Journal of Geometry and Physics 86 (2014) 112–121

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Journal of Geometry and Physics

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On weak Berwald (α, β)-metrics of scalar flag curvatureGuangzu Chen ∗, Qun He, Shengliang PanDepartment of Mathematics, Tongji University, Shanghai 200092, PR China

a r t i c l e i n f o

Article history:Received 26 October 2013Accepted 22 July 2014Available online 30 July 2014

MSC:53B4053C60

Keywords:(α, β)-metricWeak Berwald Finsler metricFlag curvatureBerwald metricMinkowskian metric

a b s t r a c t

In this paper, we studyweak Berwald (α, β)-metrics of scalar flag curvature.We prove thatthis kind of (α, β)-metrics must be Berwald metric and their flag curvatures vanish. In thiscase, they are locally Minkowskian.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

It is well known that the spray coefficients Gi of a Riemannian metric are quadratic in y ∈ TxM . It is a natural questionwhether or not there are non-Riemannian metrics whose spray coefficients Gi are quadratic in y. There are plenty of suchFinsler metrics firstly investigated by L. Berwald. Thus we call Finsler metrics whose spray coefficients are quadratic in yBerwald metrics.

Let F be a Finsler metric on an n-dimensional manifoldM and Gi be the geodesic coefficients of F , consider the followingquantity

B ij kl :=

∂3Gi

∂yj∂yk∂yl.

We obtain a well-defined tensor B := B ij kldx

j⊗ dxk ⊗ dxl ⊗ ∂i on TM/{0}. B is called Berwald curvature. It is clear that F is a

Berwald metric if and only if B = 0. Define the mean Berwald curvature E := Eijdxi ⊗ dxj by

Eij :=12B mm kl.

F is called a weak Berwald metric if E = 0. It is obvious that Berwald metrics must be weak Berwald metrics but the reverseis not true.

The flag curvature, a natural extension of the sectional curvature in Riemannian geometry, plays the central role in Finslergeometry. Generally, the flag curvature depends not only on the section but also on the flagpole. A Finsler metric is of scalar

∗ Corresponding author. Tel.: +86 13524345535.E-mail addresses: chenguangzu1@163.com (G. Chen), hequn@tongji.edu.cn (Q. He), slpan@tongji.edu.cn (S. Pan).

http://dx.doi.org/10.1016/j.geomphys.2014.07.0310393-0440/© 2014 Elsevier B.V. All rights reserved.

G. Chen et al. / Journal of Geometry and Physics 86 (2014) 112–121 113

flag curvature if its flag curvature depends only on the flagpole. It is one of hot and difficult problems to characterize Finslermetrics with scalar flag curvature.

In Finsler geometry, there is an important class of Finsler metrics expressed in the following form

F = αφ(s), s =β

α,

where α =aijyiyj is a Riemann metric and β = bi(x)yi is a 1-form with b := ∥βx∥α =

aijbibj < b0. It is proved that F =

αφ(β/α) is a positive definite Finslermetric if and only if the functionφ = φ(s) is a C∞ positive function on an open interval(−b0, b0) satisfying [1]

φ(s)− sφ′(s)+ (b2 − s2)φ′′(s) > 0, |s| ≤ b < b0.Such a metric is called an (α, β)-metric. In particular, when φ = 1 + s, Finsler metric F = α + β is Randers metric with∥β∥α < 1. More generally, when φ = k1

1 + k2s2 + k3s, where k1 > 0, k2 and k3 = 0 are constant, Finsler metrics F =

k1α2 + k2β2 + k3β are called Finsler metrics of Randers type.Let

rij :=12(bi|j + bj|i), sij :=

12(bi|j − bj|i),

bi := aijbj, si := bjsji, sij := ailslj, ri := blrli,

s0 := siyi, si0 := sijyj, r00 := rijyiyj

where ‘‘|’’ denotes the horizontal covariant derivative with respect to α.In [2], Z. Shen shows that any regular (α, β)-metrics are Berwald metric if and only if β is parallel with respect to α,

