On Special Berwald Metrics
In this paper, we study a class of Finsler metrics which contains the class of Berwald metrics as a special case. We prove that every Finsler metric in this class is a generalized Douglas-Weyl metric. Then we study isotropic flag curvature Finsler metrics in this class. Finally we show that on this...
Збережено в:
| Дата: | 2010 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2010
|
| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/146118 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On Special Berwald Metrics / E. Tayebi // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 16 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-146118 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1461182019-02-08T01:23:17Z On Special Berwald Metrics Tayebi, A. Peyghan, E. In this paper, we study a class of Finsler metrics which contains the class of Berwald metrics as a special case. We prove that every Finsler metric in this class is a generalized Douglas-Weyl metric. Then we study isotropic flag curvature Finsler metrics in this class. Finally we show that on this class of Finsler metrics, the notion of Landsberg and weakly Landsberg curvature are equivalent. 2010 Article On Special Berwald Metrics / E. Tayebi // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 16 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53C60; 53C25 http://dspace.nbuv.gov.ua/handle/123456789/146118 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In this paper, we study a class of Finsler metrics which contains the class of Berwald metrics as a special case. We prove that every Finsler metric in this class is a generalized Douglas-Weyl metric. Then we study isotropic flag curvature Finsler metrics in this class. Finally we show that on this class of Finsler metrics, the notion of Landsberg and weakly Landsberg curvature are equivalent. |
| format |
Article |
| author |
Tayebi, A. Peyghan, E. |
| spellingShingle |
Tayebi, A. Peyghan, E. On Special Berwald Metrics Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Tayebi, A. Peyghan, E. |
| author_sort |
Tayebi, A. |
| title |
On Special Berwald Metrics |
| title_short |
On Special Berwald Metrics |
| title_full |
On Special Berwald Metrics |
| title_fullStr |
On Special Berwald Metrics |
| title_full_unstemmed |
On Special Berwald Metrics |
| title_sort |
on special berwald metrics |
| publisher |
Інститут математики НАН України |
| publishDate |
2010 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/146118 |
| citation_txt |
On Special Berwald Metrics / E. Tayebi // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 16 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT tayebia onspecialberwaldmetrics AT peyghane onspecialberwaldmetrics |
| first_indexed |
2025-07-10T23:12:16Z |
| last_indexed |
2025-07-10T23:12:16Z |
| _version_ |
1837303480533909504 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 008, 9 pages
On Special Berwald Metrics
Akbar TAYEBI † and Esmaeil PEYGHAN ‡
† Department of Mathematics, Faculty of Science, Qom University, Qom, Iran
E-mail: akbar.tayebi@gmail.com
‡ Department of Mathematics, Faculty of Science, Arak University, Arak, Iran
E-mail: epeyghan@gmail.com, e-peyghan@araku.ac.ir
Received November 01, 2009, in final form January 17, 2010; Published online January 20, 2010
doi:10.3842/SIGMA.2010.008
Abstract. In this paper, we study a class of Finsler metrics which contains the class
of Berwald metrics as a special case. We prove that every Finsler metric in this class is
a generalized Douglas–Weyl metric. Then we study isotropic flag curvature Finsler metrics
in this class. Finally we show that on this class of Finsler metrics, the notion of Landsberg
and weakly Landsberg curvature are equivalent.
Key words: Randers metric; Douglas curvature; Berwald curvature
2010 Mathematics Subject Classification: 53C60; 53C25
1 Introduction
For a Finsler metric F = F (x, y), its geodesics curves are characterized by the system of diffe-
rential equations c̈i + 2Gi(ċ) = 0, where the local functions Gi = Gi(x, y) are called the spray
coefficients. A Finsler metric F is called a Berwald metric if Gi = 1
2Γi
jk(x)yjyk is quadratic in
y ∈ TxM for any x ∈ M . It is proved that on a Berwald space, the parallel translation along
any geodesic preserves the Minkowski functionals [7]. Thus Berwald spaces can be viewed as
Finsler spaces modeled on a single Minkowski space.
Recently by using the structure of Funk metric, Chen–Shen introduce the notion of isotropic
Berwald metrics [6, 16]. This motivates us to study special forms of Berwald metrics.
