On Kropina change for m-th root Finsler metrics
We study the Kropina change for m-th root Finsler metrics and establish necessary and sufficient condition under which the Kropina change of an m-th root Finsler metric is locally dually flat. Then we prove that the Kropina change of an m-th root Finsler metric is locally projectively flat if and on...
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irk-123456789-1651292020-02-24T19:10:07Z On Kropina change for m-th root Finsler metrics Tayebi, A. Tabatabaeifar, T. Peyghan, E. Короткі повідомлення We study the Kropina change for m-th root Finsler metrics and establish necessary and sufficient condition under which the Kropina change of an m-th root Finsler metric is locally dually flat. Then we prove that the Kropina change of an m-th root Finsler metric is locally projectively flat if and only if it is locally Minkowskian. Розглянуто замшу Кропіної для m-кореневих фінслерових метрик. Встановлено необхідні та достатні умови того, що заміна Кропіної для m-кореневої метрики Фінслера є локально дуально плоскою. Також доведено, що заміна Кропіної для m-кореневої метрики Фінслера є локально проективно плоскою тоді i тільки тоді, коли вона є локально мінковською. 2014 Article On Kropina change for m-th root Finsler metrics / A. Tayebi, T. Tabatabaeifar, E. Peyghan // Український математичний журнал. — 2014. — Т. 66, № 1. — С. 140–144. — Бібліогр.: 11 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165129 517.9 en Український математичний журнал Інститут математики НАН України |
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Короткі повідомлення Короткі повідомлення Tayebi, A. Tabatabaeifar, T. Peyghan, E. On Kropina change for m-th root Finsler metrics Український математичний журнал |
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We study the Kropina change for m-th root Finsler metrics and establish necessary and sufficient condition under which the Kropina change of an m-th root Finsler metric is locally dually flat. Then we prove that the Kropina change of an m-th root Finsler metric is locally projectively flat if and only if it is locally Minkowskian. |
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Article |
| author |
Tayebi, A. Tabatabaeifar, T. Peyghan, E. |
| author_facet |
Tayebi, A. Tabatabaeifar, T. Peyghan, E. |
| author_sort |
Tayebi, A. |
| title |
On Kropina change for m-th root Finsler metrics |
| title_short |
On Kropina change for m-th root Finsler metrics |
| title_full |
On Kropina change for m-th root Finsler metrics |
| title_fullStr |
On Kropina change for m-th root Finsler metrics |
| title_full_unstemmed |
On Kropina change for m-th root Finsler metrics |
| title_sort |
on kropina change for m-th root finsler metrics |
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Інститут математики НАН України |
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2014 |
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Короткі повідомлення |
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http://dspace.nbuv.gov.ua/handle/123456789/165129 |
| citation_txt |
On Kropina change for m-th root Finsler metrics / A. Tayebi, T. Tabatabaeifar, E. Peyghan // Український математичний журнал. — 2014. — Т. 66, № 1. — С. 140–144. — Бібліогр.: 11 назв. — англ. |
| series |
Український математичний журнал |
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AT tayebia onkropinachangeformthrootfinslermetrics AT tabatabaeifart onkropinachangeformthrootfinslermetrics AT peyghane onkropinachangeformthrootfinslermetrics |
| first_indexed |
2025-07-14T17:57:01Z |
| last_indexed |
2025-07-14T17:57:01Z |
| _version_ |
1837646030905016320 |
| fulltext |
UDC 517.9
A. Tayebi, T. Tabatabaeifar (Univ. Qom, Iran),
E. Peyghan (Arak Univ., Iran)
ON KROPINA CHANGE OF m-TH ROOT FINSLER METRICS
ПРО ЗАМIНУ КРОПIНОЇ ДЛЯ m-КОРЕНЕВИХ ФIНСЛЕРОВИХ МЕТРИК
We study the Kropina change for m-th root Finsler metrics. We find necessary and sufficient condition under which the
Kropina change of an mth root Finsler metric is locally dually flat. Then we prove that the Kropina change of an mth root
Finsler metric is locally projectively flat if and only if it is locally Minkowskian.
Розглянуто замiну Кропiної для m-кореневих фiнслерових метрик. Встановлено необхiднi та достатнi умови того,
що замiна Кропiної для m-кореневої метрики Фiнслера є локально дуально плоскою. Також доведено, що замiна
Кропiної дляm-кореневої метрики Фiнслера є локально проективно плоскою тодi i тiльки тодi, коли вона є локально
мiнковською.
