# On the Class of(

Authors: Ahmad Alimohammadi
Journal: ISRN Geometry
 (2014)

---

## Abstract

We study a class of Finsler metrics in the formF
=
α
+
β
q
/
α
q
-
1, where α is a Riemannian metric, β is a 1-form, and 1
<
q
<
2
.
F is called (
q
,
α
,
β
)-metrics. We find the necessary and sufficient conditions under which the class of (
q
,
α
,
β
)-metrics is locally projectively flat and Douglas metrics, respectively.


---

## Body

## 1. Introduction

It is Hilbert’s Fourth Problem in the regular case to study and characterize Finsler metrics on an open domain𝒰
⊂
ℝ
n whose geodesics are straight lines. Finsler metrics with this property are called projectively flat metrics. It is easy to see that a Finsler metric F
=
F
(
x
,
y
) on an open subset 𝒰
⊂
ℝ
n is projectively flat if and only if the spray coefficients are in the form G
i
=
P
y
i, where P is a positively homogeneous function of degree one in y. In 1903 G. Hamel found a system of partial differential equations that characterize projectively flat Finsler metrics F
=
F
(
x
,
y
) on an open subset 𝒰
⊂
ℝ
n. That is,
(1)
F
x
k
y
l
y
k
-
F
x
l
=
0
.
A natural problem is to find projectively flat metrics by solving (1). The Beltrami Theorem tells us that a Riemannian metric α
=
a
i
j
(
x
)
y
i
y
j 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(
α
,
β
)-metrics. An (
α
,
β
)-metric is defined by F
:
=
α
ϕ
(
s
), s
=
β
/
α, where ϕ
=
ϕ
(
s
) is a C
∞ positive scalar function on (
-
b
0
,
b
0
) with certain regularity, α
=
a
i
j
(
x
)
y
i
y
j is a Riemannian metric, and β
=
b
i
(
x
)
y
i is a 1-form on a manifold M. The simplest (
α
,
β
)-metric is Randers metric F
=
α
+
β. In [1]; it is proved that a Randers metric F
=
α
+
β on a manifold is locally projectively flat if and only if α is projectively flat and β is closed. The other important (
α
,
β
)-metric is Berwald metric F
=
(
α
+
β
)
2
/
α on a manifold M. In [2], Shen and Yildirim find necessary and sufficient conditions under which the Berwald metric F
=
(
α
+
β
)
2
/
α is projectively flat. They proved that the Berwald metric F is projectively flat if and only if the following conditions hold:(1)
b
i
∣
j
=
τ
{
(
1
+
2
b
2
)
-
3
b
i
b
j
},
(2)
the spray coefficientsG
α
i of α are in the form G
α
i
=
θ
y
i
-
τ
α
2
b
i,where b
:
=
∥
β
∥
α, b
i
∣
j denote the covariant derivatives of β with respect to α, τ
=
τ
(
x
) is a scalar function, and θ
=
θ
i
y
i is a 1-form on M. Also they determined local structure of F with constant flag curvature. In [3, 4], some other special (
α
,
β
)-metrics are studied.In this paper, we consider a special subclass of(
α
,
β
)-metrics in the following form:
(2)
F
=
(
α
+
β
)
q
α
q
-
1
,
where ϕ
=
(
1
+
s
)
q and 1
<
q
<
2. We call it (
q
,
α
,
β
)-metric. We prove the following.Theorem 1.
LetF
=
(
α
+
β
)
q
/
α
q
-
1
(
1
<
q
<
2
) be a (
q
,
α
,
β
)-metric on an open subset 𝒰
⊂
ℝ
n
(
n
≥
3
). F is locally projectively flat if and only if the following conditions hold:(1)
β is parallel with respect to α;
(2)
α is locally projectively flat; that is, α is of constant curvature.It is said that a Finsler metric is trivial if it satisfies the conclusion of Theorem1. Thus the above theorem tells us that in the class of (
q
,
α
,
β
)-metrics, where 1
<
q
<
2, 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(3)
B
j
k
l
i
:
=
∂
3
G
i
∂
y
j
∂
y
k
∂
y
l
,
L
j
k
l
:
=
-
1
2
F
F
y
i
∂
3
G
i
∂
y
j
∂
y
k
∂
y
l
.
Finsler metrics with B
j
k
l
i
=
0 and L
j
k
l
=
0 are called Berwald and Landsberg metrics, respectively. Clearly every Berwald metric is Landsberg, but the converse is not necessarily true.Two Finsler metricsF and F
- on a manifold M are said to be projectively equivalent if they have the same geodesics as point sets. Projective Finsler Geometry studies these metrics. Douglas curvature 𝒟 is an important projective invariant in projective Finsler Geometry. A Finsler metric is called Douglas metric if 𝒟
=
0. 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 F
=
α
+
β is a Douglas metric if and only if β is closed. For another example a Matsumoto metric F
=
α
2
/
(
α
-
β
) and the exponential metric F
=
α
exp
⁡
(
β
/
α
)
+
ε
β are Douglas metrics if and only if β is parallel with respect to α. In this case both of metrics are Berwald metrics, [4, 6, 7]. In [8], it is proved that the Berwald metric F
=
(
α
+
β
)
2
/
α is a Douglas metric if and only if
(4)
b
i
∣
j
=
τ
{
(
1
+
2
b
2
)
a
i
j
-
3
b
i
b
j
}
,
where τ
=
τ
(
x
) is scalar function.In this paper, we are going to find the conditions under which, on a manifoldM of dimension n
≥
3, the (
q
,
α
,
β
)-metric F
=
(
α
+
β
)
q
/
α
q
-
1 is a Douglas metric. More precisely, we prove following.Theorem 2.
LetF
=
(
α
+
β
)
q
/
α
q
-
1
(
1
<
q
<
2
) be a (
q
,
α
,
β
)-metric on a manifold M of dimension n
≥
3. The Finsler metric F is a Douglas metric if and only if β is parallel with respect to α.It is known that a locally projectively flat Finsler metric is a Douglas metric, but the converse is not necessarily true. By Theorems1 and 2, if F
=
(
α
+
β
)
q
/
α
q
-
1
(
1
<
q
<
2
) is a Douglas metric and α is a locally projectively flat Riemannian metric, then F is a projectively flat Finsler metric.Corollary 3.
LetF
=
(
α
+
β
)
q
/
α
q
-
1
(
1
<
q
<
2
) be a (
q
,
α
,
β
)-metric on a manifold M of dimension n
≥
3. Then F is locally projectively flat if and only if it is Douglas metric and α is locally projectively flat Riemannian metric.Corollary 4.
LetF
=
(
α
+
β
)
q
/
α
q
-
1
(
1
<
q
<
2
) be a (
q
,
α
,
β
)-metric on a manifold M of dimension n
≥
3. If F is a Douglas metric or projectively flat, then the following properties hold:(1)
F is a Landsberg metric;
(2)
F is a Berwald metric.