Let (M,F ) be a two-dimensional Finsler manifold. We refer to the Berwald’s frame (`i,mi)
where `i = yi/F (y), mi is the unit vector with `im
i = 0, `i = gij`
i and gij is the fundamental
tensor of Finsler metric F . Then the Berwald curvature is given by
Bi
jkl = F−1
(
−2I,1`
i + I2m
i
)
mjmkml, (1)
where I is 0-homogeneous function called the main scalar of Finsler metric and I2 = I,2 + I,1|2
(see [2, page 689]). By (1), we have
Bi
jkl = − 2I,1
3F 2
(
mjhkl + mkhjl + mlhjk
)
yi +
I2
3F
(
hi
jhkl + hi
khjl + hi
lhjk
)
,
where hij := mimj is called the angular metric. Using the special form of Berwald curvature
for Finsler surfaces, we define a new class of Finsler metrics on n-dimensional Finsler manifolds
which their Berwald curvature satisfy in following
Bi
jkl = (µjhkl + µkhjl + µlhjk)yi + λ
(
hi
jhkl + hi
khjl + hi
lhjk
)
, (2)
where µi = µi(x, y) and λ = λ(x, y) are homogeneous functions of degrees −2 and −1 with re-
spect to y, respectively. By definition of Berwald curvature, the function µi satisfies µiy
i=0 [12].
mailto:akbar.tayebi@gmail.com
mailto:epeyghan@gmail.com
file:e-peyghan@araku.ac.ir
http://dx.doi.org/10.3842/SIGMA.2010.008
2 A. Tayebi and E. Peyghan
The Douglas tensor is another non-Riemanian curvature defined as follows
Di
jkl :=
(
Gi − 1
n + 1
∂Gm
∂ym
yi
)
yjykyl
. (3)
Douglas curvature is a non-Riemannian projective invariant constructed from the Berwald curva-
ture. The notion of Douglas curvature was proposed by Bácsó and Matsumoto as a generalization
of Berwald curvature [4]. We show that a Finsler metric satisfies (2) with vanishing Douglas
tensor is a Randers metric (see Proposition 1). A Finsler metric is called a generalized Douglas–
Weyl (GDW) metric if the Douglas tensor satisfy in hi
αDα
jkl|mym = 0 [10]. In [5], Bácsó–Papp
show that this class of Finsler metrics is closed under projective transformation. We prove that
a Finsler metric satisfies (2) is a GDW-metric.
Theorem 1. Every Finsler metric satisfying (2) is a GDW-metric.
Theorem 1, shows that every two-dimensional Finsler metric is a generalized Douglas–Weyl
metric.
For a Finsler manifold (M,F ), the flag curvature is a function K(P, y) of tangent planes
P ⊂ TxM and directions y ∈ P . F is said to be of isotropic flag curvature if K(P, y) = K(x)
and constant flag curvature if K(P, y) = const.
Theorem 2. Let F be a Finsler metric of non-zero isotropic flag curvature K = K(x) on
a manifold M . Suppose that F satisfies (2). Then F is a Riemannian metric if and only if µi
is constant along geodesics.
Beside the Berwald curvature, there are several important Finslerian curvature. Let (M,F )
be a Finsler manifold. The second derivatives of 1
2F 2
x at y ∈ TxM0 is an inner product gy
on TxM . The third order derivatives of 1
2F 2
x at y ∈ TxM0 is a symmetric trilinear forms Cy
on TxM . We call gy and Cy the fundamental form and the Cartan torsion, respectively. The
rate of change of the Cartan torsion along geodesics is the Landsberg curvature Ly on TxM
for any y ∈ TxM0. Set Jy :=
n∑
i=1
Ly(ei, ei, ·), where {ei} is an orthonormal basis for (TxM, gy).
Jy is called the mean Landsberg curvature. F is said to be Landsbergian if L = 0, and weakly
Landsbergian if J = 0 [13, 14].
In this paper, we prove that on Finsler manifolds satisfies (2), the notions of Landsberg and
weakly Landsberg metric are equivalent.
Theorem 3. Let (M,F ) be a Finsler manifold satisfying (2). Then L = 0 if and only if J = 0.
There are many connections in Finsler geometry [15]. In this paper, we use the Berwald
connection and the h- and v-covariant derivatives of a Finsler tensor field are denoted by “|”
and “,” respectively.
