With Regard To Certain Important Of Finsler Metrics

The content of this paper is divided into four parts corresponding to fourchapters. In the first chapter, we introduce a very important class in Finslergeometry: (α,β)-metrics also called the metrics in (α,β)-type, whereαis a Rie-mannian metric andβis a 1-form. We study the classification problem on projectivley flat (α,β)-metrics with constant flag curvature and eventually concludethat there are only three types in this class. This classification can be seen asa generalization of the Beltrami theorem in Riemannian geometry. In the sec-ond chapter, we study Douglas metrics which is weaker than projectively flatmetrics. We obtain the equivalent conditions of the (α,β)-metrics to be Douglasmetrics. In the third chapter, we study an another important class in Finslergeometry, the weak Landsberg metrics. We give the equivalent conditions andfind the differential equations which should hold when a general (α,β)-metric isa weak Landsberg metric. We also prove that there are weak Landsberg met-rics which are not Landsberg metrics. And in the last chapter, we study twospecial classes in (α,β)-metrics: Matsumoto metrics and the metrics in the formF=εβ+α+3/2βarctan(β/α)+(αβ²)/2(α²+β²). We obtain the equivalent conditions ofthese metric to be projectively flat and construct some nontrivial examples.0.1 Projectively flat (α,β)-metrics with constant flagcurvatureLet us recall standard geometric definitions, notations and the background.As a more general metric geometry than Riemannian geometry, Finsler geometryis firstly introduced by Riemannian's 1854 "habilitation" though P. Finsler madeimportant contributions in it. The great mathematician S. S. Chern said thatFinsler metrics are just Riemannian metrics without quadratic restriction [1].Finsler geometry is to study the geometric properties of Finsler metric on amanifold [2]. Let Mⁿ is a n-Dim differential manifold, TM is the tangent bundle. If the function on TM, F=F(x, y): TM→R satisfies the following conditions,then it is called a Finsler metric on M:(â…°) F(x,y)is a C^∞function on TM\{0};(â…±) F(x,y)＞0, y≠0;(â…²) F(x,λy)=λF(x,y),λ＞0;(â…³) The fundamental tensor g_ij(x, y)=1/2(F²)_yⁱy^j=2/1 (?)²/(?)_yⁱ(?)_y^j[F²] is positivelydefined.From the above definition we can see that if g_ij(x, y) depends on x only, thenF is a Riemannian metric. If g_ij(x, y) depends on y only, then F is called aMinkowskian metric. Recent studies on Finsler geometry have taken on a newlook [3][4][5] and Finsler geometry can be also applied to biology, physics andpsychology [18][19][20][21][22][25][32][33]. In these researches, peoples find thatthere is a quite important metric class in which a metric consist of a Riemannianmetricα=(a_ij(x)yⁱy^j)(1/2) and a 1-formβ=b_i(x)yⁱ. We call these metrics as(α,β)-metrics or the metrics in (α,β)-type. It's difficult to discuss general Finslermetric because of the complicated computation however for the metrics in thisclass the computation can be handled though it is not easy.An (α,β)-metrics can be expressed as F=αφ(s), s=β/α,whereα=(a_ijy^ij)(1/2) is a Riemannian metric,β=b_iyⁱ is a 1-form.φ=φ(s) is aC^∞positive function on an open interval (-b_o, b_o) satisfyingφ(s)-sφ′(s)+(b²-s²)φ^″(s)＞0, (?)