## 2. Preliminaries

By definition, an(
α
,
β
)-metric is a Finsler metric expressed in the following form:
(5)
F
=
α
ϕ
(
s
)
,
s
=
β
α
,
where α
=
a
i
j
(
x
)
y
i
y
j is a Riemannian metric and β
=
b
i
(
x
)
y
i is a 1-form with b
:
=
∥
β
x
∥
<
b
0, x
∈
M. The function ϕ
=
ϕ
(
s
) is a C
∞ positive function on an open interval (
-
b
0
,
b
0
) satisfying
(6)
ϕ
(
0
)
=
1
,
ϕ
(
s
)
-
s
ϕ
′
(
s
)
+
(
b
2
-
s
2
)
ϕ
′′
(
s
)
>
0
,
ϕ
(
0
)
=
1
,
ϕ
(
s
)
-
s
ϕ
′
(
s
)
+
(
b
2
-
)
h
|
s
|
≤
b
<
b
0
.
Let G
i and G
α
i denote the spray coefficients of F and α, respectively, given by
(7)
G
i
=
1
4
g
i
j
{
[
F
2
]
x
k
y
l
y
k
-
[
F
2
]
x
l
}
,
G
α
i
=
1
4
a
i
j
{
[
α
2
]
x
k
y
l
y
k
-
[
α
2
]
x
l
}
,
where g
i
j
=
(
g
i
j
)
-
1 and a
i
j
=
(
a
i
j
)
-
1.A Finsler metricF
=
F
(
x
,
y
) on an open domain 𝒰
⊂
ℝ
n is said to be projectively flat in 𝒰 if all geodesics are straight lines. This is equivalent to G
i
=
P
(
x
,
y
)
y
i, where P
=
F
x
k
y
k
/
2
F is a 1-homogeneous function on T
M
0. In this case F is of scalar curvature with flag curvature
(8)
K
=
P
2
-
P
x
k
y
k
F
2
.
Let b
i
∣
j denote the coefficients of the covariant derivative of β with respect to α and define
(9)
r
i
j
=
1
2
(
b
i
∣
j
+
b
j
∣
i
)
,
r
00
=
r
i
j
y
i
y
j
,
r
j
=
b
i
r
i
j
,
r
0
=
r
j
y
j
,
s
i
j
=
1
2
(
b
i
∣
j
-
b
j
∣
i
)
,
s
j
i
=
a
i
m
s
m
j
,
s
0
i
=
s
j
i
y
j
,
s
j
=
b
i
s
i
j
,
s
0
=
s
j
y
j
,
where b
i
=
a
i
j
b
j and b
2
=
b
i
b
i. Clearly, β is closed if and only if s
i
j
=
0. A 1-form β is said to be parallel with respect to α if b
i
∣
j
=
0 and an (
α
,
β
)-metric F is said to be trivial if r
i
j
=
s
i
j
=
0. We have the following.Lemma 5.
The geodesic coefficients ofG
i are related to G
α
i by
(10)
G
i
=
G
α
i
+
α
Q
s
0
i
+
J
(
-
2
α
Q
s
0
+
r
00
)
y
i
α
+
H
(
-
2
α
Q
s
0
+
r
00
)
(
b
i
-
s
y
i
α
)
,
where
(11)
Q
:
=
ϕ
′
ϕ
-
s
ϕ
′
,
J
:
=
(
ϕ
-
s
ϕ
′
)
ϕ
′
2
ϕ
[
ϕ
-
s
ϕ
′
+
(
b
2
-
s
2
)
ϕ
′′
]
,
H
:
=
ϕ
′′
2
[
ϕ
-
s
ϕ
′
+
(
b
2
-
s
2
)
ϕ
′′
]
.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 α is projectively flat (
G
α
i
=
η
y
i
) and β is parallel with respect to α (r
i
j
=
0
,
s
i
j
=
0), then G
i
=
G
α
i
=
η
y
i. Thus F
=
α
ϕ
(
β
/
α
) is a projectively flat Berwald metric. If K
≠
0, then F is a Riemannian metric by Numata’s Theorem [11]. If K
=
0, then F is locally Minkowskian.Lemma 6 (see [2]).
An(
α
,
β
)-metric F
=
α
ϕ
(
s
), where s
=
β
/
α, is projectively flat on an open subset 𝒰
⊂
ℝ
n if and only if
(12)
(
a
m
l
α
2
-
y
m
y
l
)
G
α
m
+
α
3
Q
s
l
0
+
H
α
(
-
2
α
Q
s
0
+
r
00
)
(
b
l
α
-
s
y
l
)
=
0
,
where y
m
=
a
i
m
y
i and s
l
o
=
s
l
k
y
k.


