#### 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 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 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 and a -form on a manifold . Such metrics are called -metrics. An -metric can be expressed in the following form: where is a function on . It is known that is a regular Finsler metric (defined on the whole and positive definite) for any with if and only if where is a constant. If does not satisfy (2), then is singular.

Randers metrics are a special class of -metrics. It is known that a Randers metric is a Douglas metric if and only if is closed [2], and it is locally projectively flat if and only if is locally projectively flat and is closed [2, 3]. Usually we call with , where are constants, a Finsler metric of * Randers type*, which is essentially a Randers metric.

-metrics are computable, and it has been shown that -metrics have a lot of special geometric properties [4–13]. In [5, 10], the authors study and characterize -metrics which are, respectively, Douglasian and locally projectively flat in dimension . 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 , , be a regular -metric on an open subset , where . Suppose that is not parallel with respect to and is not of Randers type. Let be a Douglas metric. Then one has one of the following two cases. ** satisfies
where , and are constants satisfying
Further, must be closed.** can be written as
where are constants with . In this case, is generally not closed. *

In dimension in Theorem 1, it is proved in [5, 10] that the metric in Theorem 1 must be given by (3) with and must be closed. In Theorems 9 and 11, we give general characterizations for two-dimensional -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 and , we can determine the local structure of two-dimensional regular Douglas -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 be a two-dimensional regular Douglas -metric with . Then and can be locally written as
**
where are scalar functions such that
**
is a complex analytic function and is a scalar function satisfying if and if . *

We see that in Theorem 2 the metric is determined by the triple parametric scalar functions , where and 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 , , be a two-dimensional regular Douglas -metric with , where satisfies (3) and (4) and . Then and can be locally written as
**
where are some scalar functions which satisfy the following PDEs:
**
where , and , and is defined by
*

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 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: where are constants, and then is determined by (11).

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

*Example 4. *Let be a two-dimensional -metric. Suppose and take the form
where are some scalar functions. Define
where , and are constants with . Then is projectively flat with being not closed.

For the singular projectively flat classes in Theorem 11(iii) and Theorem 11(iv), we also construct some examples with 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 is not closed.

*Open Problem. *Determine the local structure of a two-dimensional -metric which is locally projectively flat.

#### 2. Preliminaries

Let be a Finsler metric on an -dimensional manifold . The geodesic coefficients are defined by

A Finsler metric is called a Douglas metric if the spray coefficients are in the following form: where are local functions on and is a local positively homogeneous function of degree one in . It is easy to see that is a Douglas metric if and only if is a homogeneous polynomial in of degree three, which by (16) can be written as [2]

According to G. Hamel’s result, a Finsler metric is projectively flat in if and only if The above formula implies that with given by

For a Riemannian metric and a -form on a manifold , let denote the covariant derivatives of with respect to . Put where and is the inverse of .

Consider an -metric . By (15), the spray coefficients of are given by [3, 4, 8, 10, 11] where , and , and

By (21) one can see that is a Douglas metric if and only if where , are given in (16) and .

Further, is projectively flat on if and only if where .

#### 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 as in [10, 11].

Fix an arbitrary point , and take an orthonormal basis at such that Then we change coordinates to such that where . Let We have . The following lemmas are trivial.

Lemma 5. * In the special local coordinate system at as mentioned previously, if , then at . *

Lemma 6 (see [11]). *For , suppose , where and are homogeneous polynomials in ; then .*

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

Proposition 7 (). * An -metric is a Douglas metric if and only if, at each point , there is a suitable coordinate system such that at there exist numbers which are independent of such that
*

Proposition 8 (). * An -metric is projectively flat if and only if
**
where are the connection coefficients of . *

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

#### 4. Douglas -Metrics

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

Theorem 9. *Let , , be an -metric on an open subset , where . Suppose that is not parallel with respect to and is not of Randers type. Then is a Douglas metric on if and only if lies in one of the following four classes: ** satisfy (3) with and satisfies
** and satisfy
where is a scalar function, are constants with and , and is given by
** () and satisfy
where are constants and is an integer.** () and satisfy
where are constants and is an integer. *

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

Corollary 10. * Let be a two-dimensional -metric. Then is a Douglas metric if and only if satisfies
**
where is a scalar function. Note that is regular if and only if ; is regular if and only if . *

We prove Theorem 9 using Proposition 7. The proof can be divided into two cases and .

##### 4.1.

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

First, (28) can be written in the following form: where , and are numbers independent of . By (28) and (41), it is easy to prove that if , then for some scalar , we have (see also [5]) One can see that if an -metric is not of Randers type, then .

Now we put By (41), it has been proved in [10] that if , then is of Randers type. Thus we may assume that . Then there is a scalar such that where , and are some constants determined by Note that is equivalent to . Since is not of Randers type, we get .

Plugging (45) into (29), we get where . Now plug the Taylor expansion of into (47), and let be the coefficients of in (47). By , and , we have the following cases.

If then If then . By Lemma 5 we get . If , then we get

*Case 1. *. This implies Theorem 9(i).

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

*Case 2. *. This implies Theorem 9(ii).

*Case 2A*. Assume that (48) holds. We plug (49), (50), and (51) into (47), and then we obtain
By (54) and (45), we have
Plug (55) into (54) and we get
where since (55) and . Letting and in (56), we get (33). Substituting (55) into (49) gives
Letting and and using (42), we obtain (34).

*Case 2B*. Assume that (52) holds. Then and (53) holds. It is easy for us to get
This class is a special case of Theorem 1(ii).

##### 4.2.

Since is not parallel and , we will see that from the following proof to different cases. It follows from (28) that Plugging the expressions of and into (29) yields where .