that is, bi|j = 0. However, how to characterize the weak Berwald (α, β)-metrics is still open. In 2009, X. Cheng and C. Xiangstudy a class of (α, β)-metrics in the form F = (α+ β)m+1/αm, wherem = −1, 0,−1/n and prove that F is weak Berwaldif and only if it satisfies that β is a killing 1-form with constant length with respect to α, that is, rij = si = 0 [3]. In [4],X. Cheng and C. Lu show that two kinds of weak Berwald (α, β)-metrics in the form F = α+ ϵβ+ k(β2/α) (ϵ and k = 0 areconstants) and F = α2/(α − β) are weak Berwald if and only if they satisfy that β is a killing 1-form with constant lengthwith respect to α. Further, they prove that the two kinds of weak Berwald (α, β)-metrics with scalar flag curvature must belocally Minkowskian. In this paper, we have

Theorem 1.1. Let F = αφ(s), s = β/α be a non-Riemannian (α, β)-metric on a manifold M of dimension n ≥ 3. Supposeφ = k1

1 + k2s2 + k3s for any constant k1 > 0, k2 and k3 = 0. Then F is a weak Berwald metric if and only if β is a killing

1-form with constant length with respect to α.

Further, we obtain

Theorem 1.2. Let F = αφ(s), s = β/α be a non-Riemannian (α, β)-metric of scalar flag curvature on a manifold M of dimen-sion n ≥ 3. Suppose φ = k1

1 + k2s2 + k2s for any constant k1 > 0, k2 and k3 = 0 on M. Then F is a weak Berwald metric if

and only if F is a Berwald metric and the flag curvature K = 0. In this case, F must be locally Minkowskian.

2. Preliminaries

For a given Finsler F = F(x, y), the geodesics of F are characterized locally by a system of 2nd ODEs as follows [1],

d2xi

dt2+ 2Gi

x,

dxdt

= 0,

where

Gi=

14g il

[F 2

]xmylym

− [F 2]xl

.

Gi are called the geodesic coefficients of F .There are many interesting non-Riemannian quantities in Finsler geometry. For a non-zero vector y ∈ TpM , the Cartan

torsion Cy = Cijkdxi ⊗ dxj ⊗ dxk : TpM ⊗ TpM ⊗ TpM −→ R is defined by

Cijk :=14[F 2

]yiyjyk =12∂gij∂yk

(x, y).

The mean Cartan torsion Iy = Ii(x, y)dxi : TpM −→ R is defined by

Ii := g jkCijk,

where (g ij) := (gij)−1 and gij :=12 [F

2]yiyj . It is obvious that Cijk = 0 if and only if F is Riemannian. According to Deicke’s

theorem [5], a Finsler metric is Riemannian if and only if the mean Cartan torsion vanishes.

114 G. Chen et al. / Journal of Geometry and Physics 86 (2014) 112–121

Express the volume form of F by

dVF = σ(x)dx1 · · · dxn.

For a non-zero vector y ∈ TpM , the S-curvature S(y) is defined by

S(y) :=∂Gi

∂yi(x, y)−

yi

σ(x)∂σ

∂xi(x).

From the definition, we have

Eij =12Syiyj . (1)

We say that F is of isotropic S-curvature if

S = (n + 1)cF ,

where c = c(x) is a scalar function onM .The S-curvature was first introduced by Z. Shen when he studied volume comparison. The S-curvature has important

influence on the geometric structure of Finsler metrics.

Lemma 2.1 ([6]). Let (M, F) be an n-dimensional Finsler manifold of scalar flag curvature. Suppose that the S-curvature isisotropic, i.e., S = (n + 1)c(x)F . Then the flag curvature must be of the form

K =3cxmym

F+ σ ,

where σ = σ(x) is a scalar function on M.

The Landsberg curvature L := Lijkdxi ⊗ dxj ⊗ dxk and the mean Landsberg curvature J := Jidxi are defined respectively by

Lijk := −12FFymB m

i jk, Ji := g jkLijk.

