On a Class of Two-Dimensional Douglas and Projectively Flat Finsler Metrics

Abstract

We study a class of two-dimensional Finsler metrics defined by a Riemannian metric α and a 1-form β. We characterize those metrics which are Douglasian or locally projectively flat by some equations. In particular, it shows that the known fact that β is always closed for those metrics in higher dimensions is no longer true in two-dimensional case. Further, we determine the local structures of two-dimensional (α, β)-metrics which are Douglasian, and some families of examples are given for projectively flat classes with β being not closed.

1. Introduction

Projective Finsler geometry studies equivalent Finsler metrics on the same manifold with the same geodesics as points [1]. Douglas curvature (D) is an important projective invariants in projective Finsler geometry. A Finsler metric is called Douglasian if 𝐃=0 and locally projectively flat if at every point, there are local coordinate systems in which geodesics are straight. It is known that a locally projectively flat Finsler metric can be characterized by 𝐃=0 and vanishing Weyl curvature. As we know, the locally projectively flat class of Riemannian metrics is very limited, nothing but the class of constant sectional curvature (Beltrami theorem). However, the class of locally projectively flat Finsler metrics is very rich. It is known that locally projectively flat Finsler metrics must be Douglasian, but Douglas metrics are not necessarily locally projectively flat. Therefore, it is a natural problem to study and classify Finsler metrics which are Douglasian or locally projectively flat. For this problem, we can only investigate some special classes of Finsler metrics.

In this paper, we shall consider a special class of Finsler metrics defined by a Riemannian metric 𝛼=𝑎_𝑖𝑗(𝑥)𝑦^𝑖𝑦^𝑗 and a 1-form 𝛽=𝑏_𝑖(𝑥)𝑦^𝑖 on a manifold 𝑀. Such metrics are called (𝛼,𝛽)-metrics. An (𝛼,𝛽)-metric can be expressed in the following form: (𝑠)𝛽𝐹=𝛼𝜙,𝑠=𝛼, (1) where 𝜙(𝑠)>0 is a 𝐶^∞ function on (−𝑏_𝑜,𝑏_𝑜). It is known that 𝐹 is a regular Finsler metric (defined on the whole 𝑇𝑀−{0} and positive definite) for any (𝛼,𝛽) with ‖𝛽‖_𝛼<𝑏_𝑜 if and only if 𝜙(𝑠)−𝑠𝜙^′(𝑠)+𝜌²−𝑠²𝜙^′′(𝑠)>0,|𝑠|≤𝜌<𝑏_𝑜, (2) where 𝑏_𝑜 is a constant. If 𝜙 does not satisfy (2), then 𝐹=𝛼𝜙(𝛽/𝛼) is singular.

Randers metrics are a special class of (𝛼,𝛽)-metrics. It is known that a Randers metric 𝐹=𝛼+𝛽 is a Douglas metric if and only if 𝛽 is closed [2], and it is locally projectively flat if and only if 𝛼 is locally projectively flat and 𝛽 is closed [2, 3]. Usually we call 𝐹=𝛼𝜙(𝑠) with 𝜙(𝑠)=𝜖𝑠+√1+𝑘𝑠², where 𝑘,𝜖 are constants, a Finsler metric of Randers type, which is essentially a Randers metric.

(𝛼,𝛽)-metrics are computable, and it has been shown that (𝛼,𝛽)-metrics have a lot of special geometric properties [4–13]. In [5, 10], the authors study and characterize (𝛼,𝛽)-metrics which are, respectively, Douglasian and locally projectively flat in dimension 𝑛≥3. However, the two-dimensional case remains open. In this paper, we will solve this problem in two-dimensional case and meanwhile give their local structures in part.

Theorem 1. Let 𝐹=𝛼𝜙(𝑠), 𝑠=𝛽/𝛼, be a regular (𝛼,𝛽)-metric on an open subset 𝑈⊂𝑅², where 𝜙(0)=1. Suppose that 𝛽 is not parallel with respect to 𝛼 and 𝐹 is not of Randers type. Let 𝐹 be a Douglas metric. Then one has one of the following two cases. (i)𝜙(𝑠) satisfies 1+𝑘₁+𝑘₃𝑠²+𝑘₂𝑠⁴𝜙^′′(𝑠)=𝑘₁+𝑘₂𝑠²𝜙(𝑠)−𝑠𝜙^′(𝑠), (3) where 𝑘₁,𝑘₂, and 𝑘₃ are constants satisfying 𝑘₂≠2𝑘₁+3𝑘₃3𝑘₁+2𝑘₃25,𝑘₂≠𝑘₁𝑘₃. (4) Further, 𝛽 must be closed.(ii)𝐹 can be written as 𝛽𝐹=𝛼±²𝛼,𝛼∶=𝛼²−𝑘𝛽²,𝛽∶=𝑐𝛽, (5) where 𝑘,𝑐 are constants with 𝑐≠0. In this case, 𝛽 is generally not closed.

In dimension 𝑛>2 in Theorem 1, it is proved in [5, 10] that the metric in Theorem 1 must be given by (3) with 𝑘₂≠𝑘₁𝑘₃ and 𝛽 must be closed. In Theorems 9 and 11, we give general characterizations for two-dimensional (𝛼,𝛽)-metrics (might be singular) which are Douglasian and locally projectively flat, respectively.

Next we consider the local structure of the Douglas metrics in Theorem 1. By using some deformations on 𝛼 and 𝛽, we can determine the local structure of two-dimensional regular Douglas (𝛼,𝛽)-metrics, which is shown in the following two theorems. For the local structure of the singular Douglas classes in Theorem 9(iii) and Theorem 9(iv), we will have a discussion in Section 6.

Theorem 2. Let 𝐹=𝛼±𝛽²/𝛼 be a two-dimensional regular Douglas (𝛼,𝛽)-metric with 𝛽≠0. Then 𝛼 and 𝛽 can be locally written as 𝛼²=1(1∓𝐵)³×𝐵𝑢²+𝑣²𝑦¹²+𝑦²²∓91±𝐵+𝐵²𝛽²,𝐵𝛽=(1±2𝐵)^3/2𝑢𝑦¹+𝑣𝑦²𝑢²+𝑣², (6)(7) where 𝑢=𝑢(𝑥),𝑣=𝑣(𝑥) are scalar functions such that 𝑓(𝑧)=𝑢+𝑖𝑣,𝑧=𝑥¹+𝑖𝑥², (8) is a complex analytic function and 𝐵=𝐵(𝑥) is a scalar function satisfying 0<𝐵<1 if 𝐹=𝛼+𝛽²/𝛼 and 0<𝐵<1/2 if 𝐹=𝛼−𝛽²/𝛼.

We see that in Theorem 2 the metric is determined by the triple parametric scalar functions (𝐵,𝑢,𝑣), where 𝑢 and 𝑣 are a pair of complex conjugate functions. We will prove Theorem 2 by using Corollary 10 and the result in [14] (also see [15]).

Theorem 3. Let 𝐹=𝛼𝜙(𝑠), 𝑠=𝛽/𝛼, be a two-dimensional regular Douglas (𝛼,𝛽)-metric with 𝛽≠0, where 𝜙(𝑠) satisfies (3) and (4) and 𝜙(0)=1. Then 𝛼 and 𝛽 can be locally written as √𝛼=𝐵𝑐𝑦¹²+𝑦²²𝑢²+𝑣²𝐵,𝛽=𝑢𝑦¹+𝑣𝑦²𝑐𝑢²+𝑣²,𝑐∶=𝑒^{∫𝐵0(1/2)((𝑘3+𝑘2𝑡)/(1+(𝑘1+𝑘3)𝑡+𝑘2𝑡2))𝑑𝑡}, (9) where 𝐵=𝐵(𝑥)>0,𝑢=𝑢(𝑥),𝑣=𝑣(𝑥) are some scalar functions which satisfy the following PDEs: 𝑢₁=𝑣₂,𝑢₂=−𝑣₁,𝑣₁+𝜎₁𝑣=𝑢𝜎₂, (10) where 𝑢_𝑖∶=𝑢_𝑥𝑖,𝑣_𝑖∶=𝑣_𝑥𝑖, and 𝜎_𝑖∶=𝜎_𝑥𝑖, and 𝜎 is defined by 𝑒^2𝜎𝐵∶=𝑐²𝑢²+𝑣². (11)

We will prove Theorem 3 by (32) and the result in [14] (also see [15]). The metric in Theorem 3 is determined by the triple parametric scalar functions (𝐵,𝑢,𝑣) which satisfy (10). It seems hard to obtain the complete solutions of the PDEs (10). However, we can give some special solutions of (10). For example, the following triple is a solution: 𝜎=𝑥¹,𝑢=𝑐₂sin𝑥²−𝑐₁cos𝑥²𝑒^−𝑥1,𝑣=𝑐₁sin𝑥²+𝑐₂cos𝑥²𝑒^−𝑥1, (12) where 𝑐₁,𝑐₂ are constants, and then 𝐵 is determined by (11).