Let where is a sufficiently large integer. Plugging the above Taylor series into (61), we obtain a power series . It is easily seen that the coefficient of is given by where and if . In particular, we have So if , then by , we have .

*Case I*. Suppose . We will prove that one case belongs to Theorem 9(ii) with the scalar , and other cases are excluded.

Solving the system yields the following three cases.(i)If , then we get (49), (50), and (51) by using (46).(ii)If and , then If and , then It follows from (49) or (66) that if . If (67) holds and , then we have , and thus we have by (29) since is not of Randers type (also see the proof in [5]). Therefore in this case we have .

*Case IA*. Suppose and . Then plug (66) into (61), and by using we get . This case is excluded.

*Case IB*. Suppose . Plugging (49), (50), and (51) into (61) and by using and , we obtain three ODEs on whose discussion of solutions can be divided into the following cases.

If , then in a similar way as in Section 4.1, we can easily show that this class belongs to Theorem 1(ii) with .

If , then we obtain , which is impossible since .

If , then it is easy to see that is of Randers type, which is excluded.

*Case IC*. Suppose and . Then we have (67). Note that we have since . We will show that this case is excluded.

For the function , its Taylor coefficients of () are given by where are the generalized combination coefficients. So in all ’s or ’s there exists some minimal such that

*Case IC(1)*. Assume in (69). Then plugging (67), and into (see (63)) yields
Therefore, it follows from (70) that we have

*Case IC(2)*. Assume in (69). Plugging (67) and
into (see (63)) yields
Therefore it follows from (73) that we have

Finally, plugging (67) and (71) or (74) into (61) and using we get . Thus both cases are excluded.

*Case II*. Suppose . We will obtain Theorem 9(iii) and Theorem 1(iv).

By Lemma 5 we have For the function , its Taylor coefficients of () are given by Since and is not of Randers type, in all ’s or ’s, there exists some minimal such that

Case *II(1)*. Assume in (77). Plugging (75), and into (see (63)) yields
Plugging (78) and into yields
Now plug (75), (78), and (79) into (61), and then we obtain
where is a constant determined by and . Let
Then (80) becomes
We get
where is a constant. Then we can easily get
By assumption, , we get (36). Further, since , and , we get (37). This class belongs to Theorem 9(iii).

*Case II(2)*. Assume in (77). Plugging (75) and the expressions of and into yields
Plugging (85) and the expressions of into (see (63)) yields
where and are defined by
Now plug (75), (85), and (86) into (61), and then we obtain
where is a constant. By the same argument, we obtain (38). This gives Theorem 9(iv).

#### 5. Projectively Flat -Metrics

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

Theorem 11. * Let , , be an -metric on an open subset with . Suppose that is not parallel with respect to and is not of Randers type. Then is projectively flat in with if and only if lies in one of the following four classes.** and satisfy (3) and (32), and the spray coefficients of satisfy
In this case, the projective factor is given by
** and satisfy (33) and (34), and
In this case, the projective factor is given by
where
** and satisfy (36) and (37), and
In this case, the projective factor is given by
** and satisfy (38) and (39), and
In this case, the projective factor is given by
**In the above, is a 1-form. *

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

Corollary 12. * Let be two-dimensional -metric. Then is locally projectively flat if and only if satisfies (40) and satisfy
**
In this case, the projective factor is given by
*

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

In this subsection we will show the expressions of the spray coefficients for each class in Theorem 11. Note that by , the spray of can be expressed as

*Case I*. Suppose that . It has been proved in [10] that
where are numbers independent of and is given by (32) or (34). By (50) and (51), we can get and .

If is closed (), then it follows from (44) and (101) and the expressions of and that (89) holds, where we put as . This case has been given by [10] in case of .

If is not closed, then by putting and we get (91) from (55) and (101) and the expressions of and .

*Case II*. Suppose that . Then by (30) we get

If , then we have shown in Section 4.2 that . In this case, we obtain (91) with .

If , then we get (77). If in (77), then we get from (78), (79) that Then it follows from (102) and (103) that (94) holds. If in (77), then we get from (85), (86) that 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 is closed in Theorem 11(i). So we only show the expressions of in (92), (95), and (97). In the left three classes, since may not be closed, it is not easy to show the projective factors 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 . By the above idea and a suitable affine transformation, we may assume is a local projective coordinate system satisfying that and . Then at we have

Suppose (33), (34), and (91) hold in . Then it is easy to get , and . Plug them into (21), and then at we see that , where is given by where By using we can transform (106) as (92). It is a direct computation, so the details are omitted. Since is arbitrarily chosen, (92) holds in .

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 where . Since (59) is equivalent to in two-dimensional case (see a simple proof in [7]), (59) holds if and only if is in the form where . Thus (109) and (110) give all the local solutions of (59). If we put and in the forms (109) and (110), then we have (59); that is, where is given by where , , and so forth. Further, is not closed if and only if In particular, if we put , and , where is a constant, then we have by (112) and (113) holds ( is not closed).

Proposition 13. *Define a two-dimensional -metric on by
**
where are constants with . Then is a Douglas metric if and only if and can be locally defined by (109) and (110), where , and are some scalar functions on . There are many choices for , and such that is not closed. *

Proposition 14. *Let , , be a two-dimensional -metric, where . Let be given by (36) or (38) (not given by (59)). Then is a Douglas metric if and only if and can be locally defined by (109) and (110), where , and are some scalar functions satisfying in (112). There are many choices for , and such that 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 , , be two-dimensional -metric, where . Let be given by (36) with , and define and by (109) and (110), where
Then is projectively flat with being not closed.

*Example 16. *Let , , be two-dimensional -metric, where . Let be given by (38) with , and define and by (109) and (110), where
Then is projectively flat with 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 and 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 and a 1-form by where Since is a Douglas metric, we have (40). By (117) and (40), a direct computation gives