Finsler metrics with (J = 0)L = 0 are called (weak)Landsberg metrics.Consider the (α, β)-metrics F = αφ(s), s = β/α on a manifold. Let Gi and Gi

α denote the spray coefficients of α and F ,respectively, then we have [1]

Gi= Gi

α + αQsi0 + {−2Qαs0 + r00}{Ψ bi +Θα−1yi}, (2)

where

Q :=φ′

φ − sφ′, Θ :=

Q − sQ ′

2∆, Ψ :=

Q ′

2∆(3)

and∆ := 1 + sQ + (b2 − s2)Q ′.Put

yi := aijyj, hi := αbi − syi,

Φ := −(Q − sQ ′){n∆+ 1 + sQ } − (b2 − s2)(1 + sQ )Q ′′,

Ψ1 :=

b2 − ss∆

12

√b2 − ssΦ

∆32

′

, Ψ2 := 2(n + 1)(Q − sQ ′)+ 3Φ

∆.

By a direct computation, we obtain the following formula about the mean Cartan torsion of (α, β)-metrics [7]

Ii := −Φ(φ − sφ′)

2∆φα2hi. (4)

By Deicke’s theorem, an (α, β)-metric is Riemannian if and only ifΦ ≡ 0.In [8], X. Cheng and Z. Shen have obtained the formula of S-curvature for an (α, β)-metric

S =

2Ψ −

f ′(b)bf (b)

(r0 + s0)−

Φ

2α∆2(r00 − 2αQs0), (5)

where

f (b) :=

π0 sinn−2 tT (b cos t)dt π

0 sinn−2 tdt

and T (s) := φ(φ − sφ′)n−2[(φ − sφ′)+ (b2 − s2)φ′′

].

G. Chen et al. / Journal of Geometry and Physics 86 (2014) 112–121 115

Further, the mean Landsberg curvature of an (α, β)-metric is given by [9]

Ji = −1

2∆α4

2α2

b2 − s2

Φ

∆+ (n + 1)(Q − sQ ′)

(r0 + s0)hi

+α

b2 − s2

Ψ1 + s

Φ

∆

(r00 − 2αQs0)hi + α[−αQ ′s0hi + αQ (α2si − yis0)

+α2∆si0 + α2(ri0 − 2αQsi)− (r00 − 2αQs0)yi]Φ

∆

. (6)

Contracting Ji with bi, we obtain

J := Jibi = −1

2∆α2{Ψ1(r00 − 2αQs0)+ αΨ2(r0 + s0)}.

If Finsler metric F is of scalar flag curvature, we have the following Bianchi identity [1]

Ji;mym + KF 2Ii = −1

n + 1F 2K·i,

where ‘‘;’’ denotes the horizontal covariant derivative with respect to F and K·i = Kyi . For an (α, β)-metric F = αφ(s), s =

β/α, it is easy to get [10,4]

J|0 − Jiail(rl0 + sl0)− Jl∂(Gl

− Glα)

∂yibi − 2

∂ J∂yl(Gl

− Glα)+ Kα2φ2Iibi = −

n + 13

α2φ2K·ibi, (7)

where J|0 := J|mym, rl0 := rlkyk and sl0 := slkyk.

3. Weak Berwald (α, β)-metrics

Let F = αφ(s), s = β/α be an (α, β)-metric on a manifold M , where α =aij(x)yiyj is a Riemannian metric and β =

bi(x)yi is a 1-form onM . Assuming that F is a weak Berwald metric, by (1), we have that its S-curvature satisfies S = ξ + κ ,where ξ := ξi(x)yi is a 1-form and κ := κ(x) is a scalar function on manifold M . The definition of S-curvature implies thatS(y) is positively homogeneous of degree one for any non-zero vector y ∈ TxM , then S = ξi(x)yi. It follows from (5) that

2Ψ −f ′(b)bf (b)

(r0 + s0)−

Φ

2α∆2(r00 − 2αQs0) = ξi(x)yi. (8)

To simplify the computations, we take an orthonormal basis at xwith respect to α such that

α =

ni=1

(yi)2, β = by1,

and take the following coordinate transformation [8] in TxM, ψ : (s, uA) → (yi):

y1 =s

√b2 − s2

α, yA = uA,

where α =

ni=2(uA)2. Here, our index conventions are

1 ≤ i, j, k, · · · ≤ n, 2 ≤ A, B, C, · · · ≤ n.We have

α =b

√b2 − s2

α, β =bs

√b2 − s2

α.