2 Preliminaries
Let M be a n-dimensional C∞ manifold. Denote by TxM the tangent space at x ∈ M , by
TM = ∪x∈MTxM the tangent bundle of M , and by TM0 = TM \ {0} the slit tangent bundle
on M . A Finsler metric on M is a function F : TM → [0,∞) which has the following properties:
(i) F is C∞ on TM0; (ii) F is positively 1-homogeneous on the fibers of tangent bundle TM ,
and (iii) for each y ∈ TxM , the following quadratic form gy on TxM is positive definite,
gy(u, v) :=
1
2
d2
dsdt
[
F 2(y + su + tv)
] ∣∣
s,t=0
, u, v ∈ TxM.
On Special Berwald Metrics 3
Let x ∈ M and Fx := F |TxM . To measure the non-Euclidean feature of Fx, define Cy: TxM ×
TxM × TxM → R by
Cy(u, v, w) :=
1
2
d
dt
[gy+tw(u, v)] |t=0, u, v, w ∈ TxM.
The family C := {Cy}y∈TM0 is called the Cartan torsion. It is well known that C = 0 if and
only if F is Riemannian [14]. For y ∈ TxM0, define mean Cartan torsion Iy by Iy(u) := Ii(y)ui,
where Ii := gjkCijk, gjk is the inverse of gjk and u = ui ∂
∂xi |x. By Deicke’s theorem, F is
Riemannian if and only if Iy = 0 [13].
Let α =
√
aij(x)yiyj be a Riemannian metric, and β = bi(x)yi be a 1-form on M with
b =
√
aijbibj < 1. The Finsler metric F = α + β is called a Randers metric.
Let (M,F ) be a Finsler manifold. Then for a non-zero vector y ∈ TxM0, define the Mat-
sumoto torsion My : TxM ⊗ TxM ⊗ TxM → R by My(u, v, w) := Mijk(y)uivjwk where
Mijk := Cijk − 1
n+1{Iihjk + Ijhik + Ikhij},
hij := FFyiyj = gij − 1
F 2 gipy
pgjqy
q is the angular metric and Ii := gjkCijk is the mean Cartan
torsion. By definition, we have hijy
i = 0, hi
j = δi
j − F−2yiyj , yj = gijy
i, hi
jhik = hjk and
hi
i = n− 1. A Finsler metric F is said to be C-reducible if My = 0. This quantity is introduced
by Matsumoto [8]. Matsumoto proves that every Randers metric satisfies that My = 0. Later
on, Matsumoto–Hōjō proves that the converse is true too.
Lemma 1 ([9]). A Finsler metric F on a manifold of dimension n ≥ 3 is a Randers metric if
and only if My = 0, ∀ y ∈ TM0.
Let us consider the pull-back tangent bundle π∗TM over TM0 defined by
π∗TM = {(u, v) ∈ TM0 × TM0| π(u) = π(v)} .
Let ∇ be the Berwald connection. Let {ei}n
i=1 be a local orthonormal (with respect to g) frame
field for the pulled-back bundle π∗TM such that en = `, where ` is the canonical section of π∗TM
defined by `y = y/F (y). Let {ωi}n
i=1 be its dual co-frame field. Put ∇ei = ωj
i⊗ ej , where {ωj
i}
is called the connection forms of ∇ with respect to {ei}. Put ωn+i := ωi
n + d(log F )δi
n. It is
easy to show that {ωi, ωn+i}n
i=1 is a local basis for T ∗(TM0). Since {Ωj
i} are 2-forms on TM0,
they can be expanded as
Ωj
i = 1
2Rj
iklω
k ∧ ωl + Bj
iklω
k ∧ ωn+l.
Let {ēi, ėi}n
i=1 be the local basis for T (TM0), which is dual to {ωi, ωn+i}n
i=1. The objects R
and B are called, respectively, the hh- and hv-curvature tensors of the Berwald connection
with the components R(ēk, ēl)ei = Rj
iklej and P (ēk, ėl)ei = P j
iklej [15]. With the Berwald
connection, we define covariant derivatives of quantities on TM0 in the usual way. For example,
for a scalar function f , we define f|i and f·i by
df = f|iω
i + f,iω
n+i,
where “|” and “,” denote the h- and v-covariant derivatives, respectively.
