# A Local Classification of Some Special(

Authors: Hongmei Zhu
Journal: Geometry
 (2014)

---

## Abstract

We classify some special Finsler metrics of constant flag curvature on a manifold of dimensionn
>
2.


---

## Body

## 1. Introduction

One of the important problems in Finsler geometry is to study and characterize Finsler metrics of constant flag curvature, which is the generalization of sectional curvature in Riemannian geometry. The local structure of Finsler metrics of constant flag curvature has been historically mysterious and their classification seems to be far from being solved.The(
α
,
β
)-metrics are an important class of Finsler metrics including Randers metrics as the simplest class. By making use of navigation problem, Bao et al. gave a local classification of Randers metrics with constant flag curvature [1]. Recently, Zhou has classified that the (
α
,
β
)-metrics with constant flag curvature in the following form,
(1)
F
=
(
α
+
β
)
2
α
,
are locally projectively flat [2].Lately, Shen and Zhao have studied projectively flat(
α
,
β
)-metrics
(2)
F
=
α
(
1
+
ε
s
+
2
k
s
2
-
k
2
3
s
4
)
,
s
=
β
α
,
where ε is a constant and k is a nonzero constant, and they proved such projectively flat Finsler metrics with constant flag curvature must be locally Minkowskian [3]. Hence, one natural problem is to consider the classification of such metrics with constant flag curvature. In this paper, we prove the following rigidity result.Theorem 1.
LetF
=
α
+
ε
β
+
(
2
k
β
2
/
α
)
-
(
k
2
β
4
/
3
α
3
), where the 1-form β is nonzero, ε is a constant, and k is a nonzero constant, be an (
α
,
β
)-metric on a manifold of dimension n
>
2. Suppose that F is of constant flag curvature; then it must be locally Minkowskian.


