Projectively flat Finsler metric and projective Randers metrics 

Author  ZuoYaoYong 
Tutor  ShenYiBing;BaiZhengGuo 
School  Zhejiang University 
Course  Basic mathematics 
Keywords  Finsler Metric Projectively Flat Exponential Finsler Metric Arctangent Metric Randers Metric Projectively Related 
CLC  O189.1 
Type  PhD thesis 
Year  2007 
Downloads  68 
Quotes  0 
This paper consists of three chapters.In the first chapter, we define a new Finsler metric by F=αexp（β/α）+εβ,whereα=(a_{ij}y^{i}y^{j})^{1/2} is a Riemannian metric andβ=b_{i}y^{i} is a 1form,εis aconstant, which is called an exponential Finsler metric. We give the sufficient andnecessary condition for an exponential Finsler metric F to be locally projectivelyflat and obtain the sufficient and necessary condition for an exponential Finslermetric F to be a Douglas metric.In the second chapter, we define a new Finsler metric also by F=α+εβ+βarctanβ/αwhereα=(a_{ij}y^{i}y^{j})^{1/2} is a Riemannian metric andβ= b_{i}y^{i} is a 1form,εis a constant, which is called an arctangent Finsler metric. We give the sufficientand necessary condition for an arctangent Finsler metric to be locally projectivelyflat and obtain the nontrivial special solution. We completely determine thelocally structure of projectively flat arctangent Finsler metrics with constant flagcurvature.In the third chapter, we study projectively related Randers metrics andobtain the sufficient and necessary condition for two Randers metric to be projectively related. We also consider projectively related Randers metrics withsome special curvature properties.As S.S.Chen, the international famous geometrist, said that Finsler metricsare just Riemannian metrics without quadratic restriction[1], which was firstlyintroduced by B. Riemann in 1854. Finsler geometry is to study the geometricproperties of a manifold with Finsler metric[2]. Recent studies on Finsler geometry have taken on a new look [3][4][5] and Finsler geometry can be also appliedto biology and physics [6][7][8][9][10][11][12][13], Finsler geometry has became aimportant branch of Riemannian geometry. One century ago, great German mathematician Hilbert announced his famous 23 problems. The Hilbert’s Fourth Problem is to characterize the distancefunctions on an open subset in R^{n} such that straight lines are shortest paths.Distance functions induced by a Finsler metrics are regarded as smooth ones.Thus the Hilbert’s Fourth Problem in the smooth case is to characterize Finslermetrics on an open subset in R^{n} whose geodesics are straight lines. Finsler metrics on an open domain in R^{n} with this property are said to be projectively flatmetric. One of the fundamental problems in Finsler geometry is to study thecharacteristic of projectively flat metrics on an open domain U（?）R^{n}. Projectively flat finsler metrics have been studied by many mathematicians [15][16][17].In 1903, G. Hamel [18] found a system of partial differential equations F_{xiyj}y^{i}=F_{xj}, （1）which characterize projectively flat metrics F=F（x, y） on an open subset U∈R^{n}. The wellknown Beltrami’s theorem states that a Riemannian metric islocally projectively flat if and only if it is of constant sectional curvature. Thusthis problem has been solved in Riemannian geometry. The flag curvature inFinsler geometry is a natural extension of the sectional curvature in Riemanniangeometry. In general, it also depends on the direction （flag pole） in the section（flag）. However, every locally projectively flat Finsler metric F on a manifoldM is of scalar flag curvature, i.e., the flag curvature K=K（x, y） is a scalarfunction on TM\ {0}. Moreover, there are a lot of locally projectively flat Finslermetrics which are not of constant flag curvature. Thus, the Beltrami’s theoremis no longer true for Finsler metrics. Thus One of the important problems inFinsler geometry is to study and characterize projectively flat Finsler metricswith scalar（or constant） flag curvature.The wellknown Funk metric （?）=（?）（x, y） on a strongly convex domain inR^{n} is projectively flat with constant flag curvature K=1/4. When the domainthe unit ball B^{n}∈R^{n}, the Funk metric is given by （?）=（（1x^{2}）y^{2}）+<x,y>^{2})^{1/2}/（1x^{2}）+<x,y>/（1x^{2}） which is in the form （?）=（?）+（?）,where （?）=（（1x^{2}）y^{2}）+<x,y>^{2})^{1/2}/（1x^{2}） is a Riemannian metric and （?）