Furthers1 = 0, sA = bs1A, r1 = br11, rA = br1A,s0 = siyi = sAyA = bs10,

r0 =sα

√b2 − s2

br11 + br10,

r00 =s2α2r11b2 − s2

+2sαr10

√b2 − s2

+ r00,

where

s10 :=

nA=2

s1AyA, r10 :=

nA=2

r1AyA, r00 :=

nA,B=2

rAByAyB.

116 G. Chen et al. / Journal of Geometry and Physics 86 (2014) 112–121

Then (8) is equivalent to the following two equations in the special coordinate system (s, ya)

Φ

2∆2(b2 − s2)r00 = −s

sΦ2∆2

− 2Ψ b2r11 − bt1

α2, (9)

sΦ∆2

− 2Ψ b2(r1A + s1A)− (b2Q + s)

Φ

∆2s1A − btA = 0 (10)

where t1 := −f ′(b)bf (b) r1 − ξ1 and tA := −

f ′(b)bf (b) (rA + sA)− ξA.

Lemma 3.1. Let F = αφ(s), s = β/α be a non-Riemannian (α, β)-metric on a manifold M. Suppose that F is weak Berwaldmetric, then

Φ

2∆2(k − ϵs2) = 2Ψ s(k − ϵb2)+ sυ, (11)

and in addition, if there exists A0 such that sA0 = 0, then

−2Ψ −QΦ∆2

− λ

sΦ∆2

− 2Ψ b2

= δ, (12)

where λ, k, ϵ, υ and δ are scalar functions of x.Proof. Since Φ = 0 and notice that r00 and α are independent of s, then (9) implies that there is a function k = k(x)independent of s such that

rAB = kδAB, (13)

s

sΦ2∆2

− 2Ψ b2r11 + k

Φ

2∆2(b2 − s2) = bst1. (14)

Let

r11 = k − ϵb2, t1 = bυ

where ϵ = ϵ(x), υ = υ(x) are independent of s. Then we have (11) from (14).Differentiating (12) with respect to s yields

dds

sΦ∆2

− 2Ψ b2r1A −

dds

QΦ∆2

+ 2Ψb2s1A = 0. (15)

Let

λ := −r1A0

b2s1A0.

Then it follows from (15) that

δ := −2Ψ −QΦ∆2

− λ

sΦ∆2

− 2Ψ b2

is independent of s. �

Noting that b := ∥βx∥α =aijbibj, we can obtain

∂b∂xi

=bibi|mb

=ri + si

b.

The following lemma is obvious.

Lemma 3.2. Let β be a 1-form on a Riemannian manifold (M, α). Then b := ∥βx∥α =aijbibj = constant if and only if β

satisfies the following equation

ri + si = 0.

Next we are going to show that bmust be constant under the assumption in Theorem 1.1. Firstly, we prove

Proposition 3.3. Let F = αφ(s), s = β/α be a non-Riemannian (α, β)-metric on a manifold M of dimension n ≥ 3. Supposeb = constant and φ satisfies (12). Then F must be Finsler metric of Randers type.Proof. By assumption b = constant , i.e, db = 0, then b = constant in a neighborhood. We view b as a variable in (12). Let

eq := ∆2λ

sΦ∆2

− 2Ψ b2

+ 2Ψ +QΦ∆2

+ δ

.

G. Chen et al. / Journal of Geometry and Physics 86 (2014) 112–121 117

Denote the Taylor series of Q (s) in a neighborhood of s = 0

Q (s) = a0 + a1s + a2s2 + a3s3 + a4s4 + a5s5 + a6s6 + a7s7 + a8s8 + o(s8).

