alpha, Beta) Measure And Its Related Issues

Finsler geometry is just Riemannian geometry without quadratic restriction on the metrics. Hilbert attached great importance to this field. In his famous Paris address of1900, there are two problems tightly attached to Finsler ge-ometry. The fourth Problem is to describe projectively flat Finsler metrics on U(?)Rn. From90’s of the20th century, Finsler geometry has been developed rapidly under the encouragement of S. S Chern. Meanwhile, it has been applied to physics, biology, control theory, psychology and so on ([1],[4],[6],[14]).Einstein metrics, which closely connect geometry with general relativity, play an important role in differential geometry. S. S. Chern openly asked the following question on many occasions:Does every smooth manifold admit a Finsler Einstein metric?It is still hard to describe general Finsler Einstein metrics. Recently, some mathematicians have studied some special Finsler metrics ([48],[11],[42],[18],[57],[43],[53]).In this paper we study Einstein (α,β)-metrics and related problems. The content is divided into four chapters. In the first chapter, we introduce ba-sic knowledge of Finsler geometry. In the second chapter, we studied Einstein Kropina metrics. Under the navigation description, we obtaine results similar to Einstein Randers metrics. And we get conformal rigidity for Einstein Kropina (or Einstein Randers) metrics. In the third chapter, we studied Einstein Mat-sumoto metrics. In the last chapter, projective change of Kropina metrics has been considered.0.1Einstein Kropina metricsLet F be a Finsler metric on an n-dimensional manifold M. F is called an Einstein metric with Einstein scalar a if Ric=σF2. where σ=σ(x) is a scalar function on M. In particular, F is said to be Ricci constant (resp. Ricci flat) if F satisfies () where σ=const.(resp. σ=0).An (α,β)-metric on M is expressed in the form F F=αφ(s), s=β/α, where α=(?) is a positive definite Riemannian metric, β=bi(x)yi a1-form. It is known that (α,β)-metric with‖βx‖α<b0is a Finsler metric if and only if φ=φ(s) is a positive smooth function on an open interval (-b0, b0) satisfying the following condition: φ(s)-sφ’(s)+(b2-s2)φ"(s)>0,(?)|s|≤b<b0, see [20].The (α,β)-metrics form an important class of Finsler metrics appearing iter-atively in formulating Physics, Mechanics, Seismology, Biology, Control Theory, etc.(see [5,38,54]). Recently, some progress has been made on Finsler Einstein metrics of (α,β) type. D. Bao and C. Robles have shown that every Einstein Ran-ders metric of dimension n(≥3) is necessarily Ricci constant. A3-dimensional Randers metric is Einstein if and only if it is of constant flag curvature, see [11]. For every non-Randers (α,β)-metric F=αφ(s),s=β/α with a polynomial func-tion φ(s) of degree greater than2, Cheng, Shen and Tian have proved that it is an Einstein metric if and only if it is Ricci-flat ([18]).The Kropina metric is an (α,β)-metric where φ(s)=1/s, i.e., F=α2/β, which was considered by V.K.Kropina firstly([27]). Such a metric is of physical interest in the sense that it describes the general dynamical system represented by a Lagrangian function (cf.[7]), although it has the singularity. Some recent progress on Kropina metrics has been made, e.g., see [38,54,55].Let where "|" denotes the covariant derivative with respect to the Levi-Civita con-nection of α. Denote where(aij):=(aij)-1and bi:=aijbj. Denote ri:=aijrj,si:=aijsj, ri0:=rijyj,si0:=sijyj,r00:=rijyiyj,r0:=riyi and s0:=siyi.For an (α,β)-metrics, the form β is said to be Killing (resp. closed) form if rij=0(resp. Sij=0).β is said to be a constant Killing form if it is a Killing form and has constant length with respect to a, equivalently rij=0, si=0. And accordingly, a vector field W in a Riemannian manifold (M,h) is said to be a constant Killing vector field if it is a Killing vector field and has constant length with respect to the Riemannian metric h.By the necessary and sufficient conditions for Kropina metrics to be Einstein (see Theorem2.2.1), we can get following.Theorem0.0.1Let F=α2/β be a non-Riemannian Kropina metric with constant Killing form β on an n-dimensional manifold M, n≥2. Then F is an Einstein metric if and only if a is also an Einstein metric. In this case, σ=1/4λ62≥0, where λ=λ(x) is the Einstein scalar of a. Moreover, F is Ricci constant for n≥3.Remark. B. Rezaei, etc., also discussed Einstein Kropina metrics with constant Killing forms. Unfortunately, the computation and results in [41] are wrong. Theorem0.0.1is the corrected version of Theorem4.6and Corollary4.9of [41].As is well known, a Finsler metric is of Randers type if and only if it is a solution