|s|≤b＜b_o.It is known that F is a Finsler metric if and only if ||β_x||_α＜b_o for any x∈M([2]). It's easy to see that ifφ=1, then F is a Riemannian metric. Ifφ=1+s,F=α+β, which is called Randers metric which is firstly studied in 1941 by G.Randers who is a physician and be applied in studying the electronic microscopeand navigation problems, etc. And if F=α²/(α-β)(φ=1/(1-s)), which is called Matsumoto metric. Matsumoto found that this metric is the slope-of-a-mountainmetric [30]. We also obtain some results on Matusmoto metric below.One of important problems in Finsler geometry is to study and characterizeFinsler metrics of constant flag curvature. Another problem is to study and char-acterize projectively flat Finsler metrics on an open domain in Rⁿ. In Riemanniangeometry, these two problems are essentially same. The Beltrami theorem tellsus that a Riemannian metric is locally projectively flat if and only if it is ofconstant sectional curvature. However, there are locally projectively flat Finslermetrics which are not of constant flag curvature; and there are Finsler metricsof constant flag curvature which are not locally projectively flat. In [7], Z. Shenhave given the Taylor extensions at the origin 0∈Rⁿ for x-analytic projectivelyflat metrics F=F(x, y) of constant flag curvature K. In particular, for K=-1/4,ψ=|y| andψ=1/2|y|, we can get the Funk metric (?) on the unit ball Bⁿ (?) Rⁿ: (?)=((1-|x|²)|y|²+²)(1/2)/(1-|x|²)+/(1-|x|²),where y∈T_xBⁿ≈Rⁿ. For K=0,ψ=|y| andψ=|y|, we can get Berwald'smetric B=(((1-|x|²)|y|²+²)(1/2)+)²/(1-|x|²)²((1-|x|²)|y|²+²)(1/2),where y∈T_xBⁿ≈Rⁿ. The Funk metric and Berwald's metric are related andthey can be expressed in the form (?)=(?)+(?), B-((?)+(?))²/(?),where (?):=((1-|x|²)|y|²+²)(1/2)/1-|x|², (?):/1-|x|², (?):=λ(?), (?):=λ(?),λ:=1/1-|x|².In [8], [9] and [15], Z. Shen and X. Mo and etc. have classified projectively flatmetrics in the form F=α+βor F=(α+β)²/αwith constant flag curvature.The following metrics are trivial when they are projectively flat and with constantflag curvature. In [14], Y. Shen and L. Zhao study Finsler metrics in the form F=α+εβ+2kβ²/α-k²Î²⁴/(3α³); In [43], Y. Yu studies the arctangent metricin the form F=α+εβ+arctan (β/α). And we study in Corollary 4.4 the metricF=α+εβ+3/2βarctan(β/α)+αβ²/2(α²+β²). Therefore it is a natural problem toclassify (α,β)-metrics of constant flag curvature.By using the special coordinate system, we obtain the following.Theorem 0.1.1. Let F=αφ(s), s=β/α, be an (α,β)-metric on an open subsetu in the n-dimensional Euclidean space Rⁿ (n≥3), whereα=(a_jiyⁱy^j)(1/2) andβ=b_iyⁱ≠0. Let b=||β_x||_αSuppose that db≠0 everywhere or b=constant onu. Then F is projectively flat with constant flag curvature K if and only if oneof the following holds(â…°) a is projectively flat andβis parallel with respect toα;(â…±) F=(α²+kβ²)(1/2)+εβis projectively flat with constant flag curvature K＜0,where k andε≠0 are constants;(â…²) F=(α²+kβ²)(1/2)+εβ)²/(α²+kβ²)(1/2) is projectively flat with K=0, wherek andε≠0 are constants.It is a trivial fact that ifαis locally projectively flat andβis parallel, thenF=αφ(β/α) is a projectively flat Berwald metric. Further, if the flag curvatureK=constant, then it is either Riemannian (K≠0) or locally Minkowskian(K=0). See [16].The Finsler metric in Theorem 0.1.1 (â…±) is of Randers type, i.e., F=(?)+(?),where (?):=(α²+kβ²)(1/2) and (?):=εβ. In [8], it is proved that a Finsler metric inthe form F=(?)+(?) is projectively flat with constant flag curvature if and onlyif it is locally Minkowskian or it is locally isometric to a generalized Funk metricF=c((?)