## 2. Preliminaries

LetF be a Finsler metric on an n-dimensional manifold M and G
i the geodesic coefficients of F, which are defined by
(3)
G
i
=
1
4
g
i
l
{
[
F
2
]
x
k
y
l
y
k
-
[
F
2
]
x
l
}
.
For any x
∈
M and y
∈
T
x
M
∖
{
0
}, the Riemann curvature R
y
=
R
j
i
(
∂
/
∂
x
i
)
⊗
d
x
j is defined by
(4)
R
j
i
=
2
∂
G
i
∂
x
j
-
∂
2
G
i
∂
x
k
∂
y
j
y
k
+
2
G
k
∂
2
G
i
∂
y
k
∂
y
j
-
∂
G
i
∂
y
k
∂
G
k
∂
y
j
.(
α
,
β
)-metrics were first introduced by Matsumoto [4]. They are 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. ϕ
=
ϕ
(
s
) is a smooth positive function satisfying
(6)
ϕ
(
s
)
>
0
,
ϕ
(
s
)
-
s
ϕ
′
(
s
)
+
(
b
2
-
s
2
)
ϕ
′
′
(
s
)
>
0
,
kkkkkkkkkkkkkkkkkkkkkkkkik
∀
|
s
|
≤
b
<
b
0
.For any flag(
P
,
y
), where P
=
span
⁡
{
y
,
u
}
⊂
T
x
M, the flag curvature is defined by
(7)
K
(
P
,
y
)
:
=
g
y
(
R
y
(
u
)
,
u
)
g
y
(
y
,
y
)
g
y
(
u
,
u
)
-
[
g
y
(
y
,
u
)
]
2
.
When F is Riemannian, K
(
P
,
y
)
=
K
(
P
) is independent of y
∈
P. It is just the sectional curvature of P in Riemannian geometry. F is said to be of scalar curvature if, for any y
∈
T
p
M, the flag curvature K
(
P
,
y
)
=
K
(
y
) is independent of P containing y
∈
T
p
M that is equivalent to the following system of equations in a local coordinate system (
x
i
,
y
i
) in T
M,
(8)
R
j
i
=
K
F
2
(
δ
j
i
-
F
-
1
F
y
j
y
i
)
.
If K is a constant, then F is said to be of constant flag curvature.Let(9)
r
i
j
:
=
1
2
(
b
i
|
j
+
b
j
|
i
)
,
s
i
j
:
=
1
2
(
b
i
|
j
-
b
j
|
i
)
,
where “|” denotes the covariant derivative with respect to the Levi-Civita connection of α. Clearly β is closed if and only if s
i
j
=
0. Moreover, we denote
(10)
r
j
i
:
=
a
i
k
r
k
j
,
s
j
i
:
=
a
i
k
s
k
j
,
r
00
:
=
r
i
j
y
i
y
j
,
r
i
0
:
=
r
i
j
y
j
,
s
i
:
=
b
j
s
i
j
,
s
0
:
=
s
i
y
i
,
r
:
=
r
i
j
b
i
b
j
,
s
i
0
:
=
s
i
j
y
j
,
r
i
:
=
b
j
r
j
i
,
r
i
:
=
a
i
j
r
j
,
s
i
:
=
a
i
j
s
j
,
where (
a
i
j
)
:
=
(
a
i
j
)
-
1 and b
i
:
=
a
i
j
b
j. Let G
i and α
G
i be the geodesic coefficients of F and α, respectively. Then we have the following.Lemma 2 (see [5]).
For an(
α
,
β
)-metric F
=
α
ϕ
(
s
), s
=
(
β
/
α
), the geodesic coefficients G
i are given by
(11)
G
i
=
G
α
i
+
α
Q
s
0
i
+
Θ
{
r
00
-
2
Q
α
s
0
}
l
i
+
Ψ
{
r
00
-
2
Q
α
s
0
}
b
i
,
where
(12)
Q
=
ϕ
′
ϕ
-
s
ϕ
′
,
Θ
=
ϕ
ϕ
′
-
s
(
ϕ
ϕ
′′
+
ϕ
′
ϕ
′
)
2
ϕ
[
(
ϕ
-
s
ϕ
′
)
+
(
B
-
s
2
)
ϕ
′′
]
,
Ψ
=
ϕ
′′
2
[
(
ϕ
-
s
ϕ
′
)
+
(
B
-
s
2
)
ϕ
′′
]
.
Here l
i
:
=
(
y
i
/
α
) and B
:
=
b
2.Letu
(
s
,
B
)
:
=
-
2
Θ
Q, υ
(
s
,
B
)
:
=
-
2
Ψ
Q; one has the following.Proposition 3 (see [6]).
For any(
α
,
β
)-metric F
=
α
ϕ
(
s
), s
=
(
β
/
α
), the Riemannian curvature is given by
(13)
R
j
i
=
R
α
j
i
+
T
j
i
,
where
(14)
T
j
i
=
Δ
δ
j
i
+
C
11