## 3.(
q
,
α
,
β
)-Metrics

In this section, we consider a special subclass of(
α
,
β
)-metrics in the following form:
(13)
F
=
(
α
+
β
)
q
α
q
-
1
.Letϕ
:
(
-
1,1
)
→
ℝ be a C
∞ function such that ϕ
(
s
)
=
(
1
+
s
)
q, 1
≤
q
≤
2 and b
:
=
∥
β
x
∥
<
1. It is easy to see that
(14)
ϕ
′
=
q
(
1
+
s
)
q
-
1
,
ϕ
′′
=
q
(
q
-
1
)
(
1
+
s
)
q
-
2
>
0
.
Since ϕ
(
s
)
=
(
1
+
s
)
q
>
0 and |
s
|
<
1, then ϕ
-
s
ϕ
′
=
(
1
+
s
)
q
-
1
{
1
+
s
(
1
-
q
)
}
>
0. Thus F
=
(
α
+
β
)
q
/
α
q
-
1 is a Finsler metric. We call it (
q
,
α
,
β
)-metric. When q
=
1 or q
=
2, F becomes Randers metric and Berwald metric, respectively. In this paper we assume that 1
<
q
<
2.By Lemma5, we have
(15)
Q
=
q
1
+
s
(
1
-
q
)
,
J
=
q
(
1
+
s
(
1
-
q
)
)
2
[
s
2
(
1
-
q
2
)
+
s
(
2
-
q
)
+
1
+
b
2
q
(
q
-
1
)
]
,
H
=
q
(
q
-
1
)
2
[
s
2
(
1
-
q
2
)
+
s
(
2
-
q
)
+
1
+
b
2
q
(
q
-
1
)
]
.Lemma 7.
If(
a
m
l
α
2
-
y
m
y
l
)
G
α
m
=
0, then α is locally projectively flat.Proof.
If(
a
m
l
α
2
-
y
m
y
l
)
G
α
m
=
0, then
(16)
a
m
l
α
2
G
α
m
=
y
m
y
l
G
α
m
.
Contracting (16) with a
i
l, we get
(17)
α
2
G
α
i
=
y
m
y
i
G
α
m
.
Let P
:
=
(
y
m
/
α
2
)
G
α
m, then
(18)
G
α
i
=
P
y
i
.
Thus, α is projectively flat.Theorem 8.
LetF
=
(
α
+
β
)
q
/
α
q
-
1
(
1
<
q
<
2
) be a (
q
,
α
,
β
)-metric on an open subset 𝒰
⊂
ℝ
n
(
n
≥
3
). F is locally projectively flat if and only if the following conditions hold:(1)
β is parallel with respect to α.
(2)
α is locally projectively flat; that is, α is of constant curvature.Proof.
IfF is projectively flat. First, we rewrite (12) as a polynomial in y
i and α, which is linear in α. This gives
(19)
2
[
α
2
(
1
+
b
2
q
(
q
-
1
)
)
+
α
β
(
2
-
q
)
+
β
2
(
1
-
q
2
)
]
×
[
α
+
β
(
1
-
q
)
]
[
a
l
m
α
2
-
y
m
y
l
]
G
α
m
+
2
α
4
q
[
α
2
(
1
+
b
2
q
(
q
-
1
)
)
+
α
β
(
2
-
q
)
+
2
α
4
q
w
w
+
β
2
(
1
-
q
2
)
]
s
l
0
+
α
2
q
(
q
-
1
)
×
[
-
2
α
2
q
s
0
+
r
00
(
α
+
β
(
1
-
q
)
)
]
[
b
l
α
2
-
β
y
l
]
=
0
.
Contracting (19) with b
l yields
(20)
2
[
α
2
(
1
+
b
2
q
(
q
-
1
)
)
+
α
β
(
2
-
q
)
+
β
2
(
1
-
q
2
)
]
×
[
α
+
β
(
1
-
q
)
]
[
b
m
α
2
-
y
m
β
]
G
α
m
+
2
α
4
q
[
α
2
(
1
+
b
2
q
(
q
-
1
)
)
+
α
β
(
2
-
q
)
+
2
α
4
q
w
w
+
β
2
(
1
-
q
2
)
]
s
0
+
α
2
q
(
q
-
1
)
×
[
-
2
α
2
q
s
0
+
r
00
(
α
+
β
(
1
-
q
)
)
]
[
b
2
α
2
-
β
2
]
=
0
.
Namely,
(21)
2
[
α
2
(
1
+
b
2
q
(
q
-
1
)
)
+
α
β
(
2
-
q
)
+
β
2
(
1
-
q
2
)
]
×
[
α
+
β
(
1
-
q
)
]
[
b
m
α
2
-
y
m
β
]
G
α
m
+
2
α
4
q
[
α
2
+
α
β
(
2
-
q
)
+
β
2
(
1
-
q
)
]
s
0
+
α
2
q
(
q
-
1
)
r
00
[
α
+
β
(
1
-
q
)
]
[
b
2
α
2
-
β
2
]
=
0
.
We have
(22)
[
α
2
+
α
β
(
2
-
q
)
+
β
2
(
1
-
q
)
]
=
[
α
+
β
(
1
-
q
)
]
[
