Dissertation > Mathematical sciences and chemical > Mathematics > Geometry, topology > Topology ( the situation in geometry ) > General topology

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
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This paper consists of three chapters.In the first chapter, we define a new Finsler metric by F=αexp(β/α)+εβ,whereα=(aijyiyj)1/2 is a Riemannian metric andβ=biyi is a 1-form,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=(aijyiyj)1/2 is a Riemannian metric andβ= biyi is a 1-form,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 non-trivial 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 pro-jectively 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 geom-etry 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 fa-mous 23 problems. The Hilbert’s Fourth Problem is to characterize the distancefunctions on an open subset in Rn 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 Rn whose geodesics are straight lines. Finsler met-rics on an open domain in Rn 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(?)Rn. Projec-tively flat finsler metrics have been studied by many mathematicians [15][16][17].In 1903, G. Hamel [18] found a system of partial differential equations Fxiyjyi=Fxj, (1)which characterize projectively flat metrics F=F(x, y) on an open subset U∈Rn. The well-known 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 well-known Funk metric (?)=(?)(x, y) on a strongly convex domain inRn is projectively flat with constant flag curvature K=-1/4. When the domainthe unit ball Bn∈Rn, the Funk metric is given by (?)=((1-|x|2)|y|2)+<x,y>2)1/2/(1-|x|2)+<x,y>/(1-|x|2) which is in the form (?)=(?)+(?),where (?)=((1-|x|2)|y|2)+<x,y>2)1/2/(1-|x|2) is a Riemannian metric and (?)=<x,y>/(1-|x|2) is a 1-form.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 1-form. 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 Bn, which is given by B=((1-x2)|y|2+<x,y>21/2+<x,y>2/(1-|x|22((1-|x|2)|y|2+<x,y>21/2,where y∈TxBn≡Rn. Berwald’s metric B can be expressed in B=(λ(?)+λ(?))2/λ(?),whereλ=1/(1-|x|2), (?) is Riemann metric, (?) is a 1-form.Z. Shen and G. Civil Yidirim [20] have studied and characterized locallyprojectively flat metric in the form: F=(α±k1/2β)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+kβ2/α. Recently, Y. B. Shen and L. Zhao [24] have studied and characterized locallyprojectively flat metric in the form: F=α+εβ+2kβ2/α-k2β4/3α3;and obtain the non-trivial special solution.The above metrics are all so-called (α,β)-metrics, which can be expressedas F=αφ(s), s=β/α,whereα= (aijyiyj)1/2 is a Riemannian metric,β=biyi is a 1-form.φ=φ(s) is aCpositive function on an open interval (-bo, bo) satisfyingφ(s)-sφ′(s)+(b2-s2)φ″(s)>0, (?)|s|≤b<bo.It is known that F is a Finsler metric if and only if ||βx||α<bo for any x∈M,where ||βx||αis 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 1-formβof 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+rs2)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 (xi), that is, the covariantderivatives bi|j ofβwith respect toαand the spray coefficients Gαi ofαsatisfy bi|j=2τ{(p+b2)aij+(r-1)bibj}, Gαi=θyi-τα2bi,where b=(aij(x)bi(x)bj(x))1/2,τ=τ(x) is a scalar function andθ=θi(x)yi isa 1-form. 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 Berwald-Weyl 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 closed1-form. M. Matsumoto [29] obtained that for n=dimM≥3, F=(α22)/αis aDouglas metric if and only if bi|j=τ((1+2b2)aij-3bibj)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: bi|j=τ[(1+2b2)aij-bibj], (3) Gαi=θyi-τα2bi,(4)whereτ=τ(x) is a scalar function andθ=θi(x)yi is a 1-form. In this case, Gi=(θ+τxα))yi,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(4C22(μh2-2θh-C3)), h=1/(1+μ|x|21/2{C1+<a,x>+θ|x|2/(1+(1+μ|x|21/2)},where C1, C2>0, C3,μandθare constants, andα∈Rn is a constant vector.Defineα:=eρ(?),β:=C2e3/2ρh0,where (?)=(|y|2+μ(|x|2|y|2-<x,y>2))1/2/(1+μ|x|2).Then F=α+εβ+βarctanβ/αis a non-trivial 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 thepull-back metric is pointwise projectively related to another one. In the Rieman-nian 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(?)Rn which are pointwise pro-jectively related to the standard Euclidean metric. Z.Shen has been studied theprojectively related Einstein-Finsler 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 prob-lem: 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 sec-tion 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 projec-tively related.Projectively fiat Randers metrics with constant flag curvature have beencompletely classified in[16]. So, it is natural to study projectively related Ran-ders 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 Yasuda-Shimada theorem[38], we haveTheorem 0.9 Let F=α+βand (?)=(?)+(?) have constant flag curvaturesK and (?),α=λ(?)and sj=(?)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 Einstein-Randers 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 Ein-stein 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 sj=(?)j=0. If F pointwiseprojectively related to F, then they have non-positive Ricci curvatures.In[39], Shen introduces the notion of S-curvature, It is very importantquantity. The S-curvature vanishes for Berwald metrics including Riemannianmetrics. The S-curvature has been discussed in the recent study on Finsler met-rics of scalar curvature. In this paper, we study projectively related Randersmetrics with special S-curvatures. 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.

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