l
i
l
j
+
C
12
l
i
b
j
+
C
13
l
i
s
j
+
C
14
l
i
r
j
+
C
130
l
i
s
0
j
+
C
140
l
i
r
j
0
-
(
u
α
)
l
i
s
j
|
0
+
(
2
u
α
)
l
i
s
0
|
j
-
(
u
Q
α
2
)
l
i
s
k
s
j
k
+
(
2
Θ
)
l
i
r
00
|
j
-
(
2
Θ
)
l
i
r
j
0
|
0
-
(
2
Q
α
Θ
)
l
i
r
k
0
s
j
k
+
(
4
Q
α
Θ
)
l
i
r
j
k
s
0
k
+
C
21
b
i
l
j
+
C
22
b
i
b
j
+
C
23
b
i
s
j
+
C
24
b
i
r
j
+
C
230
b
i
s
0
j
+
C
240
b
i
r
j
0
-
(
υ
α
)
b
i
s
j
|
0
+
(
2
υ
α
)
b
i
s
0
|
j
-
(
υ
Q
α
2
)
b
i
s
k
s
j
k
+
(
2
Ψ
)
b
i
r
00
|
j
-
(
2
Ψ
)
b
i
r
j
0
|
0
-
(
2
Q
α
Ψ
)
b
i
r
k
0
s
j
k
+
(
4
Q
α
Ψ
)
b
i
r
j
k
s
0
k
+
C
3
s
j
i
+
[
C
31
s
i
l
j
+
C
32
s
i
b
j
+
(
υ
Q
α
2
)
s
i
s
j
+
(
2
Q
α
Ψ
)
s
i
r
j
0
]
+
[
C
310
s
0
i
l
j
+
C
320
s
0
i
b
j
+
C
330
s
0
i
s
j
m
i
+
C
331
s
0
i
s
0
j
+
C
340
s
0
i
r
j
0
]
+
C
4
r
j
i
+
[
C
410
r
0
i
l
j
+
C
420
r
0
i
b
j
k
k
k
k
k
k
k
k
-
(
υ
α
)
r
0
i
s
j
-
(
2
Ψ
)
r
0
i
r
j
0
]
+
[
C
332
s
k
i
s
0
k
l
j
+
C
333
s
k
i
s
0
k
b
j
-
(
Q
2
α
2
)
s
k
i
s
j
k
]
+
(
2
Q
α
)
s
0
|
j
i
-
(
Q
α
)
s
j
|
0
i
+
[
C
311
s
0
|
0
i
l
j
-
Q
s
s
0
|
0
i
b
j
]
,
Δ
=
r
00
2
α
2
[
2
Θ
s
Ψ
(
B
-
s
2
)
-
Θ
s
+
Θ
2
-
2
s
Θ
Ψ
]
+
r
00
s
0
α
[
2
(
υ
Θ
s
+
u
s
Ψ
)
(
B
-
s
2
)
+
2
u
Θ
mmmm
-
2
υ
s
Θ
-
u
s
+
2
Q
Θ
s
-
2
Θ
B
2
(
υ
Θ
s
+
u
s
Ψ
)
(
B
-
s
2
)
]
+
2
r
00
r
0
α
(
2
Θ
Ψ
-
Θ
B
)
-
r
00
|
0
α
Θ
+
s
0
2
[
-
2
u
B
+
u
2
+
2
u
s
υ
(
B
-
s
2
)
+
2
Q
u
s
]
+
2
r
0
s
0
(
-
u
B
+
2
υ
Θ
)
+
2
u
Q
α
(
s
k
s
0
k
)
+
4
Q
Θ
(
r
k
0
s
0
k
)
-
u
s
0
|
0
,
C
11
=
r
00
2
α
2
[
s
2
Θ
Ψ
s
(
s
Θ
s
Ψ
s
-
4
Θ
s
Ψ
-
2
s
Θ
s
s
Ψ
)
(
B
-
s
2
)
+
6
s
Θ
Ψ
mm
+
Θ
2
+
6
s
2
Θ
s
Ψ
+
s
Θ
Θ
s
+
s
Θ
s
s
+
2
Θ
s
m
m
-
s
2
Θ
Ψ
s
]
+
r
00
s
0
α
[
(
B
-
s
2
)
(
-
2
s
Ψ
u
s
s
+
s
u
s
Ψ
s
+
s
υ
s
Θ
s
-
2
s
υ
Θ
s
s
m
1111
mi
-
5
υ
Θ
s
-
2
Ψ
u
s
)
(
B
-
s
2
)
+
2
s
Θ
B
s
+
Θ
u
mmmm
+
s
u
Θ
s
+
4
s
2
Ψ
u
s
-
5
Q
Θ
s
+
s
u
s
s
mmmm
+
s
Θ
u
s
+
s
Q
s
Θ
s
-
2
s
Q
s
Θ
s
s
-
s
2
Θ
υ
s
mmmm
+
6
s
2
υ
Θ
s
+
7
s
υ
Θ
+
u
s
+
2
Θ
B
]
+
2
r
00
r
0
α
(
Θ
B
+
s
Θ
B
s
+
s
Θ
Ψ
s
-
2
Θ
Ψ
-
2
s
Θ
s
Ψ
)
+
r
00
|
0
α
(
Θ
+
s
Θ
s
)
+
s
0
2
[
(
B
-
s
2
)
(
s
u
s
υ
s
-
3
u
s
υ
-
2
s
u
s
s
υ
)
miiiii
×
(
B
-
s
2
)
+
4
s
2
u
s
υ
-
2
Q
u
s
s
s
miimi
+
s
u
u
s
-
3
Q
u
s
+
2
s
u
B
s
+
s
u
s
Q
s
(
B
-
s
2
)
]
+
2
r
0
s
0
(
-
2
s
υ
Θ
s
+
s
Θ
υ
s
+
s
u
B
s
-
3
υ
Θ
)
+
α
s
k
s
0
k
(
s
Q
s
u
-
Q
u
-
2
s
Q
u
s
)
+
s
0
|
0
(
s
u
s
)
+
2
r
k
0
s
0
k
(
s
Q
s
Θ
-
2
s
Q
Θ
s
-
3
Q
Θ
)
,
C
12
=
r
00
2
α
2
[
(
2
Θ
s
s
Ψ
-
Θ
s
Ψ
s
)
(
B
-
s
2
)
mm
-
2
Θ
Ψ
-
Θ
Θ
s
+
s
Θ
Ψ
s
-
6
s
Θ
s
Ψ
-
Θ
s
s
(
B
-
s
2
)
]
+
r
00
s
0
α
[
(
2
υ
Θ
s
s
+
2
Ψ
u
s
s
-
u
s
Ψ
s
-
υ
s
Θ
s
)
(
B
-
s
2
)
mmmim
-
(
Q
s
+
u
+
6
s
υ
)
Θ
s
+
2
Q
Θ
s
s
-
2
Θ
B
s
mmmim
-
4
s
u
s
Ψ
+
s
υ
s
Θ
-
u
s
s
-
Θ
u
s
-
2
υ
Θ
(
B
-
s
2
)
]
+
2
r
00
r
0
α
(
-
Θ
Ψ
s
+
2
Θ
s
Ψ
-
Θ
B
s
)
+
s
0
2
[
(
2
u
s
s
υ
-
u
s
υ
s
)
(
B
-