α
+
β
]
.
Plugging (22) into (21) yields
(23)
2
[
α
2
(
1
+
b
2
q
(
q
-
1
)
)
+
α
β
(
2
-
q
)
+
β
2
(
1
-
q
2
)
]
×
[
b
m
α
2
-
y
m
β
]
G
α
m
+
2
α
4
q
[
α
+
β
]
s
0
+
α
2
q
(
q
-
1
)
r
00
[
b
2
α
2
-
β
2
]
=
0
.
Replacing y with -
y, we get
(24)
2
[
α
2
(
1
+
b
2
q
(
q
-
1
)
)
-
α
β
(
2
-
q
)
+
β
2
(
1
-
q
2
)
]
×
[
b
m
α
2
-
y
m
β
]
G
α
m
-
2
α
4
q
[
α
-
β
]
s
0
+
α
2
q
(
q
-
1
)
r
00
[
b
2
α
2
-
β
2
]
=
0
.
Equations (23)-(24) yield
(25)
4
α
β
(
2
-
q
)
[
b
m
α
2
-
y
m
β
]
G
α
m
+
4
α
5
q
s
0
=
0
.
Namely,
(26)
β
(
2
-
q
)
[
b
m
α
2
-
y
m
β
]
G
α
m
=
-
α
4
q
s
0
.
Note that  α
4 is not divisible by β and β is not divisible by α
4. Thus [
b
m
α
2
-
y
m
β
]
G
α
m is divisible by α
4 and s
0 is divisible by β. Therefore, there are two scalar functions τ
=
τ
(
x
) and λ
=
λ
(
x
) such that
(27)
[
b
m
α
2
-
y
m
β
]
G
α
m
=
τ
α
4
,
s
0
=
λ
β
.
Then (26) becomes
(28)
(
2
-
q
)
τ
=
-
λ
q
.
Plugging (27) and (28) into (23) yields
(29)
2
α
2
[
α
2
(
1
+
b
2
q
(
q
-
1
)
)
-
β
2
(
q
2
-
q
+
1
)
]
τ
=
-
q
(
q
-
1
)
r
00
[
b
2
α
2
-
β
2
]
.
Note that [
b
2
α
2
-
β
2
] is not divisible by α
2 and [
α
2
(
1
+
b
2
q
(
q
-
1
)
)
-
β
2
(
q
2
-
q
+
1
)
]. Thus τ
=
0; then λ
=
0 and
(30)
r
00
=
0
,
s
0
=
0
,
[
b
m
α
2
-
y
m
β
]
G
α
m
=
0
.
Thus β is closed and parallel with respect to α. By Lemma 7, α is locally projectively flat.
Conversely, ifβ is parallel with respect to α and α is locally projectively flat, by Lemma 5 we get G
i
=
G
α
i
=
η
y
i, where η
=
η
(
x
,
y
) is y-homogeneous function of degree one; thus F is locally projectively flat.Lemma 9.
Suppose thatF
=
(
α
+
β
)
q
/
α
q
-
1
(
1
<
q
<
2
) is locally projectively flat with constant flag curvature K
=
λ; then λ
=
0.Proof.
By Theorem8, G
i
=
η
y
i; then by (8), the equation K
=
λ yields
(31)
α
2
(
1
+
β
α
)
2
q
λ
=
η
2
-
η
x
k
y
k
.
Replacing y with -
y yields
(32)
α
2
(
1
-
β
α
)
2
q
λ
=
η
2
-
η
x
k
y
k
.
From (31) + (32), we get
(33)
α
2
λ
{
(
1
+
β
α
)
2
q
+
(
1
-
β
α
)
2
q
}
=
2
(
η
2
-
η
x
k
y
k
)
.
Using Taylor expansion of (
1
+
β
/
α
)
2
q and (
1
-
β
/
α
)
2
q, we get
(34)
2
α
2
λ
=
2
(
η
2
-
η
x
k
y
k
)
.
Because α
2 is not divisible by (
η
2
-
η
x
k
y
k
), then λ
=
0.Proposition 10.
LetF
=
(
α
+
β
)
q
/
α
q
-
1 be projectively flat with zero flag curvature; then α is flat metric and β is parallel. In this case F is locally Minkowskian.Proof.
By Theorem8, β is parallel. By Lemma 9, F has zero flag curvature; thus α has zero sectional curvature. Thus α is flat metric; that is, α is locally isometric to the Euclidean metric, so G
α
i
=
0. Thus G
i
=
0; that is, F is locally Minkowskian.