We claim a0 = 0. Suppose that a0 = 0, if ai = 0, i ≥ 2, we have Q (s) = a1s, that is

φ′

φ − sφ′= a1s.

Solving the above differential equation yields

φ(s) = k11 + a1s2.

Then F is Riemannian, it is a contraction. Thus there must be some minimal integerm(m ≥ 2) such that am = 0. ExpressingQ (s) as

Q (s) = a1s + amsm + am+1sm+1+ am+2sm+2

+ o(sm+2). (16)

Plugging (16) into eq1 and let qi be the coefficients of si in eq1. By q0 = (1+ a1b2)[−λa1b2 + a1 + (1+ a1b2)δ] = 0, we have

δ =a1(λb2 − 1)1 + a1b2

.

Plugging it into qm−1 = 0 and qm+1 = 0, we can obtain

Pλ+ P1 = 0,

Tλ+ T1 = 0, (17)

where P, P1, T , T1 are polynomials in b. By maple program, we have

P T1 − T P1 = (. . .)b8 + (. . .)b6 + (. . .)b4 + (. . .)b2 + a2mm(m − 1)(m + n + 1)

where the omitted terms in the brackets of the above are all constants. Thus (17) implies λ = 0, then

δ = −a1

1 + a1b2.

Plugging it and λ = 0 into qm−1, we have qm−1 = −m[a1b2(m− 1)− 1]am. Noting that am = 0, we get a contradiction. Thusa0 = 0.

By a direction computation, we have

q0 = (1 + a1b2)2δ − a1b2(1 + a1b2)λ+ a1 − na20 + a21b2− 2a0a2b2 − na20a1b

2− a20,

q1 = 2(1 + a1b2)(a0 + 2a2b2)δ − (1 + a1b2)[(n + 1)a0 + 4a2b2]λ+ 2a2 − na0a1 − 2a20a2b

2− (n + 1)a20 − na0a21b

2+ 2a1a2b2 − 6a0a3b2 − 2na20a2b

2.

By q0 = 0 and q1 = 0, we have

λ =E1E0, δ =

E2E0, (18)

where

E0 = (1 + a1b2)[(4a2 + (n − 1)a0a1)b2 + (n + 1)a0],E1 = [a0a1(2a0a2 − a21)n + 8a0a22 − 2a20a1a2 − 6a0a1a3 − 2a21a2]b

4

+ a0[(a20a1 + 2a0a2 − 2a21)n − a20a1 + 6a0a2 − 2a21 − 6a3]b2 + a0(a20 − a1)n + a30 − 2a0a1 + 2a2,

and

E2 = [a0a1(2a0a2 − a21)n + 8a0a22 − 2a20a1a2 − 6a0a1a3 − 2a21a2]b4

+ [a30a1n2+ a0(6a0a2 − 2a21)n − a30a1 − a0a21 + 6a20a2 − 2a1a2]b2 + a0[a20n

2+ (22

0 − a1)n + a20 − a1].

Substituting the Taylor series of Q (s) and (18) into eq, we have

E0(eq) = f2s2 + f3s3 + f4s4 + · · · ,

where fi(i = 2, 3, 4, . . .) are polynomials of b. Express f2 in the following

f2 = g0 + g2b2 + g4b4 + g6b6.

118 G. Chen et al. / Journal of Geometry and Physics 86 (2014) 112–121

By f2 = 0, we have g0 = g2 = g4 = g6 = 0. Solving the linear system yields

a2 =12a0a1, a3 = 0, a4 = −

18a0a21 (19)

or

a2 =12a0[(n + 1)a20 + a1], a3 = −

13(n + 1)a40,

a4 = −18a0[(n + 1)(n − 3)a40 + 2(n + 1)a20a1 + a21]. (20)

Case 1: (19) holds. Plugging (19) into (18), we have

λ = a20 − a1, δ =na20 + (a20 − a1)(1 + a1b2)

1 + a1b2.