s
2
)
-
u
u
s
mmii
-
Q
s
u
s
-
2
u
B
s
-
4
s
u
s
υ
+
2
Q
u
s
s
(
2
u
s
s
υ
-
u
s
υ
s
)
(
B
-
s
2
)
(
B
-
s
2
)
]
-
2
r
0
s
0
(
u
B
s
+
Θ
υ
s
-
2
υ
Θ
s
)
+
2
r
k
0
s
0
k
(
2
Q
Θ
s
-
Q
s
Θ
)
+
α
s
k
s
0
k
(
2
Q
u
s
-
Q
s
u
)
-
s
0
|
0
u
s
-
r
00
|
0
α
Θ
s
,
C
13
=
r
00
[
(
2
u
s
Ψ
-
Θ
s
υ
)
(
B
-
s
2
)
-
u
Θ
mm
+
4
Θ
B
+
s
υ
Θ
-
u
s
-
Q
Θ
s
(
B
-
s
2
)
]
+
α
s
0
[
2
u
B
-
u
2
+
Q
u
s
+
u
s
υ
(
B
-
s
2
)
]
-
α
r
0
(
2
υ
Θ
+
2
u
B
)
,
C
14
=
4
r
00
(
Θ
B
+
Θ
Ψ
)
+
4
α
s
0
(
u
B
+
υ
Θ
)
,
C
130
=
3
r
00
α
[
Q
Θ
+
(
1
+
s
Q
)
Θ
s
]
+
3
s
0
(
s
Q
+
1
)
u
s
,
C
140
=
r
00
α
[
2
Θ
s
Ψ
(
B
-
s
2
)
-
2
s
Θ
Ψ
-
Θ
s
-
2
Θ
2
]
+
s
0
[
(
4
υ
Θ
s
-
2
Ψ
u
s
)
(
B
-
s
2
)
miim
+
4
Q
Θ
s
+
u
s
-
2
u
Θ
-
4
Θ
B
-
4
s
υ
Θ
(
4
υ
Θ
s
-
2
Ψ
u
s
)
(
B
-
s
2
)
(
B
-
s
2
)
]
-
4
r
0
(
Θ
B
+
Θ
Ψ
)
,
C
21
=
r
00
2
α
2
[
(
s
Ψ
s
2
-
2
s
Ψ
Ψ
s
s
-
2
Ψ
Ψ
s
)
(
B
-
s
2
)
m
1
i
+
s
Ψ
s
s
+
Ψ
s
+
4
s
2
Ψ
Ψ
s
(
B
-
s
2
)
]
+
r
00
s
0
α
[
(
2
s
Ψ
s
υ
s
-
3
υ
Ψ
s
-
2
s
υ
Ψ
s
s
-
2
s
Ψ
υ
s
s
)
(
B
-
s
2
)
mmmim
+
s
υ
s
s
+
2
s
Ψ
B
s
-
2
Q
s
Ψ
s
s
+
5
s
2
υ
Ψ
s
mmmim
+
s
Q
s
Ψ
s
+
2
s
2
υ
s
Ψ
-
2
s
υ
Ψ
-
3
Q
Ψ
s
(
2
s
Ψ
s
υ
s
-
3
υ
Ψ
s
-
2
s
υ
Ψ
s
s
-
2
s
Ψ
υ
s
s
)
(
B
-
s
2
)
]
+
2
r
00
r
0
α
(
-
s
Ψ
Ψ
s
+
s
Ψ
B
s
)
+
s
0
2
[
(
s
υ
s
2
-
2
s
υ
υ
s
s
-
υ
υ
s
)
(
B
-
s
2
)
+
3
s
2
υ
υ
s
mim
-
2
s
Q
υ
s
s
-
3
s
υ
2
-
Q
υ
s
-
2
υ
B
+
s
Q
s
υ
s
+
2
s
υ
B
s
(
s
υ
s
2
-
2
s
υ
υ
s
s
-
υ
υ
s
)
(
B
-
s
2
)
]
+
2
s
0
r
0
(
s
υ
B
s
-
Ψ
υ
+
s
Ψ
υ
s
-
2
s
υ
Ψ
s
-
υ
B
)
+
r
00
|
0
α
(
s
Ψ
s
)
+
2
r
k
0
s
0
k
(
-
Q
Ψ
+
s
Ψ
Q
s
-
2
s
Q
Ψ
s
)
+
s
0
|
0
(
s
υ
s
-
υ
)
+
α
s
k
s
0
k
(
s
Q
s
υ
+
Q
υ
-
2
s
Q
υ
s
)
,
C
22
=
r
00
2
α
2
[
(
2
Ψ
Ψ
s
s
-
Ψ
s
2
)
(
B
-
s
2
)
-
Ψ
s
s
-
4
s
Ψ
Ψ
s
]
+
r
00
s
0
α
[
2
(
υ
Ψ
s
s
+
Ψ
υ
s
s
-
Ψ
s
υ
s
)
(
B
-
s
2
)
hhhhhhh
-
2
Ψ
B
s
-
υ
s
s
+
2
υ
Ψ
hhhhhhh
-
2
s
Ψ
υ
s
-
5
s
υ
Ψ
s
+
2
Q
Ψ
s
s
-
Q
s
Ψ
s
(
υ
Ψ
s
s
+
Ψ
υ
s
s
-
Ψ
s
υ
s
)
(
B
-
s
2
)
(
B
-
s
2
)
]
+
2
r
00
r
0
α
(
-
Ψ
B
s
+
Ψ
Ψ
s
)
+
2
s
0
r
0
(
-
Ψ
υ
s
+
2
Ψ
s
υ
-
υ
B
s
)
+
s
0
2
[
(
2
υ
υ
s
s
-
υ
s
2
)
(
B
-
s
2
)
-
Q
s
υ
s
hhhih
-
2
υ
B
s
+
2
υ
2
+
2
Q
υ
s
s
-
3
s
υ
υ
s
(
2
υ
υ
s
s
-
υ
s
2
)
(
B
-
s
2
)
]
-
r
00
|
0
α
Ψ
s
-
s
0
|
0
υ
s
+
α
s
k
s
0
k
(
2
Q
υ
s
-
Q
s
υ
)
+
2
r
k
0
s
0
k
(
2
Q
Ψ
s
-
Q
s
Ψ
)
,
C
23
=
r
00
[
(
-
υ
Ψ
s
+
2
Ψ
υ
s
)
(
B
-
s
2
)
h
h
h
+
4
Ψ
B
-
Q
Ψ
s
-
υ
s
+
2
s
υ
Ψ
(
-
υ
Ψ
s
+
2
Ψ
υ
s
)
(
B
-
s
2
)
(
B
-
s
2
)
]
+
α
s
0
[
υ
υ
s
(
B
-
s
2
)
+
2
υ
B
+
Q
υ
s
+
s
υ
2
]
-
2
α
r
0
(
Ψ
υ
+
υ
B
)
,
C
24
=
4
r
00
(
Ψ
2
+
Ψ
B
)
+
4
α
s
0
(
υ
B
+
Ψ
υ
)
,
C
230
=
3
r
00
α
(
1
+
s
Q
)
Ψ
s
+
3
s
0
[
υ
s
-
(
υ
-
s
υ
s
)
Q
]
,
C
240
=
r
00
α
[
2
Ψ
Ψ
s
(
B
-
s
2
)
-
Ψ
s
]
+
s
0
[
(
4
υ
Ψ
s
-
2
Ψ
υ
s
)
(
B
-
s
2
)
+
υ
s
hhhh
-
2
s
υ
Ψ
+
4
Q
Ψ
s
-
4
Ψ
B
(
B
-
s
2
)
]
kkkk
-
4
r
0
(
Ψ
2
+
Ψ
B
)