1. Introduction. Let M be an n-dimensional C∞ manifold, TM its tangent bundle. Let F = m
√
A
be a Finsler metric on M, where A is given by A := ai1...im(x)yi1yi2 . . . yim with ai1...im symmetric
in all its indices [3, 8 – 11]. Then F is called an m-th root Finsler metric. Suppose that Aij define a
positive definite tensor and Aij denotes its inverse. For an m-th root metric F, put
Ai =
∂A
∂yi
, Aij =
∂2A
∂yj∂yj
, Axi =
∂A
∂xi
, A0 = Axiyi.
Then the following hold:
gij =
A
2
m−2
m2
[
mAAij + (2−m)AiAj
]
,
yiAi = mA, yiAij = (m− 1)Aj , yi =
1
m
A
2
m−1Ai,
AijAjk = δik, AijAi =
1
m− 1
yj , AiAjA
ij =
m
m− 1
A.
Let (M,F ) be a Finsler manifold. For a 1-form β(x, y) = bi(x)yi on M, we have a change of Finsler
which is defined by following:
F (x, y)→ F̄ (x, y) = f(F, β),
where f(F, β) is a positively homogeneous function of F. This is called a β-change of metric. It is
easy to see that, if ‖β‖F := supF (x,y)=1
∣∣bi(x)yi
∣∣ < 1, then F̄ is again a Finsler metric [7].
In this paper, we consider a special case of β-change, namely
F̄ (x, y) =
F 2(x, y)
β(x, y)
(1)
c© A. TAYEBI, T. TABATABAEIFAR, E. PEYGHAN, 2014
140 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
ON KROPINA CHANGE OF m-TH ROOT FINSLER METRICS 141
which is called the Kropina change. If F reduces to a Riemannian metric α, then F̄ reduces to a
Kropina metric F =
α2
β
. Due to this reason, the transformation (1) has been called the Kropina
change of Finsler metrics. It is remarkable that, the Kropina metrics are closely related to physical
theories. These metrics, was introduced by Berwald in connection with a two-dimensional Finsler
space with rectilinear extremal and was investigated by Kropina [5].
In [2], Amari, Nagaoka introduced the notion of dually flat Riemannian metrics when they study
the information geometry on Riemannian manifolds. Information geometry has emerged from in-
vestigating the geometrical structure of a family of probability distributions and has been applied
successfully to various areas including statistical inference, control system theory and multi terminal
information theory [1]. In Finsler geometry, Shen extends the notion of locally dually flatness for
Finsler metrics [6]. A Finsler metric F on an open subset U ⊂ Rn is called dually flat if it satisfies
(F 2)xkyly
k = 2(F 2)xl .
In this paper, we find necessary and sufficient condition under which a Kropina change of an
m-th root metric be locally dually flat.
Theorem 1.1. Let F = m
√
A be an m-th root Finsler metric on an open subset U ⊂ Rn, where
A is irreducible. Suppose that F̄ =
F 2
β
be Kropina change of F where β = bi(x)yi. Then F̄ is
locally dually flat if and only if there exists a 1-form θ = θl(x)yl on U such that the following hold:
β0lβ − 3βlβ0 = 2ββxl , (2)
Axl =
1
3m
[mAθl + 4θAl], (3)
β0Al = −βlA0, (4)
where β0l = βxkyly
k, βxl = (bi)xlyi, β0 = βxlyi and β0l = (bl)0.
A Finsler metric is said to be locally projectively flat if at any point there is a local coordinate
system in which the geodesics are straight lines as point sets. It is known that a Finsler metric
F (x, y) on an open domain U ⊂ Rn is locally projectively flat if and only if Gi = Pyi, where
P (x, λy) = λP (x, y), λ > 0 [4].
In this paper, we prove that the Kropina change of an m-th root Finsler metric is locally projec-
tively flat if and only if it is locally Minkowskian.
Theorem 1.2. Let F = m
√
A be an m-th root Finsler metric on an open subset U ⊂ Rn, where
A is irreducible. Suppose that F̄ =
F 2
β
be Kropina change of F where β = bi(x)yi. Then F̄ is
locally projectively flat if and only if it is locally Minkowskian.
2. Proof of Theorem 1.1. A Finsler metric F = F (x, y) on a manifold M is said to be locally
dually flat if at any point there is a standard coordinate system (xi, yi) in TM such that L = F 2(x, y)
satisfies
Lxkyly
k = 2Lxl .
In this case, the coordinate (xi) is called an adapted local coordinate system. It is easy to see that
every locally Minkowskian metric satisfies in the above equation, hence is locally dually flat.