## 4. Douglas Metric

In this section, we considerF
=
(
α
+
β
)
q
/
α
q
-
1 and determine under which conditions Finsler metric F is a Douglas metric.Let(35)
D
j
k
l
i
:
=
∂
3
∂
y
j
∂
y
k
∂
y
l
(
G
i
-
1
n
+
1
∂
G
m
∂
y
m
y
i
)
.
It is easy to verify that 𝒟
=
D
j
k
l
i
d
x
j
⊗
∂
i
⊗
d
x
k
⊗
d
x
l is well-defined tensor on slit tangent bundle T
M
0. We call 𝒟 the Douglas curvature. A Finsler metric with 𝒟
=
0 is called a Douglas metric. In particular, Riemannian metrics are Douglas metrics. The Douglas tensor 𝒟 is a non-Riemannian projective invariant; namely, if two Finsler metrics F and F
¯ are projectively equivalent, G
i
=
G
¯
i
+
P
y
i, where P
=
P
(
x
,
y
) is positively y-homogeneous of degree one; then the Douglas tensor of F is the same as that of F
¯. 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 metricF is a Douglas metric if and only if
(36)
G
i
-
1
n
+
1
∂
G
m
∂
y
m
y
i
=
γ
j
k
i
(
x
)
y
j
y
k
,
for some set of local functions γ
j
k
i
(
x
).For an(
α
,
β
)-metric, the Douglas tensor is determined by
(37)
D
j
k
l
i
:
=
∂
3
∂
y
j
∂
y
k
∂
y
l
(
T
i
-
1
n
+
1
∂
T
m
∂
y
m
y
i
)
,
where
(38)
T
i
:
=
α
Q
s
0
i
+
H
(
r
00
-
2
α
Q
s
0
)
b
i
,
∂
T
m
∂
y
m
=
Q
′
s
0
+
H
′
α
-
1
(
b
2
-
s
2
)
(
r
00
-
2
α
Q
s
0
)
+
2
H
[
r
0
-
Q
′
(
b
2
-
s
2
)
s
0
-
Q
s
s
0
]
.Lemma 12 (see [8]).
Suppose thatQ
/
s is not constant. If an (
α
,
β
)-metric F
=
α
ϕ
(
s
), s
=
β
/
α, is a Douglas metric on an open subset 𝒰
⊆
ℝ
n
(
n
≥
3
) and b
=
∥
β
x
∥
α
≠
0, then β is closed.Theorem 13.
Suppose thatF
=
(
α
+
β
)
q
/
α
q
-
1
(
1
<
q
<
2
) is a (
q
,
α
,
β
)-metric on an open subset 𝒰
⊂
ℝ
n
(
n
≥
3
). The Finsler metric F is a Douglas metric in 𝒰 if and only if β is parallel with respect to α.Proof.
LetF be a Douglas metric. It is easy to verify that Q
/
s is not constant; then by Lemma 12, β is closed. Thus s
0
=
0 and s
0
i
=
0. By Lemma 11 and (37), we have
(39)
T
i
-
1
n
+
1
∂
T
m
∂
y
m
y
i
=
γ
j
k
i
(
x
)
y
j
y
k
,
where γ
j
k
i
(
x
) are some set of local scalar functions and
(40)
T
i
=
H
r
00
b
i
,
∂
T
m
∂
y
m
=
H
′
α
-
1
(
b
2
-
s
2
)
r
00
+
2
H
r
0
,
H
′
=
-
q
(
q
-
1
)
[
2
s
(
1
-
q
2
)
+
(
2
-
q
)
]
2
[
s
2
(
1
-
q
2
)
+
s
(
2
-
q
)
+
1
+
b
2
q
(
q
-
1
)
]
2
.
We rewrite (39) as a polynomial in y
i and α, which is linear in α. This gives
(41)
q
(
q
-
1
)
α
2
2
[
β
2
(
1
-
q
2
)
+
α
β
(
2
-
q
)
+
α
2
(
1
+
b
2
q
(
q
-
1
)
)
]
r
o
o
b
i
+
1
n
+
1
×
q
(
q
-
1
)
[
2
β
(
1
-
q
2
)
+
α
(
2
-
q
)
]
2
[
β
2
(
1
-
q
2
)
+
α
β
(
2
-
q
)
+
α
2
(
1
+
b
2
q
(
q
-
1
)
)
]
2
×
(
b
2
α
2
-
β
2
)
r
00
y
i
-
1
n
+
1
×
q
(
q
-
1
)
α
2
[
β
2
(
1
-
q
2
)
+
α
β
(
2
-
q
)
+
α
2
(
1
+
b
2
q
(
q
-
1
)
)
]
r
0
y
i
=
γ
j
k
i
(
x
)
y
j
y
k
.
Namely,
(42)
[
n
+
1
]
q
(
q
-
1
)
α
2
×

