#### Abstract

We study a class of Finsler metrics in the form , where is a Riemannian metric, is a -form, and is called -metrics. We find the necessary and sufficient conditions under which the class of -metrics is locally projectively flat and Douglas metrics, respectively.

#### 1. Introduction

It is Hilbert’s Fourth Problem in the regular case to study and characterize Finsler metrics on an open domain whose geodesics are straight lines. Finsler metrics with this property are called projectively flat metrics. It is easy to see that a Finsler metric on an open subset is projectively flat if and only if the spray coefficients are in the form , where is a positively homogeneous function of degree one in . In 1903 G. Hamel found a system of partial differential equations that characterize projectively flat Finsler metrics on an open subset . That is, A natural problem is to find projectively flat metrics by solving (1). The Beltrami Theorem tells us that a Riemannian metric is locally projectively flat if and only if it is of constant sectional curvature. Thus this problem has been solved in Reimannian Geometry. However for Finsler metrics, this problem is far from being solved.

In order to find examples of projectively flat Finsler metrics, we consider -metrics. An -metric is defined by , , where is a positive scalar function on with certain regularity, is a Riemannian metric, and is a -form on a manifold . The simplest -metric is Randers metric . In [1]; it is proved that a Randers metric on a manifold is locally projectively flat if and only if is projectively flat and is closed. The other important -metric is Berwald metric on a manifold . In [2], Shen and Yildirim find necessary and sufficient conditions under which the Berwald metric is projectively flat. They proved that the Berwald metric is projectively flat if and only if the following conditions hold:(1),(2)the spray coefficients of are in the form ,where , denote the covariant derivatives of with respect to , is a scalar function, and is a -form on . Also they determined local structure of with constant flag curvature. In [3, 4], some other special -metrics are studied.

In this paper, we consider a special subclass of -metrics in the following form: where and . We call it -metric. We prove the following.

Theorem 1. *Let be a -metric on an open subset . is locally projectively flat if and only if the following conditions hold:*(1)* is parallel with respect to ;*(2)* is locally projectively flat; that is, is of constant curvature.*

It is said that a Finsler metric is trivial if it satisfies the conclusion of Theorem 1. Thus the above theorem tells us that in the class of -metrics, where , there is no nontrivial projectively flat metric.

There are two important non-Riemannian quantities in Finsler Geometry, the Berwald and the Landsberg curvatures. The Berwald and Landsberg tensors are, respectively, defined by Finsler metrics with and are called Berwald and Landsberg metrics, respectively. Clearly every Berwald metric is Landsberg, but the converse is not necessarily true.

Two Finsler metrics and on a manifold are said to be projectively equivalent if they have the same geodesics as point sets. Projective Finsler Geometry studies these metrics. Douglas curvature is an important projective invariant in projective Finsler Geometry. A Finsler metric is called Douglas metric if . Douglas metrics can be viewed as generalized Berwald metrics. Every Berwald metric is Douglas metric but Douglas metrics are not necessarily Berwald. In [5], it is proved that a Randers metric is a Douglas metric if and only if is closed. For another example a Matsumoto metric and the exponential metric are Douglas metrics if and only if is parallel with respect to . In this case both of metrics are Berwald metrics, [4, 6, 7]. In [8], it is proved that the Berwald metric is a Douglas metric if and only if where is scalar function.

In this paper, we are going to find the conditions under which, on a manifold of dimension , the -metric is a Douglas metric. More precisely, we prove following.

Theorem 2. *Let be a -metric on a manifold of dimension . The Finsler metric is a Douglas metric if and only if is parallel with respect to .*

It is known that a locally projectively flat Finsler metric is a Douglas metric, but the converse is not necessarily true. By Theorems 1 and 2, if is a Douglas metric and is a locally projectively flat Riemannian metric, then is a projectively flat Finsler metric.

Corollary 3. *Let be a -metric on a manifold of dimension . Then is locally projectively flat if and only if it is Douglas metric and is locally projectively flat Riemannian metric.*

Corollary 4. *Let be a -metric on a manifold of dimension . If is a Douglas metric or projectively flat, then the following properties hold:*(1)* is a Landsberg metric;*(2)* is a Berwald metric.*

#### 2. Preliminaries

By definition, an -metric is a Finsler metric expressed in the following form: where is a Riemannian metric and is a -form with , . The function is a positive function on an open interval satisfying Let and denote the spray coefficients of and , respectively, given by where and .

A Finsler metric on an open domain is said to be projectively flat in if all geodesics are straight lines. This is equivalent to , where is a -homogeneous function on . In this case is of scalar curvature with flag curvature Let denote the coefficients of the covariant derivative of with respect to and define where and . Clearly, is closed if and only if . A -form is said to be parallel with respect to if and an -metric is said to be trivial if . We have the following.

Lemma 5. *The geodesic coefficients of are related to by
**
where
*

The formula (10) is given in [9, 10]. A different version of (10) is given in [7].

From (10), it is easy to see that if is projectively flat and is parallel with respect to (), then . Thus is a projectively flat Berwald metric. If , then is a Riemannian metric by Numata’s Theorem [11]. If , then is locally Minkowskian.

Lemma 6 (see [2]). *An -metric , where , is projectively flat on an open subset if and only if
**
where and .*

#### 3. -Metrics

In this section, we consider a special subclass of -metrics in the following form:

Let be a function such that , and . It is easy to see that Since and , then . Thus is a Finsler metric. We call it -metric. When or , becomes Randers metric and Berwald metric, respectively. In this paper we assume that .