Now we consider the local structure of the locally projectively flat metrics in Theorem 1. The local structure of 𝐹 determined by (3) with 𝑘₂≠𝑘₁𝑘₃ (for the dimension 𝑛≥2) has been solved in [16] (also see another way in [13]). However, it seems difficult to determine the local structure of 𝐹 determined by (5). By using (40) and (98) with 𝜏=0, we can construct the following example, which can be directly verified. We omit the details.

Example 4. Let 𝐹=𝛼±𝛽²/𝛼 be a two-dimensional (𝛼,𝛽)-metric. Suppose 𝛼 and 𝛽 take the form 𝛼=𝑒^𝜎(𝑥)𝑦¹²+𝑦²²,𝛽=𝑒^𝜎(𝑥)𝜉(𝑥)𝑦¹+𝜂(𝑥)𝑦², (13) where 𝜉=𝜉(𝑥),𝜂=𝜂(𝑥),𝜎=𝜎(𝑥) are some scalar functions. Define 𝜉(𝑥)1=±√2𝑥²+𝑐₂𝑐₃∓𝑥¹+𝑐₁²∓𝑥²+𝑐₂²,𝜂(𝑥)1=∓√2𝑥¹+𝑐₁𝑐₃∓𝑥¹+𝑐₁²∓𝑥²+𝑐₂²,𝜎(𝑥)=ln𝑐₃∓𝑥¹+𝑐₁²∓𝑥²+𝑐₂²+𝑐₄, (14) where 𝑐₁,𝑐₂, and 𝑐₃ are constants with 𝑐₃>0. Then 𝐹 is projectively flat with 𝛽 being not closed.

For the singular projectively flat classes in Theorem 11(iii) and Theorem 11(iv), we also construct some examples with 𝛽 being not closed (see Examples 15 and 16 below). As we have shown, it seems an obstacle to determine the local structure of the projectively flat classes when 𝛽 is not closed.

Open Problem. Determine the local structure of a two-dimensional (𝛼,𝛽)-metric 𝐹=𝛼±𝛽²/𝛼 which is locally projectively flat.

2. Preliminaries

Let 𝐹=𝐹(𝑥,𝑦) be a Finsler metric on an 𝑛-dimensional manifold 𝑀. The geodesic coefficients are defined by 𝐺^𝑖1∶=4𝑔^𝑖𝑙𝐹²_{𝑥𝑘𝑦𝑙}𝑦^𝑘−𝐹²_𝑥𝑙. (15)

A Finsler metric 𝐹=𝐹(𝑥,𝑦) is called a Douglas metric if the spray coefficients 𝐺^𝑖 are in the following form: 𝐺^𝑖=12Γ^𝑖_𝑗𝑘(𝑥)𝑦^𝑗𝑦^𝑘+𝑃𝑥,𝑦𝑦^𝑖, (16) where Γ^𝑖_𝑗𝑘(𝑥) are local functions on 𝑀 and 𝑃(𝑥,𝑦) is a local positively homogeneous function of degree one in 𝑦. It is easy to see that 𝐹 is a Douglas metric if and only if 𝐺^𝑖𝑦^𝑗−𝐺^𝑗𝑦^𝑖 is a homogeneous polynomial in (𝑦^𝑖) of degree three, which by (16) can be written as [2] 𝐺^𝑖𝑦^𝑗−𝐺^𝑗𝑦^𝑖=12Γ^𝑖_𝑘𝑙𝑦^𝑗−Γ^𝑗_𝑘𝑙𝑦^𝑖𝑦^𝑘𝑦^𝑙. (17)

According to G. Hamel’s result, a Finsler metric 𝐹 is projectively flat in 𝑈 if and only if 𝐹_{𝑥𝑚𝑦𝑙}𝑦^𝑚−𝐹_𝑥𝑙=0. (18) The above formula implies that 𝐺^𝑖=𝑃𝑦^𝑖 with 𝑃 given by 𝐹𝑃=^𝑥𝑚𝑦^𝑚.2𝐹 (19)

For a Riemannian metric 𝛼=𝑎_𝑖𝑗𝑦^𝑖𝑦^𝑗 and a 1-form 𝛽=𝑏_𝑖𝑦^𝑖 on a manifold 𝑀, let ∇𝛽=𝑏_𝑖∣𝑗𝑦^𝑖𝑑𝑥^𝑗 denote the covariant derivatives of 𝛽 with respect to 𝛼. Put 𝑟_𝑖𝑗1∶=2𝑏_𝑖∣𝑗+𝑏_𝑗∣𝑖,𝑠_𝑖𝑗1∶=2𝑏_𝑖∣𝑗−𝑏_𝑗∣𝑖,𝑟_𝑗∶=𝑏^𝑖𝑟_𝑖𝑗,𝑠_𝑗∶=𝑏^𝑖𝑠_𝑖𝑗,𝑠^𝑖∶=𝑎^𝑖𝑘𝑠_𝑘, (20) where 𝑏^𝑖∶=𝑎^𝑖𝑗𝑏_𝑗 and (𝑎^𝑖𝑗) is the inverse of (𝑎_𝑖𝑗).

Consider an (𝛼,𝛽)-metric 𝐹=𝛼𝜙(𝛽/𝛼). By (15), the spray coefficients 𝐺^𝑖 of 𝐹 are given by [3, 4, 8, 10, 11] 𝐺^𝑖=𝐺^𝑖_𝛼+𝛼𝑄𝑠^𝑖₀+𝛼⁻¹Θ−2𝛼𝑄𝑠₀+𝑟₀₀𝑦^𝑖+Ψ−2𝛼𝑄𝑠₀+𝑟₀₀𝑏^𝑖, (21) where 𝑠^𝑖_𝑗=𝑎^𝑖𝑘𝑠_𝑘𝑗,𝑠^𝑖₀=𝑠^𝑖_𝑘𝑦^𝑘,𝑠_𝑖=𝑏^𝑘𝑠_𝑘𝑖, and 𝑠₀=𝑠_𝑖𝑦^𝑖, and 𝜙𝑄∶=^′𝜙−𝑠𝜙^′,Θ∶=𝑄−𝑠𝑄^′,𝑄2ΔΨ∶=^′,𝑏2ΔΔ∶=1+𝑠𝑄+²−𝑠²𝑄^′. (22)

By (21) one can see that 𝐹=𝛼𝜙(𝛽/𝛼) is a Douglas metric if and only if 𝛼𝑄𝑠^𝑖₀𝑦^𝑗−𝑠^𝑗₀𝑦^𝑖+Ψ−2𝛼𝑄𝑠₀+𝑟₀₀𝑏^𝑖𝑦^𝑗−𝑏^𝑗𝑦^𝑖=12𝐺^𝑖_𝑘𝑙𝑦^𝑗−𝐺^𝑗_𝑘𝑙𝑦^𝑖𝑦^𝑘𝑦^𝑙, (23) where 𝐺^𝑖_𝑘𝑙∶=Γ^𝑖_𝑘𝑙−𝛾^𝑖_𝑘𝑙, Γ^𝑖_𝑘𝑙 are given in (16) and 𝛾^𝑖_𝑘𝑙∶=𝜕²𝐺^𝑖_𝛼/𝜕𝑦^𝑘𝜕𝑦^𝑙.