,
C
3
=
2
α
s
0
[
Q
s
υ
(
B
-
s
2
)
+
Q
Q
s
+
s
Q
υ
+
υ
]
+
r
00
[
2
Q
s
Ψ
(
B
-
s
2
)
+
2
Ψ
-
Q
s
+
2
s
Q
Ψ
]
,
C
31
=
r
00
(
-
2
Q
Ψ
+
2
s
Q
s
Ψ
-
s
Q
Ψ
s
)
-
α
s
0
(
υ
Q
+
s
Q
υ
s
-
2
s
Q
s
υ
)
,
C
32
=
r
00
(
-
2
Q
s
Ψ
+
Q
Ψ
s
)
-
α
s
0
(
2
Q
s
υ
-
Q
υ
s
)
,
C
310
=
r
00
α
[
(
Q
s
s
Ψ
s
-
2
s
Ψ
Q
s
s
)
(
B
-
s
2
)
h
h
i
h
+
2
Q
s
s
2
Ψ
-
2
s
Q
Ψ
+
Q
s
2
Ψ
s
+
s
Q
s
s
+
s
Ψ
s
]
+
s
0
[
(
s
Q
s
υ
s
-
Q
s
υ
-
2
s
Q
s
s
υ
)
(
B
-
s
2
)
hhhh
+
Q
s
2
υ
s
-
Q
Q
s
+
s
Q
s
2
-
3
s
Q
υ
hhhh
-
υ
+
s
υ
s
-
2
s
Q
Q
s
s
+
2
Q
s
s
2
υ
]
,
C
320
=
r
00
α
[
(
2
Q
s
s
Ψ
-
Q
s
Ψ
s
)
(
B
-
s
2
)
h
h
i
h
+
2
Q
Ψ
-
2
s
Ψ
Q
s
-
s
Q
Ψ
s
-
Q
s
s
-
Ψ
s
(
B
-
s
2
)
]
+
s
0
[
(
2
Q
s
s
υ
-
Q
s
υ
s
)
(
B
-
s
2
)
-
υ
s
-
Q
s
2
hhhh
+
2
υ
Q
-
2
s
Q
s
υ
+
2
Q
Q
s
s
-
s
Q
υ
s
(
B
-
s
2
)
]
,
C
330
=
-
α
[
υ
Q
s
(
B
-
s
2
)
+
Q
Q
s
+
υ
+
s
Q
υ
]
,
C
331
=
-
3
Q
2
+
3
s
Q
Q
s
+
3
Q
s
,
C
340
=
-
2
[
Q
s
(
B
-
s
2
)
+
s
Q
+
1
]
Ψ
+
Q
s
,
C
4
=
(
2
Ψ
)
r
00
+
(
2
υ
)
α
s
0
,
C
410
=
r
00
α
s
Ψ
s
+
s
0
(
s
υ
s
-
υ
)
,
C
420
=
-
r
00
α
Ψ
s
-
s
0
υ
s
,
C
332
=
α
(
Q
-
s
Q
s
)
Q
,
C
333
=
α
Q
Q
s
,
C
311
=
s
Q
s
-
1
.Proposition 4 (see [6]).
For an(
α
,
β
)-metric F
=
α
ϕ
(
s
)
,
s
=
(
β
/
α
), the Ricci curvature of F is related to the Ricci curvature R
i
c
α of α by
(15)
R
i
c
=
R
i
c
α
+
T
m
m
,
where
(16)
T
m
m
=
r
00
2
α
2
[
(
n
-
1
)
c
1
+
c
2
]
+
1
α
{
+
r
00
|
0
[
(
n
-
1
)
c
7
+
c
8
]
r
00
s
0
[
(
n
-
1
)
c
3
+
c
4
]
hhhh
+
r
00
r
0
[
(
n
-
1
)
c
5
+
c
6
]
hhhh
+
r
00
|
0
[
(
n
-
1
)
c
7
+
c
8
]
}
+
{
s
0
2
[
(
n
-
1
)
c
9
+
c
10
]
+
(
r
r
00
-
r
0
2
)
c
11
hhhh
+
r
0
s
0
[
(
n
-
1
)
c
12
+
c
13
]
hhhh
+
(
r
00
r
m
m
-
r
0
m
r
0
m
+
r
00
|
m
b
m
-
r
0
m
|
0
b
m
)
c
14
hhhh
+
r
0
m
s
0
m
[
(
n
-
1
)
c
15
+
c
16
]
hhhh
+
s
0
|
0
[
(
n
-
1
)
c
17
+
c
18
]
+
s
0
m
s
0
m
c
19
(
r
r
00
-
r
0
2
)
}
+
α
{
mm
-
2
s
0
|
m
b
m
+
s
m
|
0
b
m
)
c
23
+
s
0
|
m
m
c
24
r
s
0
c
20
+
s
m
s
0
m
[
(
n
-
1
)
c
21
+
c
22
]
hhhh
+
(
-
2
s
0
|
m
b
m
+
s
m
|
0
b
m
3
s
m
r
0
m
-
2
s
0
r
m
m
+
2
r
m
s
0
m
hhhhhhh
-
2
s
0
|
m
b
m
+
s
m
|
0
b
m
)
c
23
+
s
0
|
m
m
c
24
}
+
α
2
(
s
m
s
m
c
25
+
s
m
i
s
i
m
c
26
)
,
c
1
=
2
Ψ
Θ
s
(
B
-
s
2
)
-
2
s
Ψ
Θ
+
Θ
2
-
Θ
s
,
c
2
=
(
2
Ψ
Ψ
s
s
-
Ψ
s
2
)
(
B
-
s
2
)
2
-
(
6
s
Ψ
Ψ
s
+
Ψ
s
s
)
(
B
-
s
2
)
+
2
s
Ψ
s
,
c
3
=
-
4
(
2
Q
Ψ
Θ
s
+
Q
s
Ψ
Θ
)
(
B
-
s
2
)
+
2
(
Q
s
Θ
+
2
Q
Θ
s
)
+
4
Q
Θ
(
s
Ψ
-
Θ
)
-
2
Θ
B
,
c
4
=
[
(
B
-
s
2
)
2
-
4
Ψ
(
2
Q
Ψ
s
s
+
Q
s
Ψ
s
+
Q
s
s
Ψ
)
+
4
Q
Ψ
s
2
(
B
-
s
2
)
2
]
(
B
-
s
2
)
2
+
[
(
B
-
s
2
)
2
-
4
Ψ
2
(
Q
-
s
Q
s
)
+
2
(
2
Q
s
s
Ψ
+
Q
s
Ψ
s
+
2
Q
Ψ
s
s
)
m
-
2
Ψ
s
B
+
20
s
Q
Ψ
Ψ
s
(
B
-
s
2
)
2
]
(
B
-
s
2
)
+
2
Ψ
(
Q
-
s
Q
s
)
-
4
Ψ
s
-
Q
s
s
-
10
s
Q
Ψ
s
,
c
5
=
4
Ψ
Θ
-
2
Θ
B
,
c
6
=
2
(
2
Ψ
Ψ
s
-
Ψ
s
B
)
(
B
-
s
2
)
-
2
Ψ
s
,
c
7
=
-
Θ
,
c
8
=
-
Ψ
s
(
B
-
s
2
)
,
c
9
=
8
Q
Ψ
(
Q
Θ
s
+
Q
s
Θ
)
(
B
-
s
2
)
+
4
Q
2
(
Θ
2
-
Θ
s
)
+
4
Q
(
Θ
B
-
Θ
Q
s
)
,
c
10
=
[
4
Ψ
2
(
2
Q
Q
s
s
-
Q
s
2
)
+
8
Q
Ψ
(
Q
Ψ
s
s
+
Q
s
Ψ
s
)
ii
-
4
Q
2
Ψ
s
2
]
(
B
-
s
2
)
2
ii
+
[
-
16
s
Q
Ψ
(
Q
Ψ
s
+
Q
s
Ψ
)
-
4
Ψ
(
2
Q
Q
s
s
-
Q
s
2
)
hhh
-
4
Q
(
Q
Ψ
s
s
+
Q
s
Ψ
s
)
hhh
+
4
(
Q
Ψ
s
B
+
Q
s
Ψ
B
)
(
2
Q
Q
s
s
-
Q
s
2
)
]
(
B
-
s
2
)
-
12
s
2
Q
2
Ψ
2
+
4
(
2
+
3
s
Q
)
(
Q
Ψ
s
+
Q
s
Ψ
)
-
8
Q
2
Ψ
+
2
Q
Q
s
s
-
Q
s
2
+
4
s
Q
Ψ
B
+
8
B
Ψ
2
Q
2
,
c
11
=
4
Ψ
2
+
4
Ψ
B
,
c
12
=
4
Q
(
-
2
Ψ
Θ
+
Θ
B
)
,
c
13
=
[
8
Ψ
(
Q
s
Ψ
-
Q
Ψ
s
)
+
4
(
Q
Ψ
s
B
+
Q
s
Ψ
B
)
]
(
B
-
s
2
)
+
8
s
Q
Ψ
2
+
4
Q
Ψ
s
-
4
(
1
-
s
Q
)
Ψ
B
,
c
14
=
2
Ψ
,
c
15
=
4
Q
Θ
,
c
16
=
-
4
(
Q
s
Ψ
-
Q
Ψ
s
)
(
B
-
s
2
)
+
2
Q
s
-
2
(
1
+
2
s
Q
)
Ψ
,
c
17
=
2
Q
Θ
,
c
18
=
2
(
Q
s
Ψ
+
Q
Ψ
s
)
(
B
-
s
2
)
-
Q
s
+
2
s
Q
Ψ
,
c
19
=
-
2
Q
2
+
2
(
1
+
s
Q
)
Q
s
,
c
20
=
-
8
Q
(
Ψ
2
+
Ψ
B
)
,
c
21
=
-
4
Q
2
Θ
,
c
22
=
-
4
Q
2
Ψ
s
(
B
-
s
2
)
+
2
Q
Ψ
,
c
23
=
2
Q
Ψ
,
c
24
=
2
Q
,
c
25
=
-
4
Q
2
Ψ
,
c
26
=
-
Q
2
.Remark 5.
In Proposition4, we have corrected some terms in the formulas for coefficients c
9 and c
10, which are not printed.Definition 6 (see [7]).
Let(17)
D
j
k
l
i
=
∂
3
∂
y
j
∂
y
k
∂
y
l
(
G
i
-
1
n
+
1
∂
G
m
∂
y
m
y
i
)
,
where G
i are the spray coefficients of F. The tensor D
:
=
D
j
k
l
i
∂
i
⊗
d
x
j
⊗
d
x
k
⊗
d
x
l is calledDouglas tensor. A Finsler metric is called Douglas metric if the Douglas tensor vanishes.Note that an(
α
,
β
)-metric F
=
α
+
ε
β
+
(
2
k
β
2
/
α
)
-
(
k
2
β
4
/
3
α
3
) is a Douglas metric if and only if b
i
|
j
=
σ
[
(
1
+
4
k
B
)
a
i
j
-
5
k
b
i
b
j
] holds for some scalar function σ
=
σ
(
x
) [8].Definition 7 (see [7]).
Put(18)
W
k
i
:
=
A
k
i
-
1
n
+
1
∂
A
k
m
∂
y
m
y
i
,
where R
=
(
R
k
k
/
(
n
-
1
)
)
,
A
j
i
=
R
j
i
-
R
δ
j
i. Then W
=
W
k
i
(
∂
/
∂
x
i
)
⊗
d
x
k is a tensor on T
M
∖
{
0
}. it is called theWeyl curvature tensor.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. A famous theorem of Douglas is in the following.Theorem 8 (see [9]).
A Finsler metricF on a manifold M (dim M
>
2) is locally projectively flat if and only if D
=
0 and W
=
0.