Plugging them into eq, we get

(1 + a1b2)(eq) = Ξ0 + Ξ2b2 + Ξ4b4,

whereΞ0, Ξ2, Ξ4 are ODES about Q (s). By maple program, solvingΞ0 = 0,Ξ2 = 0 andΞ4 = 0, we have

Q (s) = a1s ± a01 + a1s2.

Noting that

Q (s) =φ′

φ − sφ,

we get the differential equation about φ(s)

φ′

φ − sφ= a1s ± a0

1 + a1s2.

Solving the above differential equation yields

φ(s) =

1 + a1s2 ± a0s.

Then F is Finsler metric of Randers type.Case 2: (20) holds. Plugging (20) into f3, we have

f3 = f30 + f32b2,

where f30 = −13 (n + 1)(n − 2)a50. Since a0 = 0 and n ≥ 3, it contradicts that f3 = 0. �

Lemma 3.4. Let F = αφ(s), s = β/α be a non-Riemannian (α, β)-metric on a manifold M of dimension n ≥ 3. Suppose φ =

k11 + k2s2 + k2s for any constant k1 > 0, k2 and k3 = 0. If F is weak Berwald metric, then

ri + si = 0.

Proof. Suppose that r0 + s0 = 0, by Lemma 3.2, then b = constant in a neighborhood. We view b as a variable in (11). Put

eq1 := 2∆2Φ

2∆2(k − ϵs2)− 2Ψ s(k − ϵb2)− sυ

.

Noting that k, ϵ, υ may dependent on b, Plugging the Taylor series of Q (s) in a neighborhood of s = 0

Q (s) = a0 + a1s + a2s2 + a3s3 + a4s4 + a5s5 + a6s6 + a7s7 + a8s8 + o(s8)

into eq1 and let pi be the coefficients of si in eq1, by a direction computation, we have p0 = −k[(na0a1 +2a2)b2 + (n+1)a0].In the following we prove k = 0, ϵ = 0, υ = 0.Case 1: a0 = 0. By p0 = 0, we have k = 0, then

p1 = −2a1b2 + 1

−a1ϵb2 + υa1b2 + υ

= 0.

The above equation implies

υ =a1b2ϵ

1 + a1b2.

Then we get p2 = {[(n − 2)a0a1 + 6a2]b2 + (n + 1)a0}ϵ. By p2 = 0 and a0 = 0, we have ϵ = 0. Thus k = ϵ = υ = 0.

G. Chen et al. / Journal of Geometry and Physics 86 (2014) 112–121 119

Case 2: a0 = 0. Since Q (s) = a1s, there must be some minimal integerm(m ≥ 2) such that am = 0.When m = 2, we get p0 = −2a2b2k. By p0 = 0, we have k = 0. Noting that the expression of p1 do not contain a0, we

still have

υ =a1b2ϵ

1 + a1b2.

Then by p2 = {[(n − 2)a0a1 + 6a2]b2 + (n + 1)a0}ϵ = 6a2b2ϵ = 0, we obtain k = ϵ = υ = 0.Whenm = 3, by a direction computation, we get p1 = 2a1b2(1+ a1b2)ϵ − 2[(a21 + 3a3)b2 + a1]k− 2(1+ a1b2)2υ , then

p1 = 0 implies

υ =a1b2(1 + a1b2)ϵ − [(a21 + 3a3)b2 + a1]k

(1 + a1b2)2. (21)

Plugging (21) into p3 = 0 and p5 = 0, we have the following linear system about k and ϵ

Pk + P1ϵ = 0,Tk + T1ϵ = 0, (22)

where P, P1, T , T1 are polynomials in b. By maple program, we obtainP P1T T1

= Mb4 + Nb2 − 4(n + 4)(n + 1)a23,

where

M = −(108n − 324)a23 − 600a25 + (52n − 140)a1a3a5 − 4(n − 3)(n − 5)a21a23 + 504a3a7,

and

N = 52(n + 1)a3a5 − 4(n + 1)(2n − 5)a1a23.