The horizontal covariant derivatives of C along geodesics give rise to the Landsberg curvature
Ly : TxM × TxM × TxM → R defined by
Ly(u, v, w) := Lijk(y)uivjwk,
where Lijk := Cijk|sy
s, u = ui ∂
∂xi |x, v = vi ∂
∂xi |x and w = wi ∂
∂xi |x. The family L := {Ly}y∈TM0
is called the Landsberg curvature. A Finsler metric is called a Landsberg metric if L=0. The
4 A. Tayebi and E. Peyghan
horizontal covariant derivatives of I along geodesics give rise to the mean Landsberg curvature
Jy(u) := Ji(y)ui, where Ji := gjkLijk. A Finsler metric is said to be weakly Landsbergian if
J = 0.
Given a Finsler manifold (M,F ), then a global vector field G is induced by F on TM0, which
in a standard coordinate (xi, yi) for TM0 is given by
G = yi ∂
∂xi
− 2Gi(x, y)
∂
∂yi
,
where Gi(y) are local functions on TM given by
Gi(y) :=
1
4
gil(y)
{
∂2[F 2]
∂xk∂yl
(y)yk − ∂[F 2]
∂xl
(y)
}
, y ∈ TxM.
G is called the spray associated to (M,F ). In local coordinates, a curve c(t) is a geodesic if and
only if its coordinates (ci(t)) satisfy c̈i + 2Gi(ċ) = 0.
For a tangent vector y ∈ TxM0, define By : TxM ⊗ TxM ⊗ TxM → TxM and Ey : TxM ⊗
TxM → R by By(u, v, w) := Bi
jkl(y)ujvkwl ∂
∂xi |x and Ey(u, v) := Ejk(y)ujvk where
Bi
jkl(y) :=
∂3Gi
∂yj∂yk∂yl
(y), Ejk(y) := 1
2Bm
jkm(y).
B and E are called the Berwald curvature and mean Berwald curvature, respectively. Then F
is called a Berwald metric and weakly Berwald metric if B = 0 and E = 0, respectively [14].
By definition of Berwald and mean Berwald curvatures, we have
yjBi
jkl = ykBi
jkl = ylBi
jkl = 0, yjEjk = ykEjk = 0.
The Riemann curvature Ry = Ri
kdxk ⊗ ∂
∂xi |x : TxM → TxM is a family of linear maps on
tangent spaces, defined by
Ri
k = 2
∂Gi
∂xk
− yj ∂2Gi
∂xj∂yk
+ 2Gj ∂2Gi
∂yj∂yk
− ∂Gi
∂yj
∂Gj
∂yk
.
The flag curvature in Finsler geometry is a natural extension of the sectional curvature in
Riemannian geometry was first introduced by L. Berwald [3]. For a flag P = span{y, u} ⊂ TxM
with flagpole y, the flag curvature K = K(P, y) is defined by
K(P, y) :=
gy(u,Ry(u))
gy(y, y)gy(u, u)− gy(y, u)2
.
When F is Riemannian, K = K(P ) is independent of y ∈ P , and is the sectional curvature of P .
We say that a Finsler metric F is of scalar curvature if for any y ∈ TxM , the flag curvature
K = K(x, y) is a scalar function on the slit tangent bundle TM0. If K = const, then F is said to
be of constant flag curvature. A Finsler metric F is called isotropic flag curvature, if K = K(x).
In [1], Akbar-Zadeh considered a non-Riemannian quantity H which is obtained from the
mean Berwald curvature by the covariant horizontal differentiation along geodesics. This is
a positively homogeneous scalar function of degree zero on the slit tangent bundle. The quantity
Hy = Hijdxi⊗dxj is defined as the covariant derivative of E along geodesics [11]. More precisely
Hij := Eij|mym.
In local coordinates, we have
2Hij = ym ∂4Gk
∂yi∂yj∂yk∂xm
− 2Gm ∂4Gk
∂yi∂yj∂yk∂ym
− ∂Gm
∂yi
∂3Gk
∂yj∂yk∂ym
− ∂Gm
∂yj
∂4Gk
∂yi∂yk∂ym
.