## 3. Proof of Theorem1

In this section, we will prove Theorem1. First, we will study the following lemma, because Lemma 4.1 [3] is found to have some wrong. For example, when k
=
-
(
1
/
4
b
2
), F does not satisfy the definition of (
α
,
β
)-metrics at s
=
0.Lemma 9.
F
=
α
+
ε
β
+
(
2
k
β
2
/
α
)
-
(
k
2
β
4
/
3
α
3
) is a Finsler metric if and only if one of the following holds.(a)
Ifk
>
0, then b
<
1
/
k and ε
∈
[
-
k
,
k
].
(b)
Ifk
<
0, then b
<
(
1
/
2
-
k
) and ε
∈
[
-
(
23
/
24
)
-
k
,
(
23
/
24
)
-
k
],
whereb
:
=
∥
β
x
∥
α, for any x
∈
M.Proof.
Let(19)
ϕ
=
1
+
ε
s
+
2
k
s
2
-
k
2
s
4
3
,
s
=
β
α
.
Direct computations yield(20)
ϕ
-
s
ϕ
′
=
(
k
s
2
-
1
)
2
,
ϕ
-
s
ϕ
′
+
(
b
2
-
s
2
)
ϕ
′
′
=
(
k
s
2
-
1
)
(
5
k
s
2
-
1
-
4
k
b
2
)
.
From the definition of (
α
,
β
), we know that, for all |
s
|
≤
b
<
b
0, ϕ must satisfy
(21)
ϕ
>
0
,
ϕ
-
s
ϕ
′
+
(
b
2
-
s
2
)
ϕ
′
′
>
0
.
That is,
(22)
1
+
ε
s
+
2
k
s
2
-
k
2
s
4
3
>
0
,
(
k
s
2
-
1
)
(
5
k
s
2
-
1
-
4
k
b
2
)
>
0
.
Particularly, when s
=
0 and s
=
b, it is easy to see that
(23)
1
+
4
k
b
2
>
0
,
k
b
2
-
1
≠
0
.
Now we first discuss the second inequality of (22). It is equivalent to the following two cases.
Lety
1
=
k
s
2
-
1
,
y
2
=
5
k
s
2
-
1
-
4
k
b
2; it is easy to see that y
1 and y
2 are quadratic functions with respect to s.
Case
1. When
(24)
k
s
2
-
1
>
0
,
5
k
s
2
-
1
-
4
k
b
2
>
0
,
(I)
k
>
0.
By the graph of quadratic function and (23), we find that there is no b such that k
s
2
-
1
>
0 and 5
k
s
2
-
1
-
4
k
b
2
>
0 always hold for all |
s
|
≤
b.Consider (II)k
<
0.
By the graph of quadratic function and (23), we conclude the same result with (I).
In any case, Case 1 cannot exist for all|
s
|
≤
b.
Case
2. When
(25)
k
s
2
-
1
<
0
,
5
k
s
2
-
1
-
4
k
b
2
<
0
,
(III)
k
>
0.
By the graph of quadratic function and (23), the above system of inequality always holds if and only if b
<
(
1
/
k
).Consider (IV)k
<
0.
Similarly, the above system of inequality always holds if and only ifb
<
(
1
/
2
-
k
).
In the following, we will discuss the condition satisfying the first inequality of (22). Since
(26)
1
+
ε
s
+
2
k
s
2
-
k
2
s
4
3
>
0
⟺
(
k
s
2
-
3
)
2
-
3
ε
s
-
12
<
0
,
lety
=
(
k
s
2
-
3
)
2
-
3
ε
s
-
12.Consider (1)k
>
0.
In this case, by (III), we know that|
s
|
≤
b
<
(
1
/
k
). By the graph and monotonicity of quadratic functions, we find
(27)
-
8
-
3
ε
s
<
y
≤
-
3
-
3
ε
s
.
Hence, for all |
s
|
≤
b
<
(
1
/
k
), (26) always holds if and only if 1
+
ε
s
>
0. Moreover, if ε
>
0, we have ε
≤
k. If ε
=
0, 1
+
ε
s
>
0 holds forever for all |
s
|
≤
b
<
(
1
/
k
). If ε
<
0, we get ε
≥
-
k.
In a word,ε
∈
[
-
k
,
k
].Consider (2)k
<
0.
In this case, by (IV), we know that|
s
|
≤
b
<
(
1
/
2
-
k
). By the same way as above, we have
(28)
-
3
-
3
ε
s
≤
y
<
-
23
16
-
3
ε
s
.
Hence, for all|
s
|
≤
b
<
(
1
/
2
-
k
), (26) always holds if and only if (
23
/
16
)
+
3
ε
s
≥
0. Furthermore,
ifε
>
0, we have ε
≤
(
23
/
24
)
-
k. If ε
=
0, (
23
/
16
)
+
3
ε
s
≥
0 holds forever for all |
s
|
≤
b
<
(
1
/
k
). If ε
<
0, we get ε
≤
-
(
23
/
24
)
-
k.
In any case,ε
∈
[
-
(
23
/
24
)
-
k
,
(
23
/
24
)
-
k
]. We complete the proof of the lemma.Lemma 10.
LetF
=
α
+
ε
β
+
(
2
k
β
2
/
α
)
-
(
k
2
β
4
/
3
α
3
) be a Finsler metric on a manifold M
n, where ε is a constant and k is a nonzero constant. If F is of constant flag curvature, then it must satisfy(a)
r
00
=
σ
(
1
+
4
k
B
-
5
k
s
2
)
α
2,
(b)
s
0
=
0,
(c)
s
0
k
s
0
k
=
0,
whereσ
=
σ
(
x
) is a smooth function on M.Proof.
By a direct computation, we have(29)
Q
=
3
ε
+
12
k
s
-
4
k
2
s
3
3
(
-
1
+
k
s
2
)
2
,
Ψ
=
2
k
1
+
4
k
B
-
5
k
s
2
,
Θ
=
3
ε
-
40
k
2
s
3
-
15
ε
k
s
2
+
8
k
3
s
5
2
(
1
+
4
k
B
-
5
k
s
2
)
(
3
+
3
ε
s
+
6
k
s
2
-
k
2
s
4
)
.
Since F is of constant flag curvature, F is also of constant Ricci curvature; that is, the following system of equations holds
(30)
α
R
i
i
+
T
i
i
-
K
(
n
-
1
)
F
2
=
0
.
By Proposition4, we can calculate T
i
i by Maple
(31)
T
i
i
=
r
00
2
α
2
[
(
n
-
1
)
c
-
1
A
1
3
A
3
2
+
c
-
2
A
1
4
]
+
r
00
s
0
α
[
(
n
-
1
)
c
-
3
A
1
3
A
2
2
A
3
2
+
c
-
4
A
1
4
A
2
2
]
+
1
α
{
r
00
r
0
[
(
n
-
1
)
c
-
5
A
1
2
A
3
+
c
-
6
A
1
3
]
mmii
+
r
00
|
0
[
(
n
-
1
)
c
-
7
A
1
A
3
+
c
-
8
A
1
2
]
}
+
s
0
2
[
(
n
-
1
)
c
-
9
A
1
3
A
2
4
A
3
2
+
c
-
10
A
1
4
A
2