[
β
2
(
1
-
q
2
)
+
α
β
(
2
-
q
)
+
α
2
(
1
+
b
2
q
(
q
-
1
)
)
]

r
00
b
i
+
q
(
q
-
1
)
[
2
β
(
1
-
q
2
)
+
α
(
2
-
q
)
]
×
(
b
2
α
2
-
β
2
)
r
00
y
i
-
2
q
(
q
-
1
)
α
2
×
[
β
2
(
1
-
q
2
)
+
α
β
(
2
-
q
)
+
α
2
(
1
+
b
2
q
(
q
-
1
)
)
]
r
0
y
i
=
2
γ
j
k
i
(
x
)
y
j
y
k
[
n
+
1
]
[
β
2
(
1
-
q
2
)
+
α
β
(
2
-
q
)
=
2
γ
j
k
i
(
x
)
y
j
y
k
[
n
+
1
]
w
w
+
α
2
(
1
+
b
2
q
(
q
-
1
)
)
]
2
.
Contracting (42) with b
i yields
(43)
[
n
+
1
]
q
(
q
-
1
)
α
2
×

[
β
2
(
1
-
q
2
)
+
α
β
(
2
-
q
)
+
α
2
(
1
+
b
2
q
(
q
-
1
)
)
]

r
00
b
2
+
q
(
q
-
1
)
[
2
β
(
1
-
q
2
)
+
α
(
2
-
q
)
]
×
(
b
2
α
2
-
β
2
)
r
00
β
-
2
q
(
q
-
1
)
α
2
×
[
β
2
(
1
-
q
2
)
+
α
β
(
2
-
q
)
+
α
2
(
1
+
b
2
q
(
q
-
1
)
)
]
r
0
β
-
T
[
n
+
1
]
[
β
2
(
1
-
q
2
)
+
α
β
(
2
-
q
)
-
T
[
n
+
1
]
w
w
+
α
2
(
1
+
b
2
q
(
q
-
1
)
)
]
2
=
0
,
where T
:
=
2
γ
j
k
i
(
x
)
y
j
y
k
b
i.
In the following proof we consider the following four cases.
Case  1. Suppose r
00 and T are divisible by α
2; then there are two scalar functions λ
=
λ
(
x
) and τ
=
τ
(
x
) such that
(44)
T
=
λ
α
2
,
r
00
=
τ
α
2
,
r
0
=
τ
β
.
Plugging (44) into (43), we get
(45)
A
1
α
6
+
A
2
α
5
β
+
A
3
α
4
β
2
+
A
4
α
3
β
3
+
A
5
α
2
β
4
=
0
,
where
(46)
A
1
=
[
n
+
1
]
q
(
q
-
1
)
[
1
+
b
2
q
(
q
-
1
)
]
b
2
τ
-
[
n
+
1
]
[
1
+
b
2
q
(
q
-
1
)
]
2
λ
,
A
2
=
[
n
+
2
]
q
(
q
-
1
)
(
2
-
q
)
b
2
τ
-
2
[
n
+
1
]
(
2
-
q
)
[
1
+
b
2
q
(
q
-
1
)
]
λ
,
A
3
=
[
n
+
2
]
q
(
q
-
1
)
(
1
-
q
2
)
b
2
τ
-
2
q
(
q
-
1
)
[
1
+
b
2
q
(
q
-
1
)
]
τ
-
[
n
+
1
]
(
2
-
q
)
2
λ
-
2
[
n
+
1
]
×
(
1
-
q
2
)
[
1
+
b
2
q
(
q
-
1
)
]
λ
,
A
4
=
-
3
q
(
q
-
1
)
(
2
-
q
)
τ
-
2
[
n
+
1
]
(
1
-
q
2
)
(
2
-
q
)
λ
,
A
5
=
-
4
q
(
q
-
1
)
(
1
-
q
2
)
τ
-
[
n
+
1
]
(
1
-
q
2
)
2
λ
.
Because α
even is polynomial in y
i, from (45), we obtain
(47)
A
1
α
6
+
A
3
α
4
β
2
+
A
5
α
2
β
4
=
0
,
A
2
α
5
β
+
A
4
α
3
β
3
=
0
.
By (47), we get
(48)
α
2
[
A
1
α
2
+
A
3
β
2
]
=
-
A
5
β
4
,
A
2
α
2
=
-
A
4
β
2
.
Because β
4 is not divisible by α
2 and [
A
1
α
2
+
A
3
β
2
], we get A
5
=
0
,
A
1
α
2
+
A
3
β
2
=
0. β
2 is not divisible by α
2; we get
(49)
A
1
=
A
2
=
A
3
=
A
4
=
0
.
By A
4
=
A
5
=
0, we get τ
=
λ
=
0. Thus r
00
=
r
0
=
T
=
0; that is, β is parallel with respect to α.
Case  2. Suppose that r
00 is not divisible by α
2, but T is divisible by α
2; namely, there is a scalar function λ
=
λ
(
x
) such that
(50)
T
=
λ
α
2
.
Plugging (50) into (43) yields
(51)
B
1
α
6
+
B
2
β
α
5
+
(
B
3
r
00
+
B
4
β
r
0
)
α
4
+
(
B
5
β
r
00
+
B
6
β
2
r
0
+
B
7
β
3
)
α
3
+
(
B
8
β
2
r
00
+
B
9
β
3
r
0
+
B
10
β
4
)
α
2
+
B
11
β
3
r
00
α
+
B
12
β
4
r
00
=
0
,
where
(52)
B
1
=
-
[
n
+
1
]
[
1
+
b
2
q
(
q
-
1
)
]
2
λ
,
B
2
=
-
2
[
n