Akbar-Zadeh proved the following:
Theorem 4 ([1]). Let F be a Finsler metric of scalar curvature on an n-dimensional mani-
fold M (n ≥ 3). Then the flag curvature K = const if and only if H = 0.
On Special Berwald Metrics 5
3 Proof of Theorem 1
Lemma 2. Let (M,F ) be a Finsler manifold. Suppose that the Cartan tensor satisfies in
Cijk = Bihjk + Bjhik + Bkhij with yiBi = 0. Then F is a C-reducible metric.
Proof. Suppose that the Cartan tensor of the Finsler metric F satisfies in
Cijk = Bihjk + Bjhik + Bkhij . (4)
Contracting (4) with gij yields
Ik = Bih
i
k + Bjh
j
k + (n− 1)Bk. (5)
Using (5) and Bih
i
k = Bjh
j
k = Bk, we get Ii = (n + 1)Bi. Putting this relation in (4), we
conclude that F is a C-reducible Finsler metric. �
Lemma 3. Let (M,F ) be a Finsler metric. Then F is a GDW-metric if and only if
Di
jkl|sy
s = Tjkly
i, (6)
for some tensor Tjkl on manifold M .
Proof. Let F be is a GDW-metric
hi
mDm
jkl|sy
s = 0.
This yields
Di
jkl|sy
s =
(
F−2ymDm
jkl|s
)
yi.
Therefore Tjkl := F−2ymDm
jkl|s. The proof of converse is trivial. �
Equation (6) is equivalent to the condition that, for any parallel vector fields U = U(t),
V = V (t) and W = W (t) along a geodesic c(t), there is a function T = T (t) such that
d
dt
[Dċ(U, V,W )] = T ċ.
The geometric meaning of the above identity is that the rate of change of the Douglas curvature
along a geodesic is tangent to the geodesic.
Proposition 1. Let (M,F ) be a Finsler manifold satisfies (2) with dimension n ≥ 3. Suppose
that the Douglas tensor of F vanishes. Then F is a Randers metric.
Proof. Since F satisfies (2), then by considering µiy
i = 0 we get
2Ejk = (n + 1)λhij . (7)
On the other hand, we have
hij,k = 2Cijk − F−2(yjhik + yihjk),
which implies that
2Ejk,l = (n + 1)λ,lhjk + (n + 1)λ
{
2Cjkl − F−2(ykhjl + yjhkl)
}
. (8)
6 A. Tayebi and E. Peyghan
Putting (2), (7) and (8) in (3) yields
Di
jkl = {µjhkl + µkhjl + µlhjk − 2λCjkl}yi −
(
λylF
−2 + λ,l
)
hjky
i. (9)
For the Douglas curvature, we have Di
jkl = Di
jlk. Then by (9), we conclude that
λylF
−2 + λ,l = 0. (10)
From (9) and (10) we deduce
Di
jkl = {µjhkl + µkhjl + µlhjk − 2λCjkl}yi. (11)
Since F is a Douglas metric, then
Cjkl = 1
2λ{µjhkl + µkhjl + µlhjk}.
By Lemmas 2 and 1, it follows that F is a Randers metric. �
Proof of Theorem 1. To prove the Theorem 1, we start with the equation (11):
Di
jkl = {µjhkl + µkhjl + µlhjk − 2λCjkl}yi. (12)
Taking a horizontal derivation of (12) implies that
Di
jkl|sy
s = {µ′
jhkl + µ′
khjl + µ′
lhjk − 2λ′Cjkl − 2λLjkl}yi.
where λ′ = λ|mym and µ′
i = µi|mym. By Lemma 3, F is a GDW-metric with
Tjkl = µ′
jhkl + µ′
khjl + µ′
lhjk − 2λ′Cjkl − 2λLjkl.
This completes the proof. �
The Funk metric on a strongly convex domain Bn ⊂ Rn is a non-negative function on TΩ =
Ω × Rn, which in the special case Ω = Bn (the unit ball in the Euclidean space Rn) is defined
by the following explicit formula:
F (y) :=
√
|y|2 − (|x|2|y|2 − 〈x, y〉2)
1− |x|2
+
〈x, y〉
1− |x|2
, y ∈ TxBn = Rn,
where | · | and 〈·, ·〉 denote the Euclidean norm and inner product in Rn, respectively [14]. The
Funk metric on Bn is a Randers metric. The Berwald curvature of Funk metric is given by
Bi
jkl = 1
2F
{
hi
jhkl + hi
khjl + hi
lhjk + 2Cjkly
i
}
.