5
]
-
16
k
2
A
1
2
(
r
r
00
-
r
0
2
)
+
r
0
s
0
[
(
n
-
1
)
c
-
12
A
1
2
A
2
2
A
3
+
c
-
13
A
1
3
A
2
2
]
+
4
k
A
1
(
r
00
r
i
i
-
r
0
i
r
0
i
+
r
00
|
i
b
i
-
r
0
i
|
0
b
i
)
+
r
0
i
s
0
i
[
(
n
-
1
)
c
-
15
A
1
A
2
2
A
3
+
c
-
16
A
1
2
A
2
2
]
+
s
0
|
0
[
(
n
-
1
)
c
-
17
A
1
A
2
2
A
3
+
c
-
18
A
1
2
A
2
2
]
+
c
-
19
A
2
5
s
0
i
s
0
i
+
α
{
r
s
0
c
-
20
A
1
2
A
2
2
+
s
i
s
0
i
[
(
n
-
1
)
c
-
21
A
1
A
2
4
A
3
+
c
-
22
A
1
2
A
2
4
]
}
+
α
[
c
-
23
A
1
A
2
2
(
3
s
i
r
0
i
-
2
s
0
r
i
i
+
2
r
i
s
0
i
-
2
s
0
|
i
b
i
+
s
i
|
0
b
i
)
mmi
×
c
-
23
A
1
A
2
2
+
c
-
24
A
2
2
s
0
|
i
i
]
+
α
2
(
c
-
25
A
1
A
2
4
s
i
s
i
+
c
-
26
A
2
4
s
i
j
s
j
i
)
,
where
(32)
A
1
=
1
+
4
k
B
-
5
k
s
2
,
A
2
=
-
1
+
k
s
2
,
A
3
=
3
+
3
ε
s
+
6
k
s
2
-
k
2
s
4
and c
-
i
(
i
=
1,2
,
…
,
26
) are polynomials of s.
Plugging (31) into (30) and multiplying it by A
1
4 yield
(33)
R
α
i
i
(
1
+
4
k
B
-
5
k
s
2
)
4
+
T
i
i
(
1
+
4
k
B
-
5
k
s
2
)
4
-
K
(
n
-
1
)
F
2
(
1
+
4
k
B
-
5
k
s
2
)
4
=
0
.
It is easy to see that
(34)
α
R
i
i
(
1
+
4
k
B
-
5
k
s
2
)
4
-
K
(
n
-
1
)
F
2
(
1
+
4
k
B
-
5
k
s
2
)
4
≡
0
mod
⁡
(
1
+
4
k
B
-
5
k
s
2
)
.
Hence,
(35)
T
i
i
(
1
+
4
k
B
-
5
k
s
2
)
4
≡
0
mod
⁡
(
1
+
4
k
B
-
5
k
s
2
)
.
From (31) we find that
(36)
r
00
2
α
2
c
-
2
+
r
00
s
0
α
c
-
4
A
2
2
+
c
-
10
A
2
5
s
0
2
≡
0
mod
⁡
(
1
+
4
k
B
-
5
k
s
2
)
.
By Lemma 9, we need to divide two cases.Consider (1)k
>
0.
In this case, we know thatk
∈
(
0
,
(
1
/
B
)
), where B
:
=
b
2
=
∥
β
∥
α
2. Hence, (
(
1
+
4
k
B
)
/
5
k
)
>
0.
For(37)
1
+
4
k
B
-
5
k
s
2
=
-
5
k
(
s
2
-
1
+
4
k
B
5
k
)
=
-
5
k
(
s
-
1
+
4
k
B
5
k
)
(
s
+
1
+
4
k
B
5
k
)
,
we get
(38)
r
00
2
α
2
c
-
2
+
r
00
s
0
α
c
-
4
A
2
2
+
c
-
10
A
2
5
s
0
2
≡
0
mod
⁡
(
s
+
1
+
4
k
B
5
k
)
,
r
00
2
α
2
c
-
2
+
r
00
s
0
α
c
-
4
A
2
2
+
c
-
10
A
2
5
s
0
2
≡
0
mod
⁡
(
s
-
1
+
4
k
B
5
k
)
.
Simplifying (38) by Maple yields
(39)
(
r
00
α
-
8
k
(
5
(
1
+
4
k
B
)
/
k
)
(
2
k
B
-
7
)
+
75
ε
24
(
k
B
-
1
)
2
s
0
)
2
≡
0
mod
⁡
(
s
+
1
+
4
k
B
5
k
)
,
(
r
00
α
+
8
k
(
5
(
1
+
4
k
B
)
/
k
)
(
2
k
B
-
7
)
-
75
ε
24
(
k
B
-
1
)
2
s
0
)
2
≡
0
mod
⁡
(
s
-
1
+
4
k
B
5
k
)
.
Since k
∈
(
0
,
(
1
/
B
)
), 2
k
B
-
7
≠
0. Moreover, we note that ε
≠
±
(
8
k
/
75
)
(
5
(
1
+
4
k
B
)
/
k
)
(
2
k
B
-
7
). Otherwise, by Lemma 9, F is not a Finsler metric.
From (39), and Zhou’s Lemma 4.1 [2], we obtain that
(40)
r
00
+
5
k
[
8
k
(
5
(
1
+
4
k
B
)
/
k
)
(
2
k
B
-
7
)
+
75
ε
]
24
(
k
B
-
1
)
2
1
+
4
k
B
α
s
s
0
=
σ
1
(
1
+
4
k
B
-
5
k
s
2
)
α
2
,
r
00
+
5
k
[
8
k
(
5
(
1
+
4
k
B
)
/
k
)
(
2
k
B
-
7
)
-
75
ε
]
24
(
k
B
-
1
)
2
1
+
4
k
B
α
s
s
0
=
σ
2
(
1
+
4
k
B
-
5
k
s
2
)
α
2
,
where σ
1
=
σ
1
(
x
) and σ
2
=
σ
2
(
x
) are smooth functions on a manifold M
n.
It is easy to see that(41)
s
s
0
≡
0
mod
⁡
(
1
+
4
k
B
-
5
k
s
2
)
.
The above equation holds if and only if
(42)
s
0
=
0
.
Therefore, we have
(43)
r
00
=
σ
(
1
+
4
k
B
-
5
k
s
2
)
α
2
,
where σ
=
σ
(
x
) is a smooth function on a manifold M
n.Consider (2)k
<
0.
In this case, we know thatk
∈
(
-
(
1
/
4
B
)
,
0
), where B
:
=
b
2
=
∥
β
∥
α
2. Hence, (
(
1
+
4
k
B
)
/
5
k
)
<
0. Moreover, 1
+
4
k
B
-
5
k
s
2 is irreducible with respect to s.
Simplifying (36) directly by Maple yields
(44)
[
r
00
α
+
8
(
2
k
B
-
7
)
(
5
k
s
+
5
k
(
5
k
s
2
-
4
k
B
-
1
)
)
-
75
ε
24
(
1
-
k
B
)
2
s
0
]
×
[
8
(
2
k
B
-
7
)
(
5
k
s
-
5
k
(
5
k
s
2
-
4
k
B
-
1
)
)
-
75
ε
24
(
1
-
k
B
)
2
s
0
r
00
α
+
8
(
2
k
B
-
7
)
(
5
k
s
-
5
k
(
5
k
s
2
-
4
k
B
-
1
)
)
-
75
ε
24
(
1
-
k
B
)
2
s
0
]
≡
0
mod
⁡
(
1
+
4
k
B
-
5
k
s
2
)
.
By Lemma 9, we note that 5
k
s
2
-
4
k
B
-
1
<
0. The above equation holds if and only if
(45)
r
00
+
8
(
2
k
B
-
7
)
(
5
k
s
+
5
k
(
5
k
s
2
-
4
k
B
-
1
)
)
-
75
ε
24
(
1
-
k
B
)
2
α
s
0
=
σ
3
(
1
+
4
k
B
-
5
k
s
2
)
α
2
or
(46)
r
00
+
8
(
2
k
B
-
7
)
(
5
k
s
-
5
k
(
5
k
s
2
-
4
k
B
-
1
)
)
-
75
ε
24
(
1
-
k
B
)
2
α
s
0
=
σ
4
(
1
+
4
k
B
-
5
k
s
2
)
α
2
,
where σ
3
=
σ
3
(
x
) and σ
4
=
σ
4
(
x
) are smooth functions on a manifold M
n.
In any case, it is easy to find thats
0
=
0. Hence, r
00
=
σ
(
1
+
4
k
B
-
5
k
s
2
)
α
2, where σ
=
σ
(
x
) is a smooth function on a manifold M
n.
Plugging (31) and s
0
=
0 into (30) and multiplying it by A
2
5, we have
(47)
s
0
i
s
0
i
c
-
19
≡
0
mod
⁡
(
-
1
+
k
s
2
)
,
where
(48)
c
-
19
=
-
2
9
(
20
k
5
s