=<x,y>/（1x^{2}） is a 1form.We call it a Randers metric if a Finsler metric F can be expressed in the formF=α+β, whereαis a Riemannnian metric andβis a 1form. It is known thata Randers metric F=α+βis projectively flat if and only ifαis projectively flatandβis closed. Z. Shen [16] has classified all projectively flat Randers metricswith constant flag curvature.L. Berwald [19] constructed a projectively flat metric with zero flag curvatureon the unit ball B^{n}, which is given by B=（（1x^{2}）y^{2}+<x,y>^{2}）^{1/2}+<x,y>^{2}/（1x^{2}）^{2}（（1x^{2}）y^{2}+<x,y>^{2}）^{1/2},where y∈T_{x}B^{n}≡R^{n}. Berwald’s metric B can be expressed in B=（λ（?）+λ（?））^{2}/λ（?）,whereλ=1/（1x^{2}）, （?） is Riemann metric, （?） is a 1form.Z. Shen and G. Civil Yidirim [20] have studied and characterized locallyprojectively flat metric in the form: F=(α±k^{1/2}β)^{2}/α.They have also completely determined those with constant flag curvature. Suchmetrics except Minkowskian have been constructed by X. Mo and C. Yang [21].Besides, X. Mo, Z. Shen and C. Yang [22] have also studied locally projectivelyflat metrics in the form F=α+εβ+kβ^{2}/α. Recently, Y. B. Shen and L. Zhao [24] have studied and characterized locallyprojectively flat metric in the form: F=α+εβ+2kβ^{2}/αk^{2}β^{4}/3α^{3};and obtain the nontrivial special solution.The above metrics are all socalled （α,β）metrics, which can be expressedas F=αφ（s）, s=β/α,whereα= (a_{ij}y^{i}y^{j})^{1/2} is a Riemannian metric,β=b_{i}y^{i} is a 1form.φ=φ（s） is aC^{∞}positive function on an open interval （b_{o}, b_{o}） satisfyingφ（s）sφ′（s）+（b^{2}s^{2}）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,where β_{x}_{α}is the norm ofβwith respect toαin x[2]. It is easy to see that theclass of （α,β）metrics contain Riemannian metric （φ=1） and Randers metric（φ=1+s）. It is a very important class of metrics in Finsler geometry. Ifthe Finsler metric F=αφ（β/α） is projectively flat if and only ifαis projectivelyflat andβis parallel with respect toα, then such projectively flat （α,β）metricsare called to be trivial. For example, Matsumoto metric[23]. If a （α,β）metricis projectively flat, the 1formβof projectively flat metric F=αφ（β/α） is notnecessarily parallel. The simplest one is the Randers metric F=α+βdefinedbyφ=1+s[25].In [30], Z. Shen studied a class of special （α,β）metrics f=αφ（β/α）, whereφ=φ（s） satisfiesφ（s）sφ′（s）=（p+qs+rs^{2}）φ″（s）,（2）where p, q and r are constants. He finds a sufficient condition for F=αφ（β/α）to be projectively flat in a local coordinate system （x^{i}）, that is, the covariantderivatives b_{ij} ofβwith respect toαand the spray coefficients G_{α}^{i} ofαsatisfy b_{ij}=2τ{（p+b^{2}）a_{ij}+（r1）b_{i}b_{j}}, G_{α}^{i}=θy^{i}τα^{2}b^{i},where b=(a_{ij}（x）b^{i}（x）b^{j}（x）)^{1/2},τ=τ（x） is a scalar function andθ=θ_{i}（x）y^{i} isa 1form. Putting p=1, q=1, r=0 in （2）, we getφ=εs+exp（s）. Thecorresponding metric is expressed in the following form: F=αexp（β/α）+εβ,which is called the exponential Finsler metric. We obtain the following theorem:Theorem 0.1 Let F=αexp（β/α）+εβbe an exponential Finsler metricon a manifold M. Then F is locally projectively flat if and only if the followingconditions hold:（a）βis parallel with respect toα;（b）αis locally projectively flat. That is,αis of constant curvature.By Theorem 0.1, we see that the projectively flat exponential Finsler metricis trivial.A theorem due to Douglas states that a Finsler metric F is projectively flatif and only if two special curvature tensors are zero. The first is the Douglastensor. The second is the projective Weyl tensor for n≥3, and the BerwaldWeyl tensor for n=2. It is known that the projective Weyl tensor vanishes ifand only if F has the scalar flag curvature. If the Douglas tensor of F vanishesidentically, F is called a Douglas metric. S. Bácsóand M. Matsumoto [25] provedthat a Randers metric F=α+βis a Douglas metric if and only ifβis a closed1form. M. Matsumoto [29] obtained that for n=dimM≥3, F=（α^{2}+β^{2}）/αis aDouglas metric if and only if b_{ij}=τ(（1+2b^{2}）a_{ij}3b_{i}b_{j})whereτ=τ（x） is a scalar function.In the section, we obtain the following: Theorem 0.2 Let F=αexp（β/α）+εβbe an exponential Finsler metricon a manifold M. Then the Douglas tensor of F vanishes if and only ifβisparallel with respect toα.In the second chapter, putting p=r=1/2 and q=0 in （2）, we getφ=k+εs+s arctan（s）, where k andεis two arbitrary constants. Taking k=1,we getφ=1+εs+s arctan（s）. The corresponding metric is expressed in thefollowing form: F=α+εβ+βarctan（β/α）,which is called an arctangent metric. We prove the following:Theorem 0.3 Let F=α+εβ+βarctanβ/αbe an arctangent Finslermetric on a