Thus the Funk metric is a GDW-metric which does not satisfy (2). Then by Theorem 1, we
conclude the following.
Corollary 1. The class of Finsler metrics satisfying (2) is a proper subset of the class of
generalized Douglas–Weyl metrics.
On Special Berwald Metrics 7
4 Proof of Theorem 2
To prove Theorem 2, we need the following.
Lemma 4 ([7, 11]). For the Berwald connection, the following Bianchi identities hold:
Ri
jkl|m + Ri
jlm|k + Ri
jmk|l = 0,
Bi
jml|k −Bi
jkm|l = Ri
jkl,m, (13)
Bi
jkl,m = Bi
jkm,l.
Proof of Theorem 2. We have:
Ri
jkl =
1
3
{
∂2Ri
k
∂yj∂yl
−
∂2Ri
l
∂yj∂yk
}
. (14)
Here, we assume that a Finsler metric F is of isotropic flag curvature K = K(x). In local
coordinates, Ri
k = K(x)F 2hi
k. Plugging this equation into (14) gives
Ri
jkl = K{gjlδ
i
k − gjkδ
i
l}. (15)
Differentiating (15) with respect to ym gives a formula for Ri
jkl,m expressed in terms of K and
its derivatives. Contracting (13) with yk, we obtain
Bi
jml|ky
k = 2KCjmly
i. (16)
Multiplying (16) with yi implies that
Bi
jml|ky
kyi = 2KF 2Cjml. (17)
Since F satisfies (2), then we have
Bi
jkl|mym = (µ′
jhkl + µ′
khjl + µ′
lhjk)yi + λ′(hi
jhkl + hi
khjl + hi
lhjk). (18)
By contracting (18) with yi, we have
Bi
jkl|mymyi = (µ′
jhkl + µ′
khjl + µ′
lhjk)F 2. (19)
By (17) and (19) we get
µ′
jhkl + µ′
khjl + µ′
lhjk = 2KCjkl.
Contracting with gkl yields
µ′
j =
2K
n + 1
Ij .
Since K 6= 0, then by Deicke’s theorem F is a Riemannian metric if and only if µ′
j = 0. �
Theorem 5. Let F be a Finsler metric on an n-dimensional manifold M (n ≥ 3) and satis-
fies (2). Suppose that F is of scalar flag curvature K. Then K = const if and only if λ′ = 0.
Proof. Contracting i and l in (2) yields
2Ejk = (n + 1)λhjk.
By taking a horizontal derivative of this equation, we have
2Hjk = (n + 1)λ′hjk.
Therefore Hjk = 0 if and only if λ′ = 0. By Theorem 4, we get the proof. �
8 A. Tayebi and E. Peyghan
5 Proof of Theorem 3
In this section, we are going to prove Theorem 3.
Proof of Theorem 2. Let F be a Finsler metric satisfy in following
Bi
jkl = (µjhkl + µkhjl + µlhjk)yi + λ
(
hi
jhkl + hi
khjl + hi
lhjk
)
, (20)
where µi = µi(x, y) and λ = λ(x, y) are homogeneous functions of degrees −2 and −1 with
respect to y, respectively. Contracting (20) with yi yields
yiB
i
jkl = F 2(µjhkl + µkhjl + µlhjk) + λyi
(
hi
jhkl + hi
khjl + hi
lhjk
)
. (21)
On the other hand, we have
yiB
i
jkl = −2Ljkl, (22)
yih
i
m = yi
(
δi
m − F−2yiym
)
= 0. (23)
See [14, page 84]. Using (21), (22) and (23), we get
Ljkl = −1
2F 2{µjhkl + µkhjl + µlhjk}. (24)
By (24), it is obvious that if µi = 0 then Ljkl = 0. Conversely let F be a Landsberg metric.
Then we have
µjhkl + µkhjl + µlhjk = 0. (25)
Contracting (25) with gkl yields µj = 0. Then F is a Landsberg metric if and only if µj = 0.