It follows from (22) that k = ϵ = 0. Plugging it into (21), we get υ = 0.Whenm > 3, by (21) and a3 = 0, we have

υ =a1(−k + ϵb2)

1 + a1b2, (23)

To prove k = ϵ = 0, we need to compute pm−2 and pm. Expressing Q (s) as

Q (s) = a1s + amsm + am+1sm+1+ am+2sm+2

+ o(sm+2), (24)

we have pm−2 = −m(m−1)b2amk. By pm−2 = 0, one gets k = 0. Plugging k = 0 into pm, we have pm = m(m+1)b2amϵ = 0,i.e, ϵ = 0. Then (23) implies υ = 0.

Nowwe claim that there exists A0 s.t sA0 = 0. Noting that r11 = k− ϵb2 and k = ϵ = 0, we have r1 + s1 = b(r11 + s11) =

br11 = 0. By assumption that r0 + s0 = 0, there exists A0 such that rA0 + sA0 = 0 which implies rA0 = 0 or sA0 = 0. If sA0 = 0,we have r1A0 = (1/b)rA0 = 0. By (10), the following holds

sΦ∆2

− 2Ψ b2

= γ , (25)

where γ := btA0/r1A0 . Plugging the Taylor series of Q (s)

Q (s) = a0 + a1s + a2s2 + a3s3 + a4s4 + a5s5 + a6s6 + a7s7 + a8s8 + o(s8)

into (25), we have p0 = −(1 + a1b2)[γ (1 + a1b2)+ a1b2] = 0. It implies

γ = −a1b2

1 + a1b2.

Then we have p1 = −[(n − 1)a0a1 + 4a2]b2 − (n + 1)a0 = 0. It implies a0 = 0. Since Q (s) = a1s, there is some minimalintegerm(m ≥ 2) such that am = 0. We find that the coefficient of sm−1, pm−1 = −m2b2am = 0. It is a contradiction.

By Lemma 3.1, (12) holds. Proposition 3.3 implies that F is Finslermetric of Randers type. It contradicts to the assumptionthat φ = k1

1 + k2s2 + k2s for any constant k1 > 0, k2 and k3 = 0. Hence b = constant , that is, ri + si = 0. �

Proof of Theorem 1.1. By Lemma 3.4, (9) and (10) are reduced to

Φ

2∆2r00(b2 − s2) = sbt1α2, (26)

−(s + b2Q )Φ

∆2s1A = btA. (27)

120 G. Chen et al. / Journal of Geometry and Physics 86 (2014) 112–121

Taking s = b in (26), we have t1 = 0. Then (26) becomes

Φ

2∆2r00(b2 − s2) = 0.

Noting thatΦ = 0, we have rAB = 0. Put h(s) := s + b2Q , we have

h(s) =(b2 − s2)φ′

+ sφφ − sφ′

.

Since φ(s) > 0 and φ(s)− sφ′(s) > 0, we have

h(b) =bφ(b)

φ(b)− bφ′(b)> 0, h(−b) =

−bφ(−b)φ(−b)+ bφ′(−b)

< 0.

Thus there exists c1 ∈ (−b, b) such that c1 + b2Q (c1) = 0. Taking s = c1 in (25), we get tA = 0. Then (27) becomes

Φ(s + b2Q )s1A = 0.

Assume that s1A = 0. Since Φ(s) = 0, there exists c2 ∈ (−b, b) such that Φ(c2) = 0, i.e, Φ(s) = 0 in some neighborhoodof c2 ∈ (−b, b). Then we have s + b2Q = 0 in this neighborhood. It implies that F is Riemannian. We get a contraction.Thus s1A = 0. Then sA = bs1A = 0 and s1 = bs11 = 0 imply si = 0. By si = 0 and ri + si = 0, we have r1A = (1/b)rA =

−(1/b)sA = 0. Noting that rAB = 0, we have rij = 0. �

4. (α, β)-metrics with scalar flag curvature

In this section, we prove Theorem 1.2. By Theorem 1.1 and (5), we have S = 0. Then Lemma 2.1 implies that the flagcurvature K = σ(x). In particular, by Schur theorem [5], we have the following

Proposition 4.1. Let F = αφ(s), s = β/α be a non-Riemannian (α, β)-metric of scalar flag curvature on a manifold M ofdimension n ≥ 3. Suppose φ = k1

1 + k2s2 + k2s for any constant k1 > 0, k2 and k3 = 0. If F is weak Berwald metric, then F

is of constant flag curvature.