of the navigation problem on a Riemannian manifold, see [12]. Inspired by this idea, we can prove that there is a one-to-one correspondence between a Kropina metric and a pair (h., W), where h is a Riemannian metric and W is a vector field on M with the length‖W‖h=1. And we call this pair (h,W) the navigation data of the Kropina metric.Proposition0.0.1A Finsler metric F is of Kropina type if and only if it solves the navigation problem on some Riemannian manifold (M, h), under the influence of a wind W with‖W‖h,=1. Namely, F=α2/β if and only if F=h2/2W0, where h2=e2ρα2,2W0=e2ρβ and e2ρb2=4.This new perspective allows us to characterize Einstein Kropina spaces as follows.Theorem0.0.2Let F=(α2)/β be a non-Riemannian Kropina metric on an n-dimensional manifold M,n≥2.Assume the pair(h,W)is it’s navigation data. Then F is an Einstein metric if and only if h is an Einstein metric and W is a unit Killing vector field.In this case,σ=δ≥0,where δ=δ(x)is the Einstein scalar of h.Moreover,F is Ricci constant for n≥3.By Theorem0.0.2,we can construct a vast Einstein Kropina metrics by their navigation expressions,i.e.,Riemannian Einstein metrics and unit Killing vector fields.Let h be n-dimensional Riemannian space of constant curvature μ. Denote h=‖dx‖2/H2,where H:=1+μ/4‖x‖2and‖·‖2is the standard metric in Euclidean space. Then the general solutions of Killing vector field W with respect to h are where Qij=-Qji and ci are1/2n(n+1)constants,see[44].So there exist lots of unit Killing vector fields.We list a special case here.Example Let M be an3-dimensional unit sphere with standard metric h. Let where a2+b2+c2=1and a,b,c are all non-zero constants.Define W=Wi(?)/(?)xi with the same form as in(0.0.5),where Wi=hijWj.Then‖W‖h=1.Define F=h2/(2W0),where W0=Wiyi and W0=Wi(x)yi>0.Thus F is an Einstein Kropina metric.Finsler metrics,which are of constant flag curvature,are special cases of Einstein metrics.We have following results.Corollary0.0.3[see[56]]Let F=(α2)/β be a non-Riemannian Kropina metric on an n-dimensional manifold M,n≥2. F is of constant flag curvature K if and only if the following conditions hold: 1) W is a unit Killing vector field,2) The Riemannian space (M, h) is of nonnegative constant curvature K.Remark. R. Yoshikawa. etc., also studied Kropina metrics of constant flag curvature in terms of (h,W). Their computation is tedious. Corollary0.0.4is the revised version of Theorem4of [56], which does not restrict nonnegative constant curvature K.Finally, we study the conformal rigidity for Einstein Kropina (or Einstein Randers) spaces.Liouville’s Theorem has proved that the Euclidean space can be mapped conformally on itself only by a composition of Mobius transformations. For Riemann spaces, Brinkmann has obtained general results. Little work has been done on Finsler spaces. In this paper, by navigation idea and properties of conformal map, we proved that the conformal transformation between Einstein Randers (or Einstein Kropina) spaces must be homothetic.Theorem0.0.4Conformal transformation between Einstein Randers spaces must be homothetic.We obtain the corresponding result for Kropina metrics.Theorem0.0.5Conformal transformation between Einstein Kropina spaces must be homothetic.For a Minkowski metric, a special Einstein metric, we have the following result.Corollary0.0.6Every conformally flat Einstein Randers metric (or Kropina metric) must be a Minkowskian.0.2Einstein Matsumoto metricsThe Matsumoto metric is an interesting (α,β)-metric with φ=1/(1-s), introduced by using gradient of slope, speed and gravity in [33]. This metric formulates the model of a Finsler space. Many authors ([2,33,36], etc) have studied this metric by different perspectives. Firstly,we obtain the necessary conditions for Matsumoto metrics to be Einstein.Theorem0.0.7Let F=α2/α-β be a non-Riemann Matsumoto metric on an n-dimensional manifold M,n≥2.If F is an Einstein metric,then the followings hold1)α is an Einstein metric,i.e.,Ric=λα2,2)β is a conformal form with respect to a,i.e.,r00=cα2, where Ric denotes the Ricci curvature of α,λ=λ(x),c=c(x)are functions on M. and"|"and"."denote the horizontal covariant derivative and vertical covariant derivative with respect to α,respectively.Remark. M.Rafie-Rad,etc.,also discussed Einstein Matsumoto metrics. Unfortunately,the computation and results in[40]are wrong because they ne-glected b2in ai(i=0,...,14).Theorem0.0.7is the corrected version of Theorem1in[40].We can’t obt ain the necessary and sufficient conditions for Matsumoto met-rics to be Einstein.So we study the question under some conditions.Theorem0.0.8Let F=α2/α-β be a non-Riemannian Matsumoto metric on an