+(?)) on the unit ball Bⁿ (?) Rⁿ, where c＞0 is a constant, and (?):=(1-|x|²)|y|²+²(1/2)/1-|x|² (1) (?):=±{/1-|x|²+/1+},(2) whereα∈Rⁿ is a constant vector.The Finsler metric in Theorem 0.1.1 (â…²) is in the form F=((?)+(?))²/(?),where (?):=(α²+kβ²)(1/2) and (?):=εβ. In [9] and [15], it is proved that a non-Minkowskian metric F=((?)+(?))²/(?) is projectively flat with K=0 if and onlyif it is, after scaling on x, locally isometric to a metric F=c((?)+(?))²/(?) on theunit ball Bⁿ (?) Rⁿ, where c=constant, (?)=λ(?) and (?)=λ(?), where (?) and (?) aregiven in (1) and (2), andλ:= (1+)²/1-|x|².Theorem 0.1.1 tells us that there is no other types of (α,β)-metrics whichare locally projectively flat with constant flag curvature. However, the followingproblem is still open:Is there any metric F=(β+β)²/αof constant flag curvature which is notlocally projectively flat?0.2 Douglas (α,β)-metricsIn projective Finsler geometry, we study projectively equivalent Finsler met-rics on a manifold, namely, geodesics are same up to a parametrization. J. Dou-glas introduces two projective quantities: the (projective) Douglas curvature andthe (projective) Weyl curvature ([26]). The Douglas curvature always vanishesfor Riemannian metrics and the Weyl curvature is an extension of the Weylcurvature in Riemannian geometry. In dimensions greater than two, these twocurvature both vanish if and only if the metric is locally projectively flat.In local coordinates, the geodesics of a Finsler metric F=F(x, y) are char-acterized by d²xⁱ/dt²+2Gⁱ(x,dx/dt)=0, where Gⁱ:1/4g^il{[F²]_x^ky^ly^k-[F²]_x^l}. (3)The local functions Gⁱ=Gⁱ(x, y) define a global vector field G=yⁱ (?)/(?)xⁱ-2Gⁱ (?)/(?)yⁱon TM\{0}. The Douglas curvature and the Weyl curvature of a Finsler met- ric actually depend only on G. They describe the projective properties of thegeodesics of the metric.Finsler metrics with vanishing Douglas curvature are called Douglas metrics.Douglas metrics are characterized by Gⁱ=1/2Γ_jkⁱ(x)y^jy^k+P(x, y)yⁱ, (4)whereΓ_jkⁱ(x) are local functions on M and P(x, y) is a local positively homoge-neous function of degree one. If a Finsler metric locally has the same geodesicsas a Riemannian metric, then it is a Douglas metric.Douglas metrics form an important class in Finsler geometry. All projec-tively flat Finsler metrics are Douglas metrics, because that they locally havethe same geodesics (straight lines) as an Euclidean metric. However there aremany Douglas metrics which are not projectively flat (the Weyl curvature doesnot vanish).Consider a Rander metric F=α+βwhereα=(a_ij(x)yⁱy^j)(1/2) is a Riemannianmetric andβ=b_i(x)yⁱ is a 1-form with b(x):=||β_x||_α＜1. It is known that F isa Douglas metric if and only ifβis closed [17].Below is an another example defined by a Riemannian metric and a 1-formon Rⁿ\{0}: Let F=(α+β)²/α, (5)whereα=|y| andβ=/|x|. Then F:=(α+β)²/αis a Douglas metricwhich is not projectively flat on Rⁿ\0. This motivates us to consider Douglas(α,β)-metrics.Based on [10], we study and characterize Douglas (α,β)-metrics in this pa-per. We obtain the followingTheorem 0.2.1. Let F =αφ(s), s=β/α, be an (α,β)-metric on an opensubset u in the n-dimensional Euclidean space Rⁿ (n≥3), whereφ(0)=1,α=a_ij(x)yⁱy^j andβ=b_i(x)yⁱ≠0. Let b:=||β_x||_α. Suppose that the followingconditions: (a)βis not parallel with respect toα, (b) F is not of Randers