In order to prove Theorem 1.2, we need to use (7). By Theorem 1.1, (2), (6) and (7) are simplified respectively as follows:

Gi− Gi

α = αQsi0,

Ji = −Φsi02∆α

,

J = 0.

Proof of Theorem 1.2. By a direction computation, it is easy to obtain the following

Jiail(rl0 + sl0) = −Φ

2∆αsi0si0,

Jl∂(Gl

− Glα)

∂yibi = −

Φ

2∆α[Qs + Q ′(b2 − s2)]si0si0,

J|0 = 0,∂ J∂yi(Gi

− Giα) = 0. (28)

It follows from (4) that

Iibi = −Φ

2∆αφ(φ − sφ′)(b2 − s2). (29)

Substituting (28) and (29) into (7), by Proposition 4.1, we have

si0si0 −φ(φ − sφ′)

∆α2σ(b2 − s2) = 0.

Rewrite the above equation as follows:

si0si0 − σD(s)α2= 0, (30)

where

D(s) :=φ(φ − sφ′)

∆(b2 − s2).

G. Chen et al. / Journal of Geometry and Physics 86 (2014) 112–121 121

Differentiating (30) with respect to yj, we obtain

sijsi0 + si0sij − σD′(s)(αbj − syj)− 2σD(s)yj = 0, (31)

where yj := aijyi. Contracting (31) with bj and by si = 0, we have

σ [D′(s)(b2 − s2)+ 2D(s)s] = 0.

If D′(s)(b2 − s2)+ 2D(s)s = 0, one gets

D(s) = µ(b2 − s2),

where µ = constant. Thus the following equation holds.

µ =φ(φ − sφ′)

∆=

(φ − sφ′)3

φ − sφ′ + (b2 − s2)φ′′.

Solving the above differential equation about φ(s) yields

φ(s) = k11 + k2s2 + k3s.

We get a contraction, Thus σ = 0. Then (30) implies sij = 0.In this case we have rij = sij = 0. By (2), F is Berwald metric. Nothing that K = σ = 0, we conclude that F is

Minkowskian. �

Acknowledgment

The third author was supported by the National Natural Science Foundation of China (11171254).

References

[1] S.S. Chern, Z. Shen, Rieman–Finsler Geometry, in: Nankai Tracts in Mathematics, vol. 6, World Scientific, 2005.[2] Z. Shen, On Landsberg (α, β)-metrics, Canad. J. Math. 61 (2009) 1357–1374.[3] X. Cheng, C. Xiang, On a class of weakly-Berwald (α, β)-metrics, J. Math. Res. Exposition 29 (2) (2009) 227–236.[4] X. Cheng, C. Lu, Two kinds of weak Berwald metrics of scalar flag curvature, J. Math. Res. Exposition 29 (4) (2009) 607–614.[5] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001.[6] X. Cheng, X. Mo, Z. Shen, On the flag curvature of Finsler metrics of scalar curvature, J. Lond. Math. Soc. 68 (2) (2003) 762–780.[7] X. Cheng, H. Wang, M. Wang, (α, β)-metrics with relatively isotropic mean Landsberg curvature, Publ. Math. Debrecen 72 (3–4) (2008) 475–485.[8] X. Cheng, Z. Shen, A class of Finsler metrics with isotropic S-curvature, Israel J. Math. 169 (1) (2009) 317–340.[9] B. Li, Z. Shen, On a class of weak Landsberg metrics, Sci. China. A 50 (1) (2007) 75–85.

[10] X. Cheng, On (α, β)-metrics of scalar flag curvature with constant S-curvature, Acta Math. Sin. (Engl. Ser.) 26 (9) (2010) 1701–1708.