8
-
160
k
4
s
6
-
48
ε
k
3
s
5
+
360
k
3
s
4
mmi
+
240
ε
k
2
s
3
+
45
ε
2
k
s
2
+
36
k
-
9
ε
2
)
.
Simplifying (47) by Maple, we have
(49)
s
0
i
s
0
i
(
192
ε
k
s
+
256
k
+
36
ε
2
)
≡
0
mod
⁡
(
-
1
+
k
s
2
)
.
So
(50)
s
0
i
s
0
i
=
0
.Lemma 11.
LetF
=
α
+
ε
β
+
(
2
k
β
2
/
α
)
-
(
k
2
β
4
/
3
α
3
) be a Finsler metric on an n-dimensional manifold M, where ε is a constant and k is a nonzero constant. Suppose that F is of constant flag curvature; then β is closed and K
=
0.Proof.
By Lemma10, we have
(51)
r
i
j
=
σ
[
(
1
+
4
k
B
)
a
i
j
-
5
k
b
i
b
j
]
,
s
i
=
0
,
s
k
i
s
j
k
=
0
,
where σ
=
σ
(
x
) is a smooth function on an n-dimensional manifold M.
Thus, by some computations, we have(52)
r
0
=
σ
β
(
1
-
k
B
)
,
r
=
σ
(
1
-
k
B
)
B
,
r
00
|
0
=
σ
0
α
2
(
1
+
4
k
B
-
5
k
s
2
)
+
2
k
σ
α
(
4
α
r
0
-
5
s
r
00
)
,
r
00
|
j
b
j
=
σ
b
α
2
(
1
+
4
k
B
-
5
k
s
2
)
+
2
k
σ
α
(
4
α
r
-
5
s
r
0
)
,
r
j
0
|
0
b
j
=
σ
0
(
1
-
k
B
)
β
+
k
σ
(
3
β
r
0
-
5
B
r
00
)
,
r
i
i
=
σ
[
(
1
+
4
k
B
)
n
-
5
k
B
]
,
r
i
0
r
0
i
=
σ
2
α
2
[
(
1
+
4
k
B
)
2
-
5
k
s
2
(
2
+
3
k
B
)
]
,
r
j
0
=
σ
α
[
(
1
+
4
k
B
)
l
j
-
5
k
s
b
j
]
,
r
j
=
σ
(
1
-
k
B
)
b
j
,
r
j
0
|
0
=
σ
0
α
[
(
1
+
4
k
B
)
l
j
-
5
k
s
b
j
]
+
σ
k
α
(
8
r
0
l
j
-
5
r
00
α
b
j
-
5
s
r
j
0
+
5
s
s
0
j
)
,
r
00
|
j
=
σ
j
α
2
(
1
+
4
k
B
-
5
k
s
2
)
+
2
σ
k
α
(
4
r
j
α
-
5
s
r
j
0
-
5
s
s
0
j
)
,
r
0
i
=
σ
α
[
(
1
+
4
k
B
)
l
i
-
5
k
s
b
i
]
,
r
j
i
=
σ
[
(
1
+
4
k
B
)
δ
j
i
-
5
k
b
i
b
j
]
,
where σ
0
:
=
σ
x
i
y
i
,
σ
b
:
=
σ
x
i
b
i
,
r
0
:
=
r
i
y
i
,
l
j
:
=
(
y
j
/
α
)
,
l
i
:
=
a
i
j
l
j
=
(
y
i
/
α
)
,
(
a
i
j
)
:
=
(
a
i
j
)
-
1.
Case
1 (ε
≠
0). Plugging the above equations into the expression of T
j
i in Proposition 3 and simplifying it by Maple, we have
(53)
T
j
i
=
(
T
1
A
3
σ
2
+
T
2
σ
0
)
α
2
A
3
δ
j
i
+
[
T
3
A
3
2
α
σ
2
+
(
T
4
A
3
α
+
T
5
)
σ
0
A
1
]
α
A
3
l
i
l
j
+
[
T
6
A
3
2
α
σ
2
+
(
T
7
A
3
α
+
T
8
)
σ
0
A
1
]
α
A
3
l
i
b
j
+
α
A
2
2
A
3
(
T
9
σ
+
T
10
)
l
i
s
0
j
+
T
11
A
3
α
2
l
i
σ
j
+
4
α
k
(
5
α
k
s
2
-
4
k
B
-
1
A
1
σ
0
+
α
k
s
σ
2
)
b
i
l
j
+
4
α
k
2
[
5
s
(
1
-
α
)
A
1
σ
0
-
α
σ
2
]
b
i
b
j
+
α
σ
T
12
A
2
2
b
i
s
0
j
+
4
k
α
2
b
i
σ
j
-
20
k
2
s
A
3
A
2
3
(
s
l
j
-
b
j
)
s
0
i
+
T
13
A
2
5
s
0
i
s
0
j
+
T
14
α
A
2
2
(
2
s
0
|
j
i
-
s
j
|
0
i
)
+
T
15
A
2
3
s
0
|
0
i
l
j
+
T
16
A
2
3
s
0
|
0
i
b
j
,
where T
i
(
i
=
1
,
…
,
16
) are polynomials of s, A
1
=
1
+
4
k
B
-
5
k
s
2, A
2
=
-
1
+
k
s
2, and A
3
=
3
+
3
ε
s
+
6
k
s
2
-
k
2
s
4.
BecauseF has constant flag curvature, it is equivalent to
(54)
α
R
j
i
+
T
j
i
=
K
F
2
(
δ
j
i
-
y
i
F
F
y
j
)
.
Substitute (53) into (54) and multiplying it by A
2
5, we find
(55)
s
0
i
s
0
j
T
13
≡
0
mod
⁡
(
k
s
2
-
1
)
,
where T
14
=
-
(
1
/
3
)
(
20
k
5
s
8
-
160
k
4
s
6
-
48
ε
k
3
s
5
+
360
k
3
s
4
+
240
ε
k
2
s
3
+
45
ε
2
k
s
2
+
36
k
-
9
ε
2
).
By the same way as Lemma10, we conclude s
0
i
=
0 or s
0
j
=
0; that is, β is closed.
Case
2  (
ε
=
0
). By the same approach as the above, we still conclude that β is closed.
Now we will proveK
=
0.
Letϕ
=
1
+
ε
s
+
2
k
s
2
-
(
k
2
s
4
/
3
). If F
=
α
ϕ
(
s
), s
=
β
/
α, has constant flag curvature K, then F has constant Ricci curvature. That means that R
i
c
=
(
n
-
1
)
K
F
2. Hence, F is an Einstein metric with Ricci constant K.
Note thatϕ
=
1
+
ε
s
+
2
k
s
2
-
(
k
2
s
4
/
3
) is a polynomial in s of degree 4. According to Theorem 1.1 (see [6]), we know that it is Ricci flat. So K
=
0.Proof of Theorem1.
By Lemma10, we have
(56)
r
i
j
=
σ
[
(
1
+
4
k
B
)
a
i
j
-
5
k
b
i
b
j
]
,
where σ
=
σ
(
x
) is a smooth function on an n-dimensional manifold.
Moreover, by Lemma11, we know that β is closed; that is, s
i
j
=
0, Hence,
(57)
b
i
|
j
=
σ
[
(
1
+
4
k
B
)
a
i
j
-
5
k
b
i
b
j
]
.
(1)
Ifσ
=
0, then b
i
|
j
=
0; that is, β is parallel with respect to α. By the proof of Lemma 11, we find that the sectional curvature of α equals 0. That means that α is flat. Therefore, F is locally Minkowskian.
(2)
Ifσ
≠
0.
By (57), we obtain that F is a Douglas metric. That means that the Douglas curvature tensor D
=
0.
SinceF has constant flag curvature, the Weyl curvature tensor W
=
0. By Theorem 8, we know that F must be locally projectively flat. Hence, by Theorem 1.3 (see [3]), F is locally Minkowskian.



---

*Source: 931319-2014-08-17.xml*