By Lemma 5, we have

Lemma 7. *If , then is locally projectively flat.*

*Proof. *If , then
Contracting (16) with , we get
Let , then
Thus, is projectively flat.

Theorem 8. *Let be a -metric on an open subset . is locally projectively flat if and only if the following conditions hold:*(1)* is parallel with respect to .*(2)* is locally projectively flat; that is, is of constant curvature.*

*Proof. *If is projectively flat. First, we rewrite (12) as a polynomial in and , which is linear in . This gives
Contracting (19) with yields
Namely,
We have
Plugging (22) into (21) yields
Replacing with , we get
Equations (23)-(24) yield
Namely,
Note that is not divisible by and is not divisible by . Thus is divisible by and is divisible by . Therefore, there are two scalar functions and such that
Then (26) becomes
Plugging (27) and (28) into (23) yields
Note that is not divisible by and . Thus ; then and
Thus is closed and parallel with respect to . By Lemma 7, is locally projectively flat.

Conversely, if is parallel with respect to and is locally projectively flat, by Lemma 5 we get , where is -homogeneous function of degree one; thus is locally projectively flat.

Lemma 9. *Suppose that is locally projectively flat with constant flag curvature ; then .*

*Proof. *By Theorem 8, ; then by (8), the equation yields
Replacing with yields
From (31) + (32), we get
Using Taylor expansion of and , we get
Because is not divisible by , then .

Proposition 10. *Let be projectively flat with zero flag curvature; then is flat metric and is parallel. In this case is locally Minkowskian.*

*Proof. *By Theorem 8, is parallel. By Lemma 9, has zero flag curvature; thus has zero sectional curvature. Thus is flat metric; that is, is locally isometric to the Euclidean metric, so . Thus ; that is, is locally Minkowskian.

#### 4. Douglas Metric

In this section, we consider and determine under which conditions Finsler metric is a Douglas metric.

Let It is easy to verify that is well-defined tensor on slit tangent bundle . We call the Douglas curvature. A Finsler metric with is called a Douglas metric. In particular, Riemannian metrics are Douglas metrics. The Douglas tensor is a non-Riemannian projective invariant; namely, if two Finsler metrics and are projectively equivalent, , where is positively -homogeneous of degree one; then the Douglas tensor of is the same as that of . The notion of Douglas metric was proposed by Bácsó and Matsumoto as a generalization of Berwald metrics [5].

It is easy to verify the following lemma.

Lemma 11. *A Finsle metric is a Douglas metric if and only if
**
for some set of local functions .*

For an -metric, the Douglas tensor is determined by where

Lemma 12 (see [8]). *Suppose that is not constant. If an -metric , , is a Douglas metric on an open subset and , then is closed.*

Theorem 13. *Suppose that is a -metric on an open subset . The Finsler metric is a Douglas metric in if and only if is parallel with respect to .*

*Proof. *Let be a Douglas metric. It is easy to verify that is not constant; then by Lemma 12, is closed. Thus and . By Lemma 11 and (37), we have
where are some set of local scalar functions and
We rewrite (39) as a polynomial in and , which is linear in . This gives
Namely,
Contracting (42) with yields
where .

In the following proof we consider the following four cases.*Case 1*. Suppose and are divisible by ; then there are two scalar functions and such that
Plugging (44) into (43), we get
where
Because is polynomial in , from (45), we obtain
By (47), we get
Because is not divisible by and , we get . is not divisible by ; we get
By , we get . Thus ; that is, is parallel with respect to .*Case 2*. Suppose that is not divisible by , but is divisible by ; namely, there is a scalar function such that
Plugging (50) into (43) yields
where
Because is polynomial in , from (51), we obtain
From (54), we have
Because is not divisible by , must be divisible by . This contradicts our assumption. Hence, this case is impossible.*Case 3*. Suppose that is not divisible by , but is divisible by ; namely, there is a scalar function such that
Plugging (56) into (43) yields
where
Because is polynomial in , from (57), we obtain
From (60), we have
Because is not divisible by , then must be divisible by . This contradicts our assumption. Hence, this case is impossible, too.*Case 4*. Suppose that and are not divisible by . From (43), we have
where
Because is polynomial in , from (62), we get
We get
Because and are not divisible by , and are divisible by ; hence there are two scalar functions and such that
Let
We have ; then, we get
From (66) and (68), we obtain
Thus and are divisible by . This contradicts our assumption; hence, this case is impossible too. So only Case 1 is possible and in this case we proved that ; that is, is parallel with respect to .

Conversely, if is parallel with respect to , by Lemma 5, we get . Because is a Riemannian metric then .

By Theorems 1 and 2, we have following Corollary.

Corollary 14. *Let be a -metric. Then is locally projectively flat if and only if it is Douglas metric and is locally projectively flat Riemannian metric.*

By Theorems 1 and 2 and Lemma 5, we have following corollary.

Corollary 15. *Let be a -metric. If is a Douglas metric or projectively flat then the following properties hold:*(1)* is a Landsberg metric;*(2)* is a Berwald metric.*

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The problem “*On the class of **-metrics*” was suggested by Dr. A. Tayebi to the author. The author wants to thank Dr. A. Tayebi for encouragement and valuable suggestion.