manifold M. Then F is locally projectively flat if and only if thefollowing conditions hold: b_{ij}=τ[（1+2b^{2}）a_{ij}b_{i}b_{j}], （3） G_{α}^{i}=θy^{i}τα^{2}b^{i},（4）whereτ=τ（x） is a scalar function andθ=θ_{i}（x）y^{i} is a 1form. In this case, G^{i}=（θ+τxα）)y^{i},where x=（ε+arctans）/2[1+（ε+arctans）s], s=β/α.And then, we study projectively flat arctangent Finsler metric with constantflag curvature. We havePropostion 0.4 Suppose that F=α+εβ+βarctanβ/αis a projectivelyflat arctangent Finsler metric with constant flag curvature K=λ. Thenλ=0. Propostion 0.5 Let F=α+εβ+βarctanβ/αis projectively flat arctangentFinsler metric with zero flag curvature. Thenαis a flat metric andβis parallelwith respect toα. In this case, F is locally Minkowshian.Finally, we give special solutions of （3） and （4）.Theorem 0.6 Let F=α+εβ+βarctanβ/αbe an arctangent Finslermetric, whereεis a constant. Assume thatρ:=ρ（h） and h:=h（x） are asfollows:ρ=ln（4C_{2}^{2}（μh^{2}2θhC_{3}））, h=1/（1+μx^{2}）^{1/2}{C_{1}+<a,x>+θx^{2}/(1+（1+μx^{2}）^{1/2})},where C_{1}, C_{2}＞0, C_{3},μandθare constants, andα∈R^{n} is a constant vector.Defineα:=e^{ρ}（?）,β:=C_{2}e^{3/2ρ}h_{0},where （?）=（y^{2}+μ（x^{2}y^{2}<x,y>^{2}））^{1/2}/（1+μx^{2}）.Then F=α+εβ+βarctanβ/αis a nontrivial projectively flat Finsler metric.In the third chapter, we study projectively related Randers metrics. Tworegular metrics on a manifold are said to be pointwise projectively related if theyhave the same geodesics as point sets. Two regular metric space are said tobe projectively related if there is a diffeomorphism between them such that thepullback metric is pointwise projectively related to another one. In the Riemannian geometry, two spaces of constant sectional curvature are always projectivelyrelated according to the Beltrami theorem. In Finsler geometry, there are manyFinsler metrics on a strongly convex subsetΩ（?）R^{n} which are pointwise projectively related to the standard Euclidean metric. Z.Shen has been studied theprojectively related EinsteinFinsler metrics in [42]. As well known, Randers metrics are interesting and important in Finslergeometry, which represent a medium such that Riemannian geometry interfaceswith Finsler geometry proper. In the third chapter, we study the following problem: Given a Randers metric on a manifold M, discuss all Randers metrics whichare pointwise projectively related to the given one. The main results in this section are as follows.Theorem 0.7 A Randers metric F=α+βis pointwise projectivelyrelated to another Randers metric（?）=（?）+（?） if and only if they have the sameDouglas tensors and the corresponding Riemannian metricsαand （?） are projectively related.Projectively fiat Randers metrics with constant flag curvature have beencompletely classified in[16]. So, it is natural to study projectively related Randers metrics with constant flag curvature. We haveTheorem 0.8 Let F=α+βbe of constant flag curvature K andα≠λ（x）（?）. Then F=（?）+（?） is pointwise projectively related to F if andonly if F and （?） are projectively fiat and K≤0.By using the YasudaShimada theorem[38], we haveTheorem 0.9 Let F=α+βand （?）=（?）+（?） have constant flag curvaturesK and （?）,α=λ（?）and s_{j}=（?）_{j}=0. Then F is pointwise projectively related to（?） if and only if one of the following cases holds:（1）K＞0, F=λ（?）;（2） F and （?） are locally Minkowskian, i.e., K=（?）=0;（3） F and （?） are locally isometric to Funk metric. In this case, K=（?）= 1/4.We also study the EinsteinRanders metrics and obtain the following Theorem 0.10 Let F=α+βbe an Einstein metric andα≠λ（x）（?）. If（?）=（?）+（?）is pointwise projectively related to F=α+β, thenαand （?）are Einstein metrics,αhas nonpositive scalar curvature and F have nonpositive Riccicurvature.Theorem 0.11 Let F=α+βand （?）=（?）+（?）be two Einstein metricswithα=γ（?）. Assume thatβis not divisible byβand s_{j}=（?）_{j}=0. If F pointwiseprojectively related to F, then they have nonpositive Ricci curvatures.In[39], Shen introduces the notion of Scurvature, It is very importantquantity. The Scurvature vanishes for Berwald metrics including Riemannianmetrics. The Scurvature has been discussed in the recent study on Finsler metrics of scalar curvature. In this paper, we study projectively related Randersmetrics with special Scurvatures. We obtain the followingTheorem 0.12 Let F=α+βand（?）=（?）+（?） have scalar flag curvaturesand S=σ（x）（?）,σ（x）≠0,α≠λ（?）. Then （?） is pointwise projectively related toF if and only if their flag curvatures are zero andβand （?） are parallel.