Further, 𝐹=𝛼𝜙(𝛽/𝛼) is projectively flat on 𝑈⊂𝑅^𝑛 if and only if 𝑎_𝑚𝑙𝛼²−𝑦_𝑚𝑦_𝑙𝐺^𝑚_𝛼+𝛼³𝑄𝑠_𝑙0+Ψ𝛼−2𝛼𝑄𝑠₀+𝑟₀₀𝛼𝑏_𝑙−𝑠𝑦_𝑙=0, (24) where 𝑦_𝑙=𝑎_𝑚𝑙𝑦^𝑚.

3. Equations in a Special Coordinate System

In order to prove Theorems 9 and 11, one has to simplify (23) and (24). The main technique is to fix a point and choose a special coordinate system (𝑠,𝑦^𝑎) as in [10, 11].

Fix an arbitrary point 𝑥∈𝑀, and take an orthonormal basis {𝑒_𝑖} at 𝑥 such that 𝛼=^𝑛_𝑖=1𝑦^𝑖2,𝛽=𝑏𝑦¹. (25) Then we change coordinates (𝑦^𝑖) to (𝑠,𝑦^𝑎) such that 𝑏𝛼=𝑏²−𝑠²𝛼,𝛽=𝑏𝑠𝑏²−𝑠²𝛼, (26) where 𝛼=∑^𝑛_𝑎=2(𝑦^𝑎)². Let 𝑟₁₀∶=𝑟_1𝑎𝑦^𝑎,𝑟₀₀∶=𝑟_𝑎𝑏𝑦^𝑎𝑦^𝑏,𝑠₀∶=𝑠_𝑎𝑦^𝑎. (27) We have 𝑠₀=𝑏𝑠₁₀,𝑠₁=𝑏𝑠₁₁=0. The following lemmas are trivial.

Lemma 5. In the special local coordinate system at 𝑥 as mentioned previously, if 𝑏=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, then 𝑟₁₁=0,𝑟_1𝑎+𝑠_1𝑎=0 at 𝑥.

Lemma 6 (see [11]). For 𝑛≥2, suppose 𝑝+𝑞𝛼=0, where 𝑝=𝑝(𝑦) and 𝑞=𝑞(𝑦) are homogeneous polynomials in 𝑦=(𝑦^𝑎); then 𝑝=0,𝑞=0.

The following two propositions are simple corollaries from [5, 10] (also see (23) and (24)).

Proposition 7 (𝑛=2). An (𝛼,𝛽)-metric 𝐹=𝛼𝜙(𝛽/𝛼) is a Douglas metric if and only if, at each point 𝑥, there is a suitable coordinate system such that at 𝑥 there exist numbers 𝐺^𝑖_𝑗𝑘  (𝑖,𝑗,𝑘=1,2) which are independent of 𝑠 such that 𝑠²2𝑏²−𝑠²𝐺¹₁₁−𝐺²₁₂−𝐺²₂₁+12𝐺¹₂₂𝑠=𝑏Ψ²𝑏²−𝑠²𝑟₁₁+𝑟₂₂,1𝑏²−𝑠²2Ψ𝑏²−𝑠²−1𝑏³𝑄𝑠₁₂−2𝑏Ψ𝑟₁₂𝑠=𝐺²₁₁2𝑏²−𝑠²𝑠³+12𝐺²₂₂−𝐺¹₁₂−𝐺¹₂₁𝑠. (28)(29)

Proposition 8 (𝑛=2). An (𝛼,𝛽)-metric 𝐹=𝛼𝜙(𝛽/𝛼) is projectively flat if and only if 𝑠²2𝑏²−𝑠²−𝐺¹₁₁+2𝐺²₁₂−12𝐺¹₂₂𝑠=𝑏Ψ²𝑏²−𝑠²𝑟₁₁+𝑟₂₂,1𝑏²−𝑠²2Ψ𝑏²−𝑠²−1𝑏³𝑄𝑠₁₂−2𝑏Ψ𝑟₁₂𝑠𝐺=−²¹¹2𝑏²−𝑠²𝑠³+12−𝐺²₂₂+2𝐺¹₁₂𝑠, (30)(31) where 𝐺^𝑖_𝑗𝑘∶=𝜕²𝐺^𝑖_𝛼/𝜕𝑦^𝑗𝜕𝑦^𝑘 are the connection coefficients of 𝛼.

Comparing (28) and (30), (29) and (31), it is easy to see that if 𝐺^𝑖_𝑗𝑘=𝐺^𝑖_𝑘𝑗, then 𝐺^𝑖_𝑗𝑘=−𝐺^𝑖_𝑗𝑘. So if we can solve 𝐺^𝑖_𝑗𝑘 from (28) and (29), then we can solve 𝐺^𝑖_𝑗𝑘 from (30) and (31). In the following we only consider (28) and (29), from which we will solve 𝐺^𝑖_𝑗𝑘.

4. Douglas (𝛼,𝛽)-Metrics

In this section, we characterize two-dimensional (𝛼,𝛽)-metrics (might be singular) which are Douglas metrics. We have the following theorem.

Theorem 9. Let 𝐹=𝛼𝜙(𝑠), 𝑠=𝛽/𝛼, be an (𝛼,𝛽)-metric on an open subset 𝑈⊂𝑅², where 𝜙(0)=1. Suppose that 𝛽 is not parallel with respect to 𝛼 and 𝐹 is not of Randers type. Then 𝐹 is a Douglas metric on 𝑈 if and only if 𝐹 lies in one of the following four classes: (i)𝜙(𝑠) satisfy (3) with 𝑘₂≠𝑘₁𝑘₃ and 𝛽 satisfies 𝑏_𝑖∣𝑗=2𝜏1+𝑘₁𝑏²𝑎_𝑖𝑗+𝑘₂𝑏²+𝑘₃𝑏_𝑖𝑏_𝑗, (32)(ii)𝜙(𝑠) and 𝛽 satisfy 𝜙(𝑠)=√1−𝑘𝑠²+𝑐𝑠²√1−𝑘𝑠²,𝑟_𝑖𝑗=2𝜏1+(2𝑐−𝑘)𝑏²𝑎_𝑖𝑗−𝑘+3𝑐−(𝑘+𝑐)𝑘𝑏²𝑏_𝑖𝑏_𝑗+𝑑𝑏_𝑖𝑠_𝑗+𝑏_𝑗𝑠_𝑖, (33)(34) where 𝜏=𝜏(𝑥) is a scalar function, 𝑘,𝑐 are constants with 𝑐≠0 and 1−𝑘𝑏²≥0, and 𝑑=𝑑(𝑥) is given by ()𝑑=3𝑐−𝑘−2𝑐−𝑘𝑘𝑏²1−(𝑘+𝑐)𝑏². (35)(iii) (𝑏=constant) 𝜙(𝑠) and 𝛽 satisfy 𝜙(𝑠)=𝑏²−𝑠²𝑏+𝑏²−𝑠^2𝑠₀𝑐𝑏²−𝑡^23/2𝑡²1−𝑘𝑡^2𝑚𝑟𝑑𝑡,_𝑖𝑗1=−𝑏²𝑏_𝑖𝑠_𝑗+𝑏_𝑗𝑠_𝑖, (36)(37) where 𝑐,𝑘 are constants and 𝑚≥1 is an integer.(iv) (𝑏=constant) 𝜙(𝑠) and 𝛽 satisfy 𝜙(𝑠)=𝑏²−𝑠²𝑏+𝑏²−𝑠^2𝑠₀𝑐𝑏²−𝑡^23/2𝑡²1−𝑘𝑡^2𝑚−1/2𝑟𝑑𝑡,_𝑖𝑗1=−𝑏²𝑏_𝑖𝑠_𝑗+𝑏_𝑗𝑠_𝑖, (38)(39) where 𝑐,𝑘 are constants and 𝑚≥1 is an integer.

By Theorem 9(ii), we can easily get the following corollary.