+
1
]
(
2
-
q
)
[
1
+
b
2
q
(
q
-
1
)
]
λ
,
B
3
=
[
n
+
1
]
q
(
q
-
1
)
[
1
+
b
2
q
(
q
-
1
)
]
b
2
,
B
4
=
-
2
q
(
q
-
1
)
[
1
+
b
2
q
(
q
-
1
)
]
,
B
5
=
[
n
+
2
]
q
(
q
-
1
)
(
2
-
q
)
b
2
,
B
6
=
-
2
q
(
q
-
1
)
(
2
-
q
)
,
B
7
=
-
2
[
n
+
1
]
(
2
-
q
)
(
1
-
q
2
)
λ
,
B
8
=
[
n
+
3
]
q
(
q
-
1
)
(
1
-
q
2
)
b
2
,
B
9
=
-
2
q
(
q
-
1
)
(
1
-
q
2
)
,
B
10
=
-
[
n
+
1
]
(
1
-
q
2
)
2
λ
,
B
11
=
-
q
(
q
-
1
)
(
2
-
q
)
,
B
12
=
-
q
(
q
-
1
)
(
1
-
q
2
)
.
Because α
even is polynomial in y
i, from (51), we obtain
(53)
B
1
α
6
+
(
B
3
r
00
+
B
4
β
r
0
)
α
4
wwii
+
(
B
8
β
2
r
00
+
B
9
β
3
r
0
+
B
10
β
4
)
α
2
wwii
+
B
12
β
4
r
00
=
0
,
(54)
B
2
α
5
β
+
(
B
5
β
r
00
+
B
6
β
2
r
0
+
B
7
β
3
)
α
3
+
B
11
β
3
r
00
α
=
0
.
From (54), we have
(55)
α
2
[
B
2
α
2
+
(
B
5
r
00
+
B
6
β
r
0
+
B
7
β
2
)
]
=
-
B
11
β
2
r
00
.
Because β
2 is not divisible by α
2, r
00 must be divisible by α
2. This contradicts our assumption. Hence, this case is impossible.
Case  3. Suppose that T is not divisible by α
2, but r
00 is divisible by α
2; namely, there is a scalar function τ
=
τ
(
x
) such that
(56)
r
00
=
τ
α
2
,
r
0
=
τ
β
.
Plugging (56) into (43) yields
(57)
C
1
α
6
+
C
2
β
α
5
+
(
C
3
β
2
+
C
4
T
)
α
4
wwii
+
(
C
5
β
3
+
C
6
β
T
)
α
3
+
(
C
7
β
4
+
C
8
β
2
T
)
α
2
wwii
+
C
9
β
3
T
α
+
C
10
β
4
T
=
0
,
where
(58)
C
1
=
[
n
+
1
]
q
(
q
-
1
)
[
1
+
b
2
q
(
q
-
1
)
]
b
2
τ
,
C
2
=
[
n
+
2
]
q
(
q
-
1
)
(
2
-
q
)
b
2
τ
,
C
3
=
[
n
+
3
]
q
(
q
-
1
)
(
1
-
q
2
)
b
2
τ
-
2
q
(
q
-
1
)
[
1
+
b
2
q
(
q
-
1
)
]
τ
,
C
4
=
[
n
+
1
]
[
1
+
b
2
q
(
q
-
1
)
]
2
,
C
5
=
-
3
q
(
q
-
1
)
(
2
-
q
)
τ
,
C
6
=
-
2
[
n
+
1
]
(
2
-
q
)
[
1
+
b
2
q
(
q
-
1
)
]
,
C
7
=
-
4
q
(
q
-
1
)
(
1
-
q
2
)
τ
,
C
8
=
-
2
[
n
+
1
]
(
1
-
q
2
)
[
1
+
b
2
q
(
q
-
1
)
]
,
C
9
=
-
2
[
n
+
1
]
(
2
-
q
)
(
1
-
q
2
)
,
C
10
=
[
n
+
1
]
(
1
-
q
2
)
2
.
Because α
even is polynomial in y
i, from (57), we obtain
(59)
C
1
α
6
+
(
C
3
β
2
+
C
4
T
)
α
4
+
(
C
7
β
4
+
C
8
β
2
T
)
α
2
wwii
+
C
10
β
4
T
=
0
,
(60)
C
2
β
α
5
+
(
C
5
β
3
+
C
6
β
T
)
α
3
+
C
9
β
3
T
α
=
0
.
From (60), we have
(61)
α
2
[
C
2
α
2
+
(
C
5
β
2
+
C
6
T
)
]
=
-
C
9
β
2
T
.
Because β
2 is not divisible by α
2, then T must be divisible by α
2. This contradicts our assumption. Hence, this case is impossible, too.
Case  4. Suppose that T and r
00 are not divisible by α
2. From (43), we have
(62)
(
D
1
r
00
+
D
2
r
0
β
+
D
3
T
)
α
4
+
(
D
4
r
00
β
+
D
5
r
0
β
2
+
D
6
β
T
)
α
3
+
(
D
7
r
00
β
2
+
D
8
r
0
β
3
+
D
9
β
2
T
)
α
2
+
(
D
10
r
00
β
3
+
D
11
β
3
T
)
α
+
(
D
12
r
00
β
4
+
D
13
β
4
T
)
=
0
,
where
(63)
D
1
=
[
n
+
1
]
q
(
q
-
1
)
[
1
+
b
2
q
(
q
-
1
)
]
b
2
,
D
2
=
-
2
q
(
q
-
1
)
[
1
+
b
2