In this section, we are going to prove the Theorem 1.1. To prove it, we need the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
142 A. TAYEBI, T. TABATABAEIFAR, E. PEYGHAN
Lemma 2.1. Suppose that the equation ΦA2 + ΨA+ Θ = 0 holds, where Φ, Ψ, Θ are polyno-
mials in y and m > 2. Then Φ = Ψ = Θ = 0.
Proof of Theorem 1.1. Let F̄ be a locally dually flat metric. We have
F̄ 2 =
A
4
m
β2
,
(
F̄ 2
)
xk =
1
β2
4
m
A
4
m−1Axk −
2
β3
A
4
mβxk ,
(F̄ 2)xkyly
k =
1
β2
[
4
m
A
4
m
−1A0l +
(
4
m
)(
4
m
− 1
)
A
4
m−2A0Al
]
−
− 2
β3
[
4
m
A
4
m−1Alβ0 +
4
m
A
4
m−1A0βl +A
4
mβ0l +
6
β4
A
4
mβ0βl
]
.
Thus, we get
A
4
m−2
β4
[
4
m
β2
((
4
m
− 1
)
A0Al +AA0l − 2AAxl
)
− 8
m
Aβ(Alβ0 +A0βl)+
+2A2
(
3β0βl + 2ββxl − ββ0l
)]
= 0.
By Lemma 2.1, we obtain (
4
m
− 1
)
AlA0 +AA0l = 2AAxl , (5)
β0Al = −A0βl, (6)
β0lβ − 3βlβ0 = 2βxlβ. (7)
One can rewrite (5) as follows:
A(2Axl −A0l) =
(
4
m
− 1
)
AlA0. (8)
Irreducibility of A and deg(Al) = m− 1 imply that there exists a 1-form θ = θly
l on U such that
A0 = θA. (9)
Plugging (9) into (8), yields
A0l = Aθl + θAl −Axl . (10)
Substituting (9) and (10) into (8) yields (3). The converse is a direct computation.
Theorem 1.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
ON KROPINA CHANGE OF m-TH ROOT FINSLER METRICS 143
3. Proof of Theorem 1.2. A Finsler metric F (x, y) on an open domain U ⊂ Rn is said to be
locally projectively flat if its geodesic coefficients Gi are in the form Gi(x, y) = P (x, y)yi, where
P : TU = U × Rn → R is positively homogeneous with degree one, P (x, λy) = λP (x, y), λ > 0.
We call P (x, y) the projective factor of F.
In this section, we are going to prove the Theorem 1.2. To prove it, we need the following
proposition.
Proposition 3.1. Let F = A
1
m be an m-th root Finsler metric on an open subset U ⊂ Rn,
n ≥ 3, where A is irreducible. Suppose that F̄ =
F 2
β
be Kropina change of F where β = bi(x)yi.
If F̄ is projectively flat metric then it reduces to a Berwald metric.
Proof. Let F̄ be projectively flat metric. We have
F̄xk =
2
mβ
A
2
m−1Axk −
1
β2
A
2
mβxk ,
F̄xkyly
k =
1
β
[
2
m
A
2
m−1A0l +
(
2
m
)(
2
m
− 1
)
A
2
m−2A0Al
]
−
− 1
β2
[
2
m
A
2
m−1Alβ0 +
2
m
A
2
m−1A0βl +A
2
mβ0l +
2
β3
A
2
mβ0βl
]
.
Thus, we get
A
2
m
−2
β3
[
2
m
β2
((
2
m
− 1
)
A0Al +AA0l −AAxl
)
−
− 2
m
Aβ
(
Alβ0 +A0βl
)
+A2
(
2β0βl + ββxl − ββ0l
)]
= 0.
By Lemma 2.1, we obtain
mA(A0l −Axl) = (m− 2)A0Al.
Then irreducibility of A and deg(Al) = m − 1 < deg(A) implies that A0 is divisible by A. This
means that, there is a 1-form θ = θly
l on U such that the following holds A0 = 2mAθ. Then
Gi = Pyi, where P = θ. Then F is a Berwald metric.
Proposition 3.1 is proved.
Proof of Theorem 1.2. By Proposition 3.1, if F is projectively flat then it reduces to a Berwald
metric. Now, if n ≥ 3 then by Numata’s theorem every Berwald metric of non-zero scalar flag
curvature K must be Riemaniann. This is contradicts with our assumption. Then K = 0, and in this
case F reduces to a locally Minkowskian metric.
Theorem 1.2 is proved.
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Received 31.05.12
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
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