Corollary 10. Let 𝐹=𝛼±𝛽²/𝛼 be a two-dimensional (𝛼,𝛽)-metric. Then 𝐹 is a Douglas metric if and only if 𝛽 satisfies 𝑟_𝑖𝑗=2𝜏1±2𝑏²𝑎_𝑖𝑗∓3𝑏_𝑖𝑏_𝑗+3±1−𝑏²𝑏_𝑖𝑠_𝑗+𝑏_𝑗𝑠_𝑖, (40) where 𝜏=𝜏(𝑥) is a scalar function. Note that 𝐹=𝛼+𝛽²/𝛼 is regular if and only if 𝑏<1; 𝐹=𝛼−𝛽²/𝛼 is regular if and only if 𝑏<1/2.

We prove Theorem 9 using Proposition 7. The proof can be divided into two cases (𝑟₁₁,𝑟₂₂)≠(0,0) and (𝑟₁₁,𝑟₂₂)=(0,0).

4.1. (𝑟₁₁,𝑟₂₂)≠(0,0)

In this case, we will obtain two classes: Theorem 9(i) and Theorem 9(ii).

First, (28) can be written in the following form: 2Ψ=𝜆𝑠²+𝜇𝑏²−𝑠²𝛿𝑠²+𝜂𝑏²−𝑠², (41) where 𝜆,𝜇,𝛿, and 𝜂 are numbers independent of 𝑠. By (28) and (41), it is easy to prove that if 𝜆𝜂−𝜇𝛿≠0, then for some scalar 𝜏=𝜏(𝑥), we have (see also [5]) 𝑟₁₁=2𝑏²𝛿𝜏,𝑟₂₂=2𝑏²𝜂𝜏. (42) One can see that if an (𝛼,𝛽)-metric 𝐹=𝛼𝜙(𝛽/𝛼) is not of Randers type, then 𝜆𝜂−𝜇𝛿≠0.

Now we put 𝑎₀∶=1,𝑎_𝑖𝜙∶=^(𝑖)(0)𝑖!,𝑖=1,2,…. (43) By (41), it has been proved in [10] that if 2𝑎₄+𝑎²₂=0, then 𝐹 is of Randers type. Thus we may assume that 2𝑎₄+𝑎²₂≠0. Then there is a scalar 𝜖=𝜖(𝑥)≠0 such that 𝜇=𝑘₁𝜖,𝜂=1+𝑘₁𝑏²𝜖,𝜆=𝑘₁+𝑘₂𝑏²𝑘𝜖,𝛿=1+₁+𝑘₃𝑏²+𝑘₂𝑏⁴𝑘𝜖,2Ψ=¹+𝑘₂𝑠²1+𝑘₁𝑏²+𝑘₃+𝑘₂𝑏²𝑠², (44)(45) where 𝑘₁,𝑘₂, and 𝑘₃ are some constants determined by 𝑘₁=2𝑎₂,𝑘₂=2𝑎₄𝑎²₂−5𝑎₂𝑎₆+12𝑎²₄2𝑎₄+𝑎²₂,𝑘₃=−11𝑎₂𝑎₄+5𝑎₆+3𝑎³₂2𝑎₄+𝑎²₂. (46) Note that 𝑘₂−𝑘₁𝑘₃≠0 is equivalent to 2𝑎₄+𝑎²₂≠0. Since 𝐹 is not of Randers type, we get 𝑘₂−𝑘₁𝑘₃≠0.

Plugging (45) into (29), we get −2𝑏𝑠𝑏²−𝑠²𝑘₁+𝑘₂𝑠²𝑟₁₂−2𝑏³1+𝑘₁𝑠²+𝑘₃𝑠²+𝑘₂𝑠⁴𝑄𝑠₁₂+𝑠1+𝑘₁𝑏²+𝑘₃𝑠²+𝑘₂𝑏²𝑠²×𝑏²−𝑠²𝜉−𝐺²₁₁𝑠²=0, (47) where 𝜉∶=𝐺¹₁₂+𝐺¹₂₁−𝐺²₂₂. Now plug the Taylor expansion of 𝜙(𝑠) into (47), and let 𝑝_𝑖 be the coefficients of 𝑠^𝑖 in (47). By 𝑝₁=0,𝑝₃=0, and 𝑝₅=0, we have the following cases.

(i) If 1+𝑘₁+𝑘₃𝑏²+𝑘₂𝑏⁴≠0, (48) then 𝑟₁₂=𝑏²𝑘15¹3𝑘²₁+2𝑘₁𝑘₃+10𝑘₂𝑏⁴+21𝑘²₁+14𝑘₁𝑘₃−5𝑘₂𝑏²+18𝑘₁−3𝑘₃×1+𝑘₁+𝑘₃𝑏²+𝑘₂𝑏⁴⁻¹𝑠₁₂,𝐺²₁₁=2𝑏153𝑘²₁𝑘₃+2𝑘₁𝑘²₃−2𝑘₂𝑘₃−18𝑘₁𝑘₂𝑏⁴−53𝑘²₁+𝑘₂+2𝑘₁𝑘₃𝑏²−15𝑘₁×1+(𝑘₁+𝑘₃)𝑏²+𝑘₂𝑏⁴⁻¹𝑠₁₂,𝜉=2𝑏𝑘₁153𝑘²₁+2𝑘₁𝑘₃+10𝑘₂𝑏⁴+63𝑘₁+2𝑘₃𝑏²×+151+(𝑘₁+𝑘₃)𝑏²+𝑘₂𝑏⁴⁻¹𝑠₁₂. (49)(50)(51)(ii) If 1+𝑘₁+𝑘₃𝑏²+𝑘₂𝑏⁴=0, (52)then 𝑏=constant. By Lemma 5 we get 𝑟₁₂+𝑠₁₂=0. If 𝑠₁₂≠0, then we get 𝑘₂=3+𝑘₁𝑏²2𝑏⁴,𝑘₃=−5+3𝑘₁𝑏²2𝑏². (53)

Case 1. 𝑠₁₂=0. This implies Theorem 9(i).

If (48) holds, then 𝑟₁₂=0 by (49). If (52) holds, we also get 𝑟₁₂=0. In both cases, we have 𝐺²₁₁=𝜉=0 by plugging 𝑟₁₂=𝑠₁₂=0 into (29). Thus (29) becomes trivial. By 𝑟₁₂=0, (42) and (44), we get the expression of 𝑏_𝑖∣𝑗 in (32). Further, (45) can be written in the form (3) with 𝑘₂≠𝑘₁𝑘₃. This class belongs to Theorem 9(i).

Case 2. 𝑠₁₂≠0. This implies Theorem 9(ii).

Case 2A. Assume that (48) holds. We plug (49), (50), and (51) into (47), and then we obtain 1𝑄=−153𝑘²₁𝑘₃+2𝑘₁𝑘²₃−2𝑘₂𝑘₃−18𝑘₁𝑘₂𝑠⁴−53𝑘²₁+𝑘₂+2𝑘₁𝑘₃𝑠²−15𝑘₁𝑠×1+𝑘₁+𝑘₃𝑠²+𝑘₂𝑠⁴⁻¹. (54) By (54) and (45), we have 𝑘₂=2𝑘₁+3𝑘₃3𝑘₁+2𝑘₃.25 (55) Plug (55) into (54) and we get 𝜙(𝑠)=1√55+4𝑘₁+𝑘₃𝑠²5+3𝑘₁+2𝑘₃𝑠², (56) where 𝑘₁≠𝑘₃ since (55) and 𝑘₂≠𝑘₁𝑘₃. Letting 𝑘₁=2𝑐−𝑘 and 𝑘₃=−3𝑐−𝑘 in (56), we get (33). Substituting (55) into (49) gives 𝑟₁₂=𝑏²𝑘₁3𝑘₁+2𝑘₃𝑏²+6𝑘₁−𝑘₃5+2𝑘₁+3𝑘₃𝑏²𝑠₁₂. (57) Letting 𝑘₁=2𝑐−𝑘 and 𝑘₃=−3𝑐−𝑘 and using (42), we obtain (34).

Case 2B. Assume that (52) holds. Then 𝑟₁₂=−𝑠₁₂ and (53) holds. It is easy for us to get 𝜙(𝑠)=𝑏+𝑐𝑠̃²𝑏²−𝑠²,𝑟_𝑖𝑗=2𝜏𝑏²𝑎_𝑖𝑗−𝑏_𝑖𝑏_𝑗−1𝑏²𝑏_𝑖𝑠_𝑗+𝑏_𝑗𝑠_𝑖. (58)(59) This class is a special case of Theorem 1(ii).