type,and (c) db≠0 everywhere or b=constant on u. Then F is a Douglas metric on u if and only if the functionφ=φ(s) satisfies the following ODE: {1+(k₁+K₂s²)s²+k₃s²}φ″(s)=(k₁+k₂s²){φ(s)-sφ′(s)} (6)and the covariant derivative▽β=b_ijyⁱdx^j ofβwith respect toαsatisfies thefollowing equation: b_i|j=2Ï„{(1+k₁b²)a_ij+(k₂b²+k₃)b_ib_j}, (7)whereÏ„=Ï„(x) is a scalar function on u and k₁, k₂ and k₃ are constants with(k₂, k₃)≠(0, 0).Theorem 0.2.1 holds good in dimension n≥3. When n=2, the classifica-tion is still unknown.Equation (7) implies thatβis closed. Note that the Finsler metric in (5)can be expressed as F=αφ(s), whereφ=(1+s)² and s=β/α. It is easyto see thatφsatisfies (6) andβsatisfies (7) with K₁=2, k₂=0, K₃=-3 andÏ„=1/(6|x|).0.3 Weak Landsberg (α,β)-metricsThe study of Landsberg metrics in Finsler geometry has a long history. Inlate 20′s last century, L. Berwald studied a class of Finsler metrics F=F(x, y)on a manifold M, whose geodesics are determined by some second order ODEssimilar to the Riemannian case. More precisely, the geodesics in local coordinatessatisfy d²xⁱ/dt²+2Gⁱ(x,dx/dt)=0,where Gⁱ(x,y)=1/2Γ_jkⁱ(x)yⁱy^k are quadratic in y=yⁱ (?)/(?)xⁱ|x∈T_xM. Finslermetrics with this property are called Berwald metrics. It can be shown thatBerwald manifolds are modeled on a single norm space, i.e., all the tangentspaces T_xM with the induced norm F_x are linearly isometric to each other.There is a weaker notion of metrics defined by a non-Riemannian quantity,L=L_ijkdxⁱ (?) dx^j (?) dx^k, on the slit tangent bundle TM\{0}, whereL_iyk:=-1/2FF_y^m[G^m]_yⁱy^jy^k. Finsler metrics with L_ijk=0 are called Landsberg metrics. Clearly, any Berwaldmetric is a Landsberg metric. It is known that on a Landsberg manifold, thetangent spaces T_xM with the induced Riemannian metric (?)_x:=g_ij(x, y)dyⁱ (?)dy^j, where g_ij(x,y):=1/2[F²]_yⁱy^j(x,y), are all isometric. It might be possiblethat two tangent Riemannian spaces (T_x M, (?)_x) and (T_x′ M, (?)_x′) are isometric, but(T_xM, F_x) and (T_x′M, F_x′) are not linearly isometric. Thus a natural questionarises:Is there a Landsberg metric which is not a Berwald metric?This question has a satisfied answer now. First, G. Asanov proved that hismetrics arising from Finslerian General Relativity are actually Landsberg metricsbut not Berwald metrics ([23], [24]). Shortly after Asanov's preprints appeared,Z. Shen made a classification of Landsberg metrics defined by a Riemannianmetric and a 1-form. He obtained a two-parameter family of Landsberg metricsincluding Asanov's examples ([11]).On a Landsberg manifold, the volume function Vol(x) of the unit tangentsphere S_x M (?) (T_xM,(?)_x) is a constant. The constancy of Vol(x) is requiredto establish a Gauss-Bonnet theorem for Finsler manifolds [35]. The volumefunction Vol(x) is closely related to a weaker non-Riemannian quantity, J=J_kdx^k, where J_k:=g^ijL_ijk.Finsler metrics with J=0 are called weak Landsberg metrics. Clearly, in di-mension two, any weak Landsberg metric must be a Landsberg metric. It hasbeen shown that on a weak Landsberg manifold, the volume function Vol(x) is aconstant ([36]). Some rigidity problems also lead to weak Landsberg manifolds.For example, for a closed Finsler manifold of nonpositive flag curvature, if theS-curvature is a constant, then it is weak Landsbergian [13]. Apparently, weakLandsberg manifolds deserve further investigation. Another natural