q
(
q
-
1
)
]
,
D
3
=
-
[
n
+
1
]
[
1
+
b
2
q
(
q
-
1
)
]
2
,
D
4
=
[
n
+
2
]
q
(
q
-
1
)
(
2
-
q
)
b
2
,
D
5
=
-
2
q
(
q
-
1
)
(
2
-
q
)
,
D
6
=
-
2
[
n
+
1
]
[
1
+
b
2
q
(
q
-
1
)
]
,
D
7
=
[
n
+
3
]
q
(
q
-
1
)
(
1
-
q
2
)
,
D
8
=
-
2
q
(
q
-
1
)
(
1
-
q
2
)
,
D
9
=
-
[
n
+
1
]
[
2
(
1
-
q
2
)
[
1
+
b
2
q
(
q
-
1
)
]
+
(
2
-
q
)
2
]
,
D
10
=
-
q
(
q
-
1
)
(
2
-
q
)
,
D
11
=
-
2
[
n
+
1
]
(
2
-
q
)
(
1
-
q
2
)
,
D
12
=
-
2
q
(
q
-
1
)
(
1
-
q
2
)
,
D
13
=
-
[
n
+
1
]
(
1
-
q
2
)
2
.
Because α
even is polynomial in y
i, from (62), we get
(64)
(
D
1
r
00
+
D
2
r
0
β
+
D
3
T
)
α
4
+
(
D
7
r
00
β
2
+
D
8
r
0
β
3
+
D
9
β
2
T
)
α
2
+
(
D
12
r
00
β
4
+
D
13
β
4
T
)
=
0
,
(
D
4
r
00
β
+
D
5
r
0
β
2
+
D
6
β
T
)
α
3
+
(
D
10
r
00
β
3
+
D
11
β
3
T
)
α
=
0
.
We get
(65)
α
2
[
(
D
1
r
00
+
D
2
r
0
β
+
D
3
T
)
α
2
α
2
[
+
(
D
7
r
00
β
2
+
D
8
r
0
β
3
+
D
9
β
2
T
)
]
=
-
β
4
[
D
12
r
00
+
D
13
T
]
,
α
2
[
D
4
r
00
+
D
5
r
0
β
+
D
6
T
]
=
-
β
2
[
D
10
r
00
+
D
11
T
]
.
Because β
4 and β
2 are not divisible by α
2, [
D
12
r
00
+
D
13
T
] and [
D
10
r
00
+
D
11
T
] are divisible by α
2; hence there are two scalar functions K
1
=
K
1
(
x
) and K
2
=
K
2
(
x
) such that
(66)
D
12
r
00
+
D
13
T
=
K
1
α
2
,
D
10
r
00
+
D
11
T
=
K
2
α
2
.
Let
(67)
A
:
=
[
D
12
D
13
D
10
D
11
]
.
We have det
⁡
A
=
3
[
n
+
1
]
q
(
q
-
1
)
(
2
-
q
)
(
1
-
q
2
)
2
≠
0; then, we get
(68)
A
-
1
≔
[
-
2
3
q
(
q
-
1
)
(
1
-
q
2
)
1
3
q
(
q
-
1
)
(
2
-
q
)
1
3
[
n
+
1
]
(
1
-
q
2
)
2
-
2
3
[
n
+
1
]
(
2
-
q
)
(
1
-
q
2
)
]
.
From (66) and (68), we obtain
(69)
r
00
=
[
-
2
3
q
(
q
-
1
)
(
1
-
q
2
)
K
1
+
1
3
[
n
+
1
]
(
1
-
q
2
)
2
K
2
]
α
2
,
T
=
[
1
3
q
(
q
-
1
)
(
2
-
q
)
K
1
T
=
[
+
-
2
3
[
n
+
1
]
(
2
-
q
)
(
1
-
q
2
)
K
2
]
α
2
.
Thus r
00 and T are divisible by α
2. This contradicts our assumption; hence, this case is impossible too. So only Case 1 is possible and in this case we proved that r
00
=
0; that is, β is parallel with respect to α.
Conversely, ifβ is parallel with respect to α, by Lemma 5, we get G
i
=
G
α
i. Because α is a Riemannian metric then D
j
k
l
i
=
0.By Theorems1 and 2, we have following Corollary.Corollary 14.
LetF
=
(
α
+
β
)
q
/
α
q
-
1
(
1
<
q
<
2
) be a (
q
,
α
,
β
)-metric. Then F is locally projectively flat if and only if it is Douglas metric and α is locally projectively flat Riemannian metric.By Theorems1 and 2 and Lemma 5, we have following corollary.Corollary 15.
LetF
=
(
α
+
β
)
q
/
α
q
-
1
(
1
<
q
<
2
) be a (
q
,
α
,
β
)-metric. If F is a Douglas metric or projectively flat then the following properties hold:(1)
F is a Landsberg metric;
(2)
F is a Berwald metric.



---

*Source: 596090-2014-03-10.xml*