4.2. (𝑟₁₁,𝑟₂₂)=(0,0)

Since 𝛽 is not parallel and (𝑟₁₁,𝑟₂₂)=(0,0), we will see that 𝑠₁₂≠0 from the following proof to different cases. It follows from (28) that 𝐺¹₂₂=0,𝐺¹₁₁=𝐺²₁₂+𝐺²₂₁. (60) Plugging the expressions of 𝑄 and Ψ into (29) yields 𝑠𝑏²−𝑠²2𝑏𝑟₁₂+𝐺²₁₁𝑠²−𝑏²−𝑠²𝜉𝜙^′′+𝑠𝐺²₁₁𝑠²−𝑏²−𝑠²𝜉𝜙−𝑠𝜙^′+2𝑏³𝑠₁₂𝜙^′=0, (61) where 𝜉∶=𝐺¹₁₂+𝐺¹₂₁−𝐺²₂₂.

Let 𝜙=𝑎₀+^ℎ_𝑖=1𝑎_𝑖𝑠^𝑖+𝑜𝑠^ℎ,𝑎₀=1, (62) where ℎ is a sufficiently large integer. Plugging the above Taylor series into (61), we obtain a power series ∑_𝑘𝑝_𝑘𝑠^𝑘=0. It is easily seen that the coefficient 𝑝_𝑘 of 𝑠^𝑘 is given by 𝑝_𝑘=𝐴₁𝑟₁₂+𝐴₂𝐺²₁₁+𝐴₃𝜉+𝐴₄𝑠₁₂, (63) where 𝐴₁∶=𝑏(𝑘−1)(𝑘−2)𝑎_𝑘−1−𝑘(𝑘+1)𝑎_𝑘+1𝑏²,𝐴₂1∶=2(𝑘−2)(𝑘−4)𝑎_𝑘−3−(𝑘−1)𝑎_𝑘−1𝑏²,𝐴₃1∶=2𝑘(𝑘+1)𝑎_𝑘+1𝑏⁴−(2𝑘−1)(𝑘−2)𝑎_𝑘−1𝑏²+(𝑘−2)(𝑘−4)𝑎_𝑘−3,𝐴₄∶=−(𝑘+1)𝑎_𝑘+1𝑏³, (64) and 𝑎_𝑖=0 if 𝑖<0. In particular, we have 𝑝₀=−𝑎₁𝑏³𝑠₁₂. (65) So if 𝑠₁₂≠0, then by 𝑝₀=0, we have 𝑎₁=0.

Case I. Suppose 𝑑𝑏≠0. We will prove that one case belongs to Theorem 9(ii) with the scalar 𝜏=𝜏(𝑥)=0, and other cases are excluded.

Solving the system 𝑝₁=0,𝑝₃=0,𝑝₅=0 yields the following three cases.(i)If 2𝑎₄≠−𝑎²₂, then we get (49), (50), and (51) by using (46).(ii)If 2𝑎₄=−𝑎²₂ and 2𝑎₆≠𝑎³₂, then 𝑟₁₂=158𝑎₂𝑏²−1𝑠₁₂,𝐺²₁₁=−4𝑎₂𝑏𝑠₁₂,𝜉=165𝑎₂𝑏𝑠₁₂. (66)(iii) If 2𝑎₄=−𝑎²₂ and 2𝑎₆=𝑎³₂, then 𝐺²₁₁=−4𝑎₂𝑏𝑠₁₂,𝜉=4𝑎₂𝑏1+2𝑎₂𝑏²𝑟₁₂+𝑠₁₂. (67)It follows from (49) or (66) that 𝑟₁₂=0 if 𝑠₁₂=0. If (67) holds and 𝑠₁₂=0, then we have 𝐺²₁₁=0, and thus we have 𝑟₁₂=0 by (29) since 𝐹 is not of Randers type (also see the proof in [5]). Therefore in this case we have 𝑠₁₂≠0.

Case IA. Suppose 2𝑎₄=−𝑎²₂ and 2𝑎₆≠𝑎³₂. Then plug (66) into (61), and by using 𝑑𝑏≠0 we get 𝜙(𝑠)=1+2𝑎₂𝑠². This case is excluded.

Case IB. Suppose 𝑎₄≠−(1/2)𝑎²₂. Plugging (49), (50), and (51) into (61) and by using 𝑑𝑏≠0 and 𝑠₁₂≠0, we obtain three ODEs on 𝜙(𝑠) whose discussion of solutions can be divided into the following cases.

If 𝑘₃≠𝑘₁, then in a similar way as in Section 4.1, we can easily show that this class belongs to Theorem 1(ii) with 𝜏=𝜏(𝑥)=0.

If 𝑘₃=𝑘₁≠0, then we obtain 𝑘₂=𝑘²₁, which is impossible since 𝑘₂≠𝑘₁𝑘₃.

If 𝑘₁=𝑘₃=0, then it is easy to see that 𝐹 is of Randers type, which is excluded.

Case IC. Suppose 𝑎₄=−(1/2)𝑎²₂ and 𝑎₆=(1/2)𝑎³₂. Then we have (67). Note that we have 𝑎₁=0 since 𝑠₁₂≠0. We will show that this case is excluded.

For the function 𝑓(𝑠)=1+2𝑎₂𝑠², its Taylor coefficients 𝑐_𝑖 of 𝑠^𝑖 (𝑖≥0) are given by 𝑐_2𝑖+1=0,𝑐_2𝑖=𝐶^𝑖_1/22𝑎₂^𝑖, (68) where 𝐶^𝑖_𝜇 are the generalized combination coefficients. So in all 𝑎_2𝑖+1’s or 𝑎_2𝑖’s there exists some minimal 𝑚 such that 𝑎_2𝑚+1()𝑎≠0,𝑚≥1,or_2𝑚≠𝐶^𝑚_1/22𝑎₂^𝑚,(𝑚≥4). (69)

Case IC(1). Assume 𝑎_2𝑚+1≠0 in (69). Then plugging (67), 𝑎_2𝑚−3=0 and 𝑎_2𝑚−1=0 into 𝑝_2𝑚=0 (see (63)) yields 𝑎_1+2𝑚−2𝑚𝑟₁₂+4𝑚𝑎₂𝑏²−2𝑎₂𝑏²−1𝑠₁₂=0. (70) Therefore, it follows from (70) that we have 𝑟₁₂=4𝑚𝑎₂𝑏²−2𝑎₂𝑏²−1𝑠2𝑚¹². (71)

Case IC(2). Assume 𝑎_2𝑚≠𝐶^𝑚_1/2(2𝑎₂)^𝑚 in (69). Plugging (67) and 𝑎_2𝑚−4=𝐶^𝑚−2_1/22𝑎₂^𝑚−2,𝑎_2𝑚−2=𝐶^𝑚−1_1/22𝑎₂^𝑚−1 (72) into 𝑝_2𝑚−1=0 (see (63)) yields 𝑎_2𝑚−𝐶^𝑚_1/22𝑎₂^𝑚×(1−2𝑚)𝑟₁₂+4𝑚𝑎₂𝑏²−4𝑎₂𝑏²−1𝑠₁₂=0. (73) Therefore it follows from (73) that we have 𝑟₁₂=4𝑚𝑎₂𝑏²−4𝑎₂𝑏²−1𝑠2𝑚−1¹². (74)

Finally, plugging (67) and (71) or (74) into (61) and using 𝑑𝑏≠0 we get 𝜙(𝑠)=1+2𝑎₂𝑠². Thus both cases are excluded.

Case II. Suppose 𝑑𝑏=0. We will obtain Theorem 9(iii) and Theorem 1(iv).