questionarises:Is there a weak Landsberg metric which is not a Landsberg metric? To find answers to the above questions, we also consider weak Landsberg(α,β)-metrics. In the previous definition of (α,β)-metrics, if we consider a 1-formβwith ||β_x||_α:=(a^ijb_ib_j)(1/2)≤b_o. then F=αφ(β/α) might be singular ata point x with b(x)=b_o. Such metrics are called almost regular (α,β)-metrics.We prove the followingTheorem 0.3.1. Let F=αφ(β/α) be an almost regular (α,β)-metric on ann-dimensional manifold M (n≥3), whereα=(a_ijyⁱy^j)(1/2) andβ=b_iyⁱ. Supposeβis not parallel with respect toαandφ≠k₁ (1+k₂s²)(1/2) for any constants k₁ andk₂. Let b(x):=||β_x||_α≠0,Δ:=1+sQ+(b²-s²)Q′and Q:=φ′/(φ-sφ′).Then F is a weak Landsberg metric if and only if one of the following holds(â…°)βsatisfies r_ij=k(b²a_ij-b_ib_j), s_ij=0, (8)where k=k(x) is a number depending on x andφ=φ(s) satisfies (Q-sQ′){nâ–³+1+sQ}+(b²-s²)(1+sQ)Q″=λ/(b²-s²)(1/2)Δ^3/2 (9)whereλis a constant.(â…±)βsatisfies r_j+s_j=0, (10)andφ=φ(s) satisfies (Q-sQ′){nâ–³+1+sQ}+(b²-s²)(1+sQ)Q″=0. (11)In both cases, b=b_o.Note thatφ=1+s does not satisfy (9) for any numberλ. Thus a Randersmetric F=α+βis a weak Landsberg metric if and only ifβis parallel withrespect toα. This has been proved in [28] and [42].Note that (8) implies (10), and the length b(x):=||β_x||_αsatisfies thatdb(x)=0 at a point x∈M if and only if (10) holds at x. Thus b=constant onan open set u if and only ifβsatisfies (10) on u. It is shown in [11] that in dimension n≥3, F=αφ(β/α) is a Landsbergmetric if and only ifβsatisfies (8) and Q:=φ′/(φ- sφ′) is given by Q=q₀(1-(s/b)²)(1/2)+q₁s, (12)where q₀ and q₁ are constants, provided thatβis not parallel with respect toα.Letβsatisfy (8). If Q=Q(s) satisfies (9), then F=αφ(β/α) is a weakLandsberg metric by Theorem 0.3.1. However, ifλ≠nq₀b/(1+q₁b²)(1/2),where q₀=Q(0) and q₁=Q′(0), then Q=Q(s) can not be expressed by (12).Thus F=αφ(β/α) is not a Landsberg metric. We have the following example.Example 0.3.1. At a point x=(x¹, x², x³)∈R³ and in the direction y=(y¹,y²,y³)∈T_xR³, defineα=α(x, y) andβ=β(x,y) byα:((y¹)²+e^2x1((y²)²+(y³)²))(1/2)β:=y¹.Thenαandβsatisfy (8) with b=||β_x||_α≡1 and k=1. Letφ=φ(s) satisfy (9)with b=1. Then F=αφ(β/α) is an almost regular weak Landsberg metric. Ifλ≠nQ(0)/(1+Q′(0))(1/2),then F is not a Landsberg metric.Because in dimension two, any weak Landsberg metric must be a Landsbergmetric, we also discuss the two-dimensional case. We shall prove the followingTheorem 0.3.2. (n=2) Let F=αφ(β/α) be an almost regular (α,β)-metric ona surface M, whereφ=φ(s) is defined on (-b_o, b_o) such thatφ≠k₁(1+k₂s²)(1/2) forany constants k₁ and k₂. Let b(x):=||β_x||_α≠0. Then F is a Landsberg metricon M if and only if eitherβis parallel with respect toα, orβhas constant length, b=b_o. andφis given byφ(s)=exp[∫Q/(1+sQ) ds],(13)where Q:=-s/b²+(1-(s/b)²)(1/2){q₀+(1/b²+q₁)s/(1-(s/b)²)(1/2)+(q₀-λ/2(1/b²+q₁)(1/2))s},(14) where q₀, q₁ andλare constants. In fact, F is a Berwald metric.One can verify that the function Q=Q(s) in (14) satisfies (9) when n=2.In 1973, Hashiguchi-H(?)j(?)