n-dimensional manifold M,n≥3.Suppose the length of β with respect to α is constant.Then F is an Einstein metric if and only if α is Ricci-flat and β is parallel with respect to α.In this case,F is Ricci-flat.Theorem0.0.9Let F=α2/α-β be a non-Riemannian Matsumoto metric on an n-dimensional manifold M,n≥3. Suppose β#,which is dual to β,is a homothetic vector field,i.e.,r00=cα2,where c=constant. Then F is an Einstein metric if and only if α is Ricci-flat and β is parallel with respect to a. In t his case,F is Ricci-flat.For an(α,β)-metric,the form β is said to be Killing(resp.closed)form if rij=0(resp. Sij=0). β is said to be a constant Killing form if it is a Killing form and has constant length with respect to α, equivalently rij=0and si=0.Remark. B. Rezaei, etc., discussed Einstein Matsumoto metrics with con-stant Killing form in [41]. Meanwhile, they got wrong results. Theorem0.0.8and Theorem0.0.9generalize their study.For the S-curvature with respect to the Busemann-Hausdorff volume form ([20]), we have followingTheorem0.0.10Let F=α2/α-β be a non-Riemannian Matsumoto metric on an n-dimensional manifold M, n≥2. Then S-curvature vanishes if and only if β is a constant Killing form.From above theorems, we can easily get the followingCorollary0.0.11Let F=α2/α-β be a non-Riemannian Matsumoto metric on an n-dimensional manifold M, n≥3. Suppose F is an Einstein metric. Then S-curvature vanishes if and only if a is Ricci-flat and β is parallel with respect to a. In this case, F is Ricci-flat.0.3projective change of Kropina metricsA simple fact is that a Finsler metric F=F(x, y) on an open subset U(?) Rn is projectively flat if and only if the spray coefficients are in the form Gi=Pyi, where Gi is defined as followingsAccording to the Beltrami Theorem, a Riemannian metric is projectively flat if and only if it is of constant sectional curvature. The well-known Funk metric F=F(x,y) on a strongly convex domain in Rn is projectively flat with the constant flag curvature K=-1/4. A Randers metric F=α+β is locally projectively flat if and only if a is locally projectively flat and β is closed. R. Bryant has classified projectively flat Finsler metrics on Sn with constant curva-ture K=1. Recently, some special projectively flat metrics have been studied such as the class of(α,β)-metrics by many geometers([19],[45]1[35],[60],[25]). Shen has studied and characterized projectively flat(α,β)-metrics,where φ sat-isfiesφ(0)=1,φ(s)>0,φ(s)-sφ’(s)+(b2-s2)φ"(s)>0in[51].Projectively flat metrics must be Douglas metrics.Theorem0.0.12Let F=α2/β be a non-Riemannian Kropina metric on an n≥2-dimensional manifold M.Then F is of Douglas type if and only if0=b2sij-sibi+sibj.Example Take any function f=f(x)on M.let β=df.Define F=α2/β. Then F must be a Douglas metric.We can make change for Kropina metric F=α2/β followings: Let e2ρ=1/b2.Then‖β‖α=1.Thus without loss of generality,we always assume that Kropina metric F=α2/β satisfy b:=‖β‖α=1.Theorem0.0.13Let F=α2/β be a non-R,iemannian Kropina metric on an n≥2-dimensional manifold M,and satis‖β‖α=1.F is projectively flat if and only if F is of Douglas type and satisfies the followings Where Gi denotes the Spray coefficients of α and ζ:=1/n+1Gykk is a one form.Theorem0.0.14Let F=α2/β be a non-Riemannian Kropina metric on an n≥2-dimensional manifold M,and satisfy‖β‖α=1.Let F be of weakly isotropic S-curvature.Then F is projectively flat if and only if a is of constant curvature and β is parallel with respect to α.In this case,F is of constant curvature.Theorem0.0.15Let F=α2/β be a non-Riemannian Kropina metric on an n≥3-dimensional manifold M,and satisfy‖β‖α=1.Let F be Einstein.Then F is projectively flat if and only if α is of constant curvature and β is parallel with respect to α.In this case,F is of constant curvature. Corollary0.0.16Let F=α2/β be a non-Riemannian Kropina metric on an n≥3-dimensional manifold M,and satisfy‖β‖α=1.Let F be of constant curvature.Then F is projectively flat if and only if a is of constant curvature and β is parallel with respect to α.In this case,F is of constant curvature.Let(M,α)be an n-dimensional Riemannian space of constant curvature μ. Denote α=‖dx‖2/H2,where H:=1+μ/4‖x‖2and‖·‖2is the standard metric in Euclidean space. Then the general solutions of Killing vector field W with respect to α are where Wi:=aijWj,Qij=-Qji and ci are1/2n(n+1)constants,see[44].So there exist lots of one form β,which is parallel respect to α and listed as following Where ci are n constants.Example Let(M,α)be n-dimensional Riemannian space of constant cur-vature μ.Denote α=‖dx‖2/H2.Define β=bi(x)dxi with the same form as in (0.0.7).Define F=α2/β.Thus F is projectively flat.