By Lemma 5 we have 𝑟₁₂=−𝑠₁₂. (75) For the function 𝑓(𝑠)=(1/𝑏)𝑏²−𝑠², its Taylor coefficients 𝑐_𝑖 of 𝑠^𝑖 (𝑖≥0) are given by 𝑐_2𝑖+1=0,𝑐_2𝑖=𝐶^𝑖_1/2−1𝑏^2𝑖. (76) Since 𝑎₁=0 and 𝐹 is not of Randers type, in all 𝑎_2𝑖+1’s or 𝑎_2𝑖’s, there exists some minimal 𝑚 such that 𝑎_2𝑚+1()𝑎≠0,𝑚≥1,or_2𝑚≠𝐶^𝑚_1/2−1𝑏^2𝑚,(𝑚≥1). (77)

Case II(1). Assume 𝑎_2𝑚+1≠0 in (77). Plugging (75), 𝑎_2𝑚−3=0 and 𝑎_2𝑚−1=0 into 𝑝_2𝑚=0 (see (63)) yields 𝜉=−2𝑚−1𝑠𝑚𝑏¹². (78) Plugging (78) and 𝑎_2𝑚−1=0 into 𝑝_2𝑚+2=0 yields 𝐺²₁₁=1𝑚()()2𝑚+12𝑚+3𝑏𝑎_2𝑚+3𝑚𝑎_2𝑚+1+4𝑚²−3𝑏𝑠₁₂. (79) Now plug (75), (78), and (79) into (61), and then we obtain 𝜙−𝑠𝜙^′+𝑏²−𝑠²𝜙^′′𝑠𝜙+𝑏²−𝑠²𝜙^′=2𝑚𝑠1−𝑘𝑠², (80) where 𝑘 is a constant determined by 𝑎_2𝑚+1 and 𝑎_2𝑚+3. Let Φ∶=𝑠𝜙(𝑠)+𝑏²−𝑠²𝜙^′(𝑠). (81) Then (80) becomes Φ^′Φ=2𝑚𝑠1−𝑘𝑠². (82) We get 𝑠Φ=𝑐²1−𝑘𝑠^2𝑚, (83) where 𝑐 is a constant. Then we can easily get 𝜙=𝑏²−𝑠²𝑐𝑏²−𝑠^23/2𝑠²1−𝑘𝑠^2𝑚𝑑𝑠. (84) By assumption, 𝜙(0)=1, we get (36). Further, since 𝑟₁₁=0,𝑟₂₂=0, and 𝑟₁₂=−𝑠₁₂, we get (37). This class belongs to Theorem 9(iii).

Case II(2). Assume 𝑎_2𝑚≠𝐶^𝑚_1/2(−1/𝑏²)^𝑚 in (77). Plugging (75) and the expressions of 𝑎_2𝑚−4 and 𝑎_2𝑚−2 into 𝑝_2𝑚−1=0 yields 4()𝜉=−𝑚−1()𝑏𝑠2𝑚−1¹². (85) Plugging (85) and the expressions of 𝑎_2𝑚−2 into 𝑝_2𝑚+1=0 (see (63)) yields 𝐺²₁₁=𝑇₁𝑇₂𝑠₁₂, (86) where 𝑇₁ and 𝑇₂ are defined by 𝑇₁∶=4𝑚(2𝑚−1)(𝑚−1)𝐶^𝑚_1/2−1𝑏^2𝑚−1+2(2𝑚−1)2𝑚²−2𝑚−1𝑏²𝑎_2𝑚+4(𝑚+1)𝑏⁴𝑎_2𝑚+2,𝑇₂∶=𝑚(2𝑚−1)²𝑏³𝑎_2𝑚−𝐶^𝑚_1/2−1𝑏^2𝑚. (87) Now plug (75), (85), and (86) into (61), and then we obtain 𝜙−𝑠𝜙^′+𝑏²−𝑠²𝜙^′′𝑠𝜙+𝑏²−𝑠²𝜙^′=2𝑚−1𝑠1−𝑘𝑠², (88) where 𝑘 is a constant. By the same argument, we obtain (38). This gives Theorem 9(iv).

5. Projectively Flat (𝛼,𝛽)-Metrics

In this section, we characterize two-dimensional (𝛼,𝛽)-metrics (might be singular) which are projectively flat. We have the following theorem.

Theorem 11. Let 𝐹=𝛼𝜙(𝑠), 𝑠=𝛽/𝛼, be an (𝛼,𝛽)-metric on an open subset 𝑈⊂𝑅² with 𝜙(0)=1. Suppose that 𝛽 is not parallel with respect to 𝛼 and 𝐹 is not of Randers type. Then 𝐹 is projectively flat in 𝑈 with 𝐺^𝑖=𝑃(𝑥,𝑦)𝑦^𝑖 if and only if 𝐹 lies in one of the following four classes.(i)𝜙(𝑠) and 𝛽 satisfy (3) and (32), and the spray coefficients 𝐺^𝑖_𝛼 of 𝛼 satisfy 𝐺^𝑖_𝛼=𝜌𝑦^𝑖−𝜏𝑘₁𝛼²+𝑘₂𝛽²𝑏^𝑖. (89) In this case, the projective factor 𝑃 is given by 𝑃=𝜌+𝜏𝛼1+𝑘₁+𝑘₃𝑠²+𝑘₂𝑠⁴𝜙^′𝜙−𝑘₁+𝑘₂𝑠²𝑠. (90)(ii)𝜙(𝑠) and 𝛽 satisfy (33) and (34), and 𝐺^𝑖_𝛼=𝜌𝑦^𝑖−𝜏(2𝑐−𝑘)𝛼²+(𝑐+𝑘)𝑘𝛽²𝑏^𝑖+(2𝑐−𝑘)1−𝑘𝑏²𝛼²+𝑐𝑘𝛽²1−(𝑐+𝑘)𝑏²𝑠^𝑖. (91) In this case, the projective factor 𝑃 is given by 𝑃=𝜌−4𝑐²𝑠³1+(𝑐−𝑘)𝑠²𝜎𝜏𝛼+¹𝜎₂𝑠₀, (92) where 𝜎₁∶=(𝑘−2𝑐)(𝑐−𝑘)𝑘𝑏²+4𝑐²−3𝑘𝑐+𝑘²𝑠²+(2𝑐−𝑘)1−𝑘𝑏²,𝜎₂∶=1−(𝑐+𝑘)𝑏²1+(𝑐−𝑘)𝑠². (93)(iii)𝜙(𝑠) and 𝛽 satisfy (36) and (37), and 𝐺^𝑖_𝛼=𝜌𝑦^𝑖−(2𝑚−1)𝛼²+𝑘𝛽²2𝑚𝑏²𝑠^𝑖. (94) In this case, the projective factor 𝑃 is given by 𝑠𝑃=𝜌−1−𝑘𝑠²𝜙^′+(2𝑚−1)+𝑘𝑠²𝜙2𝑚𝑏²𝜙𝑠₀. (95)(iv)𝜙(𝑠) and 𝛽 satisfy (38) and (39), and 𝐺^𝑖_𝛼=𝜌𝑦^𝑖−2(𝑚−1)𝛼²+𝑘𝛽²(2𝑚−1)𝑏²𝑠^𝑖. (96) In this case, the projective factor 𝑃 is given by 𝑠𝑃=𝜌−1−𝑘𝑏²𝜙^′+2(𝑚−1)+𝑘𝑠²𝜙(2𝑚−1)𝑏²𝜙𝑠₀. (97)In the above, 𝜌=𝑐₁(𝑥)𝑦¹+𝑐₂(𝑥)𝑦² is a 1-form.

By Theorem 11(ii), we can easily get the following corollary.

Corollary 12. Let 𝐹=𝛼±𝛽²/𝛼 be two-dimensional (𝛼,𝛽)-metric. Then 𝐹 is locally projectively flat if and only if 𝛽 satisfies (40) and 𝐺^𝑖_𝛼 satisfy 𝐺^𝑖_𝛼=𝜌𝑦^𝑖∓2𝜏𝛼²𝑏^𝑖−2𝛼²𝑏²𝑠∓1^𝑖. (98) In this case, the projective factor 𝑃 is given by 𝑃=𝜌−4𝑠³1±𝑠²2𝜏𝛼−2𝑠²±1𝑏²∓1𝑠²𝑠±1⁰. (99)

To prove Theorem 11, it follows from comparing Propositions 7 and 8 that we only need to give the expressions (89)–(97) for each class in Theorem 11.