Now, contracting (24) with gkl yields
Jj = −1
2(n + 1)F 2µj . (26)
By (26), Jj = 0 if and only if µj = 0. Then L = 0 if and only if J = 0. �
By using the notion of Landsberg curvature, we define the stretch curvature Σy : TxM ⊗
TxM ⊗ TxM ⊗ TxM → R by Σy(u, v, w, z) := Σijkl(y)uivjwkzl where
Σijkl := 2(Lijk|l − Lijl|k).
In [3], L. Berwald has introduce the stretch curvature tensor Σ and showed that this tensor
vanishes if and only if the length of a vector remains unchanged under the parallel displacement
along an infinitesimal parallelogram.
Theorem 6. Let (M,F ) be a Finsler manifold on which (2) holds. Suppose that F is a stretch
metric. Then µj is constant along any Finslerian geodesics.
Proof. Taking a horizontal derivation of (24) yields
Lijk|l = −1
2F 2{µi|lhjk + µj|lhki + µk|lhij}.
Suppose that Σ = 0. Then by Lijk|l = Lijl|k, we get
µi|lhjk + µj|lhki + µk|lhij = µi|khjl + µj|khli + µl|khij . (27)
Multiplying (27) with yl implies that
µ′
ihjk + µ′
jhki + µ′
khij = 0. (28)
By contracting (28) with gjk, we conclude the following
(n + 1)µ′
i = 0.
Then on a stretch Finsler spaces, µi is constant along any geodesics. �
On Special Berwald Metrics 9
References
[1] Akbar-Zadeh H., Sur les espaces de Finsler à courbures sectionnelles constantes, Acad. Roy. Belg. Bull. Cl.
Sci. (5) 74 (1988), no. 10, 271–322.
[2] Antonelli P.L., Handbook of Finsler geometry, Kluwer Academic Publishers, Dordrecht, 2003.
[3] Berwald L., Über Parallelübertragung in Räumen mit allgemeiner Massbestimmung, Jahresbericht D.M.V.
34 (1926), 213–220.
[4] Bácsó S., Matsumoto M., On Finsler spaces of Douglas type – a generalization of notion of Berwald space,
Publ. Math. Debrecen 51 (1997), 385–406.
[5] Bácsó S., Papp I., A note on a generalized Douglas space, Period. Math. Hungar. 48 (2004), 181–184.
[6] Chen X., Shen Z., On Douglas metrics, Publ. Math. Debrecen 66 (2005), 503–512.
[7] Ichijyō Y., Finsler manifolds modeled on a Minkowski space, J. Math. Kyoto Univ. 16 (1976), 639–652.
[8] Matsumoto M., On C-reducible Finsler spaces, Tensor (N.S.) 24 (1972), 29–37.
[9] Matsumoto M., Hōjō S., A conclusive theorem for C-reducible Finsler spaces, Tensor (N.S.) 32 (1978),
225–230.
[10] Najafi B., Shen Z., Tayebi A., On a projective class of Finsler metrics, Publ. Math. Debrecen 70 (2007),
211–219.
[11] Najafi B., Shen Z., Tayebi A., Finsler metrics of scalar flag curvature with special non-Riemannian curvature
properties, Geom. Dedicata 131 (2008), 87–97.
[12] Pande H.D., Tripathi P.N., Prasad B.N., On a special form of the hv-curvature tensor of Berwald’s connection
BΓ of Finsler space, Indian J. Pure. Appl. Math. 25 (1994), 1275–1280.
[13] Shen Z., Lectures on Finsler geometry, World Scientific Publishing Co., Singapore, 2001.
[14] Shen Z., Differential geometry of spray and Finsler spaces, Kluwer Academic Publishers, Dordrecht, 2001.
[15] Tayebi A., Azizpour E., Esrafilian E., On a family of connections in Finsler geometry, Publ. Math. Debrecen
72 (2008), 1–15.
[16] Tayebi A., Rafie Rad M., S-curvature of isotropic Berwald metrics, Sci. China Ser. A 51 (2008), 2198–2204.
http://dx.doi.org/10.1023/B:MAHU.0000038974.24588.83
http://dx.doi.org/10.1007/s10711-007-9218-9
http://dx.doi.org/10.1007/s11425-008-0095-y
1 Introduction
2 Preliminaries
3 Proof of Theorem 1
4 Proof of Theorem 2
5 Proof of Theorem 3
References
|