-Matsumoto wrote a paper on two-dimensionalLandsberg (α,β)-metrics [37], trying to find two-dimensional Landsberg met-tics. Later on, they wrote another paper with the same title [38], pointing out anerror in [37]. The main result in [38] is that if a two-dimensional Randers metricor Kropina metrics is a Landsberg metric, then it must be Berwaldian. In The-orem 0.3.2, we give a complete characterization of two-dimensional Landsberg(α,β)-metrics.0.4 Two special classes of projectively flat metricsIt is a fundamental problem in Finsler geometry to study projectively flatmetrics on an open subset in Rⁿ. By projectively flat Finsler metrics on U wemean their geodesics are straight lines. This is the Hilbert's 4th problem inregular case [40]. In 1903, G. Hamel[41] found a system of partial differentialequations F_xⁱy^jyⁱ=F_x^j,which characterize projectively fiat metrics F=F(x, y) on an open subset U∈Rⁿ.As mentioned above Matsumoto metric is an important metric in Finslergeometry. It is the Matsumoto's slope-of-a-mountain metric [30]. In the Mat-sumoto metric, the 1-formβ=b_iyⁱ was originally to be induced by earth gravity.Hence we could regard b_i as the the infinitesimals. Then it induces the approx-imate Matsumoto metric if we neglect all powers≥k of b_i for some positiveinteger k. For Matsumoto metric, we have the following.Theorem 0.4.1. The Matsumoto metric F=α²/(α-β) is locally projectively flat ifand only if(â…°)βis parallel with respect toα,(â…±)αis locally projectively flat, i.e. of constant curvature.And in the approximation case, we have similar result.Theorem 0.4.2. The (k-1)-th (k≥3) approximate Matsumoto metric F_k=α{sum from n=0 to k sⁿ} where s=β/αis locally projectively flat if and only if(â…°)βis parallel with respect toα,(â…±)αis locally projectively flat, i.e. of constant curvature.From the above theorems we can see that projectively flat Matsumoto metricand its approximation are both trivial, i.e. the 1-formβis parallel with respectto the Riemannian metricαandαis projectively flat. Thus, the trivial metricmust be a Berwaldian metric. By Beltrami Theorem we know that projectivelyflat Riemannian metric must be constant curvature metric, which have beenclassified. Therefore it is not difficult to construct these metrics. Though thereare many trivial metrics [43], there are many nontrivial ones [10].For the existence of these nontrivial metrics, the Finsler geometry becomesmore colorful. We consider a special (α,β)-metric F=εβ+α+3/2βarctan(β/α)+αβ²/2(α²+β²). And we find the sufficient and necessary conditions for it to be projec-tively flat.Theorem 0.4.3. F=α(εs+1+3s/2 arctan(s)+s²/2(1+s²)), where s=β/αis locallyprojectively flat if and only if b_i;j=1/2Ï„((1+4b²)a_ij-3b_ib_j) and G_αⁱ=θyⁱ-τα²bⁱwhereÏ„=Ï„(x) andθ=a_i(x)yⁱ. In this case, Gⁱ={θ+Ï„xα}yⁱ,where x=ε(1+s²)²+3/2 arctan(s)(1+s²)²+3/2s(1+s²)+s/2(2εs(1+s²)+2(1+s²)+3s arctan(s)(1+s²)+s²)-s,s=β/α.Thus the metrics in above theorem are not trivial and it's not easy to con-struct the examples. Based on the construction of examples in [6], we have thefollowing example.Example 0.4.1. Defineα:=e^ρ(h)α_μ,β:=C_2e^5/4ρ(h)dh,whereα_μ=(|y|²+μ(|x|²|y|²-＜x,y＞²))(1/2)/1+μ|x|², h:=1/(1+μ|x|²)(1/2){C₁+＜a,x＞+η|x|²/1+(1+η|x|²)(1/2)},ρ=ρ(t) be given byρ(t)=ln[-2(C₂)²(C₃+μt-1/2μt²)]^-2,ηand C_i are constants (C₂＞0) and a∈Rⁿ is a constant vector.By Theorem 0.4.3, we can completely determine the local structure of aprojectively flat Finsler metric F in this form which is of constant flag curvature.Corollary 0.4.1. Let F=α(εs+1+3s/2 arctan(s)+s²/2(1+s²)), where s=β/α.Suppose that F is a locally projectively flat metric with constant flag curvatureλ. Thenλ=0,αis a flat metric andβis parallel with respect toα. Thus F islocally Minkowskian.