5.1. The Spray Coefficients of 𝛼

In this subsection we will show the expressions of the spray coefficients 𝐺^𝑖_𝛼 for each class in Theorem 11. Note that by 𝐺^𝑖_𝑗𝑘=𝜕²𝐺^𝑖_𝛼/𝜕𝑦^𝑗𝜕𝑦^𝑘, the spray 𝐺^𝑖_𝛼 of 𝛼 can be expressed as 𝐺^𝑖_𝛼=12𝐺^𝑖_𝑗𝑘𝑦^𝑗𝑦^𝑘. (100)

Case I. Suppose that (𝑟₁₁,𝑟₂₂)≠(0,0). It has been proved in [10] that 𝐺¹₁₁=2𝑡₁−2𝜆𝑏³𝜏,𝐺¹₂₂=−2𝜇𝑏³𝐺𝜏,₁₁₂=𝑡₂,𝐺²₁₂=𝑡₁, (101) where 𝑡₁,𝑡₂ are numbers independent of 𝑠 and 𝜏 is given by (32) or (34). By (50) and (51), we can get 𝐺²₁₁ and 𝐺²₂₂.

If 𝛽 is closed (𝑠₁₂=0), then it follows from (44) and (101) and the expressions of 𝐺²₁₁ and 𝐺²₂₂ that (89) holds, where we put 𝜌 as 𝜌=𝑡_𝑖𝑦^𝑖. This case has been given by [10] in case of 𝑛≥3.

If 𝛽 is not closed, then by putting 𝑘₁=2𝑐−𝑘 and 𝑘₃=−3𝑐−𝑘 we get (91) from (55) and (101) and the expressions of 𝐺²₁₁ and 𝐺²₂₂.

Case II. Suppose that (𝑟₁₁,𝑟₂₂)=(0,0). Then by (30) we get 𝐺¹₂₂=0,𝐺¹₁₁=2𝐺²₁₂=2𝑡₁,𝐺²₁₂=𝑡₁,𝐺¹₁₂=𝑡₂. (102)

If 𝑑𝑏≠0, then we have shown in Section 4.2 that 𝑎₄≠−(1/2)𝑎²₂. In this case, we obtain (91) with 𝜏=0.

If 𝑑𝑏=0, then we get (77). If 𝑎_2𝑚+1≠0 in (77), then we get from (78), (79) that 𝐺²₁₁=−2𝑚−1+𝑘𝑏²𝑠𝑚𝑏¹²,𝐺²₂₂=2𝐺¹₁₂−2𝑚−1𝑠𝑚𝑏¹². (103) Then it follows from (102) and (103) that (94) holds. If 𝑎_2𝑚≠𝐶^𝑚_1/2(−1/𝑏²)^𝑚 in (77), then we get from (85), (86) that 𝐺²₁₁2=−2𝑚²−2𝑚−𝑘𝑚()𝑏𝑠2𝑚−1¹²,𝐺²₂₂=2𝐺¹₁₂−4(𝑚−1)()𝑏𝑠2𝑚−1¹². (104) Then it follows from (102) and (104) that (96) holds.

5.2. The Projective Factors

In this subsection, we are going to find the expression for the projective factor for each class in Theorem 11.

Actually, (90) has been proved in [10], since 𝛽 is closed in Theorem 11(i). So we only show the expressions of 𝑃 in (92), (95), and (97). In the left three classes, since 𝛽 may not be closed, it is not easy to show the projective factors 𝑃 in the initial local projective coordinate system (in such a coordinate system, geodesics are straight lines). However, it is easy to be solved by choosing another local projective coordinate system and then returning to the initial local projective coordinate system, just as that in [10].

Fix an arbitrary point 𝑥_𝑜∈𝑈⊂𝑅². By the above idea and a suitable affine transformation, we may assume (𝑈,𝑥^𝑖) is a local projective coordinate system satisfying that 𝛼_𝑥𝑜=(𝑦¹)²+(𝑦²)² and 𝛽_𝑥𝑜=𝑏𝑦¹. Then at 𝑥_𝑜 we have 𝑠¹=𝑠₁=0,𝑠²=𝑠₂=𝑏𝑠₁₂,𝑠₀=𝑏𝑠₁₂𝑦²,𝑏¹=𝑏₁=𝑏,𝑏²=𝑏₂=0. (105)

Suppose (33), (34), and (91) hold in 𝑈. Then it is easy to get 𝑟₀₀,𝑠^𝑖₀,𝑄,Θ, and Ψ. Plug them into (21), and then at 𝑥_𝑜 we see that 𝐺^𝑖=𝑃𝑦^𝑖, where 𝑃 is given by 𝑃=𝜌+𝐴₁𝜏+𝐴₂𝑏𝑠₁₂𝑦², (106) where 𝐴₁∶=−4𝑐²𝑏𝑦¹³1+(𝑐−𝑘)𝑏²𝑦¹²+𝑦²²,𝐴₂∶=1+(2𝑐−𝑘)𝑏²(2𝑐−𝑘)−𝑘(𝑐−𝑘)𝑏²×𝑦¹²+(2𝑐−𝑘)1−𝑘𝑏²𝑦²²×1−(𝑐+𝑘)𝑏²×1+(𝑐−𝑘)𝑏²𝑦¹²+𝑦²²⁻¹. (107) By using 𝑏𝑠₁₂𝑦²=𝑠₀,𝑦¹²+𝑦²²=𝛼²,𝑏𝑦¹𝛽=𝛽,𝛼=𝑠, (108) we can transform (106) as (92). It is a direct computation, so the details are omitted. Since 𝑥_𝑜 is arbitrarily chosen, (92) holds in 𝑈.

The left proofs are similar. So the details are omitted.

We have found the projective factor for each class in Theorem 11. This also gives a proof to the inverse of Theorem 11.

6. Singular Classes in Theorems 9 and 11

In this section, we will firstly discuss the local structures of the singular classes in Theorem 9(iii) and Theorem 9(iv), and then construct some examples for Theorem 11(iii) and Theorem 11(iv).

Since every two-dimensional Riemann metric is locally conformally flat, we may put 𝛼=𝑒^𝜎(𝑥)𝑦¹²+𝑦²², (109) where 𝑥=(𝑥¹,𝑥²). Since (59) is equivalent to 𝑏²=‖𝛽‖²_𝛼=constant in two-dimensional case (see a simple proof in [7]), (59) holds if and only if 𝛽 is in the form 𝛽=𝑏𝑒^𝜎(𝑥)𝜉(𝑥)𝑦¹+𝜂(𝑥)𝑦²𝜉(𝑥)²+𝜂(𝑥)², (110) where 𝑏=‖𝛽‖_𝛼=constant. Thus (109) and (110) give all the local solutions of (59). If we put 𝛼 and 𝛽 in the forms (109) and (110), then we have (59); that is, 𝑟_𝑖𝑗=2𝜏𝑏²𝑎_𝑖𝑗−𝑏_𝑖𝑏_𝑗−1𝑏²𝑏_𝑖𝑠_𝑗+𝑏_𝑗𝑠_𝑖, (111) where 𝜏 is given by 2𝜉𝜏=²+𝜂²𝜉𝜎₁+𝜂𝜎₂−𝜉𝜂𝜂₁+𝜉²𝜂₂+𝜂²𝜉₁−𝜉𝜂𝜉₂𝑏𝑒^𝜎𝜉²+𝜂^23/2, (112) where 𝜎₁∶=𝜕𝜎/𝜕𝑥¹, 𝜎₂∶=𝜕𝜎/𝜕𝑥², and so forth. Further, 𝛽 is not closed if and only if 𝜉²+𝜂²𝜉𝜎₂−𝜂𝜎₁−𝜉²𝜂₁−𝜉𝜂𝜂₂+𝜉𝜂𝜉₁+𝜂²𝜉₂≠0. (113) In particular, if we put 𝜉=𝑥²,𝜂=−𝑥¹, and 𝜎=𝑐[(𝑥¹)²+(𝑥²)²], where 𝑐≠0 is a constant, then we have 𝜏=0 by (112) and (113) holds (𝛽 is not closed).

Proposition 13. Define a two-dimensional (𝛼,𝛽)-metric 𝐹 on 𝑅² by 𝐹=𝑏𝛼²+𝑘𝛽²𝑏²𝛼²−𝛽², (114) where 𝑘,𝑏 are constants with 𝑘≠−1/𝑏. Then 𝐹 is a Douglas metric if and only if 𝛼 and 𝛽 can be locally defined by (109) and (110), where 𝜉,𝜂, and 𝜎 are some scalar functions on 𝑅². There are many choices for 𝜉,𝜂, and 𝜎 such that 𝛽 is not closed.

Proposition 14. Let 𝐹=𝛼𝜙(𝑠), 𝑠=𝛽/𝛼, be a two-dimensional (𝛼,𝛽)-metric, where 𝜙(0)=1. Let 𝜙(𝑠) be given by (36) or (38) (not given by (59)). Then 𝐹 is a Douglas metric if and only if 𝛼 and 𝛽 can be locally defined by (109) and (110), where 𝜉,𝜂, and 𝜎 are some scalar functions satisfying 𝜏=0 in (112). There are many choices for 𝜉,𝜂, and 𝜎 such that 𝛽 is not closed.

Next we construct some singular examples for Theorem 11(iii) and Theorem 11(iv) which are projectively flat. One can directly verify the following two examples.

Example 15. Let 𝐹=𝛼𝜙(𝑠), 𝑠=𝛽/𝛼, be two-dimensional (𝛼,𝛽)-metric, where 𝜙(0)=1. Let 𝜙(𝑠) be given by (36) with 𝑘=0, and define 𝛼 and 𝛽 by (109) and (110), where 𝜉=𝑥²,𝜂=−𝑥¹1,𝜎=𝑚−2ln𝑥¹²+𝑥²². (115) Then 𝐹 is projectively flat with 𝛽 being not closed.

Example 16. Let 𝐹=𝛼𝜙(𝑠), 𝑠=𝛽/𝛼, be two-dimensional (𝛼,𝛽)-metric, where 𝜙(0)=1. Let 𝜙(𝑠) be given by (38) with 𝑘=0, and define 𝛼 and 𝛽 by (109) and (110), where 𝜉=𝑥²,𝜂=−𝑥¹,𝜎=(𝑚−1)ln𝑥¹²+𝑥²². (116) Then 𝐹 is projectively flat with 𝛽 being not closed.

It might be also an interesting problem to show the local structures of the two classes of Theorem 11(iii) and Theorem 11(iv). This problem is still open.

7. Proof of Theorem 2

In the following proof, our idea is to choose a suitable deformation on 𝛼 and 𝛽 such that we can obtain a conformal form on a Riemannian space. Then using the result in [14], we can complete our proof.

Define a new Riemann metric 𝛼 and a 1-form 𝛽 by 𝛼∶=𝜉𝛼²+𝜂𝛽²,𝛽∶=𝛽, (117) where 𝜉∶=1∓𝑏²³1±2𝑏^23/2,9𝜂∶=8𝑏²1±2𝑏^23/2−1∓2𝑏²+4𝑏⁴1±2𝑏^23/2. (118) Since 𝐹=𝛼±𝛽²/𝛼 is a Douglas metric, we have (40). By (117) and (40), a direct computation gives 𝑟̃_𝑖𝑗=2𝜏1∓𝑏²²1±2𝑏^25/2𝑎_𝑖𝑗. (119) So 𝛽=𝛽 is a conformal 1-form with respect to 𝛼.

Since 𝛼 is a two-dimensional Riemann metric, we can express 𝛼 locally as 𝛼∶=𝑒^𝜎𝑦¹²+𝑦²², (120) where 𝜎=𝜎(𝑥) is a scalar function. We can obtain the local expression of 𝛽=𝛽 by (119) and (120) (see [14]). By the result in [14], we have 𝛽=𝑏̃₁𝑦¹+𝑏̃₂𝑦²=𝑒^2𝜎𝑢𝑦¹+𝑣𝑦², (121) where 𝑢=𝑢(𝑥),𝑣=𝑣(𝑥) are a pair of scalar functions such that 𝑓(𝑧)=𝑢+𝑖𝑣,𝑧=𝑥¹+𝑖𝑥², (122) is a complex analytic function.

We can express 𝜎 using 𝑢,𝑣,𝑏² by computing the quantity ‖𝛽‖𝛼². Firstly, by (120) and (121) we get 𝛽𝛼²=𝑒^2𝜎𝑢²+𝑣². (123) On the other hand, by the definition of 𝛼 in (117), the inverse 𝑎^𝑖𝑗 of 𝑎_𝑖𝑗 is given by 𝑎^𝑖𝑗=1𝜉𝑎^𝑖𝑗−𝜂𝑏^𝑖𝑏^𝑗𝜉+𝜂𝑏². (124) Now plug 𝜉 and 𝜂 into the above, and we obtain 𝛽𝛼²=𝑎^𝑖𝑗𝑏_𝑖𝑏_𝑗=𝑏²1±2𝑏^23/2. (125) Thus by (123) and (125) we get 𝑒^2𝜎=1𝑢²+𝑣²𝑏²1±2𝑏^23/2. (126)

Finally, by plugging (126), (120), and (121) into (117), we easily get 𝛼 and 𝛽 given by (6) and (7), respectively, where we define 𝐵∶=𝑏².

The Riemann metric 𝛼 expressed in (6) is generally not in the conformally flat form. We show another example which is expressed in a different form. Put 𝛼=𝑒^𝜎𝑦¹²+𝑦²²,𝛽=𝑒^𝜎𝜉𝑦¹+𝜂𝑦²,3𝜎=−2ln1±2𝑥¹²±2𝑥²²+𝑐,𝜉=𝑥²,𝜂=−𝑥¹, (127) where 𝑐 is a constant. Then 𝛽 is not closed and satisfies (40) with 𝜏=0. Therefore, 𝐹=𝛼±𝛽²/𝛼 is a Douglas metric.

8. Proof of Theorem 3

This proof is similar to that in Theorem 2. In the proof of Theorem 2, we make a deformation on 𝛼 and keep 𝛽 unchanged. For the proof of Theorem 3 in the following, we will give a deformation on 𝛽 but keep 𝛼 unchanged.

Define a Riemannian metric 𝛼 and 1-form 𝛽 by 𝛽𝛼∶=𝛼,𝛽∶=𝑐, (128) where 𝑐=𝑐(𝑏²) is defined in (9), where we define 𝐵∶=𝑏². By (32) and a direct computation, we can obtain 𝑏̃_𝑖∣𝑗=2𝜏1+𝑘₁𝑏²𝑐𝑎_𝑖𝑗=2𝜏1+𝑘₁𝑏²𝑐𝑎_𝑖𝑗. (129) So 𝛽 is a closed 1-form conformal with respect to 𝛼.

Now we express 𝛼 locally as 𝛼∶=𝑒^𝜎𝑦¹²+𝑦²², (130) where 𝜎=𝜎(𝑥) is a scalar function. Then by the result in [14], we have 𝛽=𝑏̃₁𝑦¹+𝑏̃₂𝑦²=𝑒^2𝜎𝑢𝑦¹+𝑣𝑦², (131) where 𝑢=𝑢(𝑥),𝑣=𝑣(𝑥) are a pair of scalar functions such that 𝑓(𝑧)=𝑢+𝑖𝑣,𝑧=𝑥¹+𝑖𝑥², (132) is a complex analytic function.

By the property of 𝑢,𝑣 and the fact that 𝛽 given in (131) is closed, it is easily seen that 𝑢,𝑣, and 𝜎 satisfy the PDEs (10).

Now we determine 𝜎 in terms of the triple (𝐵,𝑢,𝑣), where 𝐵∶=𝑏². Firstly by (128) and then by (130) and (131) we get 𝛽²_𝛼=𝑏²𝑐⁻²,𝛽²_𝛼=𝑒^2𝜎𝑢²+𝑣². (133) Therefore, by (133), we get (11).

Finally, we can easily get 𝛼 and 𝛽 given by (9) from (11), (128), (130), and (131).

Conflict of Interests

The author does not have a direct financial relation that might lead to a conflict of interests.