PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 101(115) (2017), 183–190 https://doi.org/10.2298/PIM1715183T

ON MATSUMOTO CHANGE OF m-th ROOT FINSLER METRICS

Akbar Tayebi and Mohammad Shahbazi Nia

Abstract. We consider Matsumoto change of Finsler metrics. First, we find a condition under which the Matsumoto change of a Finsler metric is projectively related to it. Then, considering the subspace of m-th root Finsler metrics, if F̄ is the Matsumoto change of F, we prove that F̄ is locally projectively flat if and only if it is locally dually flat. In this case, F and F̄ reduce to locally Minkowskian metrics.

1. Introduction

Let (M, F) be a Finsler space. In 1984, C. Shibata studied the properties of Finsler space (M, F̄) whose fundamental metric function F̄ is obtained from F by the relation F (x, y) → F̄(x, y) = f(F, β), where β(x, y) = b_{i}(x)y^{i} is a 1-form on M and f = f(F, β) is a positively homogeneous function of F and β. This change of Finsler metric function has been called a β-change. He studied some geometrical properties of tensors being invariant by β-change of the metric [9]. If ||β||_{F} := sup_{F (x,y)=1} |β| < 1, then F̄ is again a Finsler metric.

There is a special case of β-change, namely

(1.1)  F̄(x, y) = \frac{F^{2}}{F − β}

which is called the Matsumoto change of F. If F reduces to a Riemannian metric α, then F̄ reduces to the Matsumoto metric F = \frac{α^{2}}{α−β} . Due to this reason, transformation (1.1) is called the Matsumoto change of Finsler metrics. The Matsumoto metric is an important metric in the Finsler geometry which is the Matsumoto’s slope-of-a-mountain metric. This metric was introduced by Matsumoto as a realization of Finsler’s idea “a slope measure of a mountain with respect to a time measure”.

Two Finsler metrics F and F̄ on a manifold M are called projectively related if any geodesic of the first is also geodesic for the second and vice versa. In this case, there is a scalar function P = P (x, y) defined on T M_{0} such that Ḡ^{i} = G^{i} + P y^{i}, where Ḡ^{i} and G^{i} are the geodesic spray coefficients of F̄ and F, respectively. In this paper, we find a condition under which the Matsumoto change of a Finsler metric is projectively related to it. Let (M, F) be a Finsler manifold and β = b_{i}(x)y^{i} a 1-form on M. Put r_{ij} := \frac{1}{2}(b_{i|j} + b_{j|i} ), s_{ij} := \frac{1}{2}(b_{i|j} − b_{j|i} ), r_{00} = r_{ij} y^{i}y^{j}, where | denotes the horizontal derivation with respect to the Berwald connection of F. Then, we have the following.

2010 Mathematics Subject Classification: Primary 53B40; Secondary 53C60.

Key words and phrases: Matsumoto change, locally dually flat metric, projectively flat metric, m-th root metric.

Communicated by Stevan Pilipović.

Theorem 1.1. Let (M, F) be a Finsler manifold. Suppose that F̄ = F^{2}/(F −β) be the Matsumoto change of F. Then F̄ is projectively related F if only if β satisfies

s_{ij} = [A_{j} r_{ik} − A_{i}r_{jk} ]y^{k}, where A_{i} := \frac{2(F_{i} − b_{i})F − 2F_{i}(F − β)}{F (F − β)} .

In this case, the projective factor is given by P = \frac{1}{2(F −β)} r_{00}.

Let M be an n-dimensional C^{∞} manifold, T M its tangent bundle. Let F = \sqrt[m]{A} be a Finsler metric on M, where A := a_{i_{1} ...i_{m}} (x)y^{i_{1}} y^{i_{2}} …y^{i_{m}} with a_{i_{1} ...i_{m}} symmetric in all its indices. Then F is called an m-th root Finsler metric [10–18]. The special m-th root metric in the form F = \sqrt[m]{y^{1}y^{2} …y^{m}} is called the Berwald– Moór metric [1–3, 5, 6].

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 ⊂ ℝ^{n} is locally projectively flat if and only if G^{i} = P y^{i}, where P = P (x, y) is called the projective factor and is a C^{∞} scalar function on T M_{0} satisfying P (x, λy) = λP (x, y) for all λ > 0.

Theorem 1.2. Let F = \sqrt[m]{A} (m > 2), be an m-th root Finsler metric on an open subset U ⊂ ℝ^{n}. Suppose that F̄ = F^{2}/(F − β) be the Matsumoto change of F. Then F̄ is locally projectively flat if and only if A_{x^{l}} = 0 and b_{i} = constant.

A Finsler metric F on a manifold M is said to be locally dually flat if at any point there is a coordinate system (x^{i}) in which the spray coefficients are in the form G^{i} = −\frac{1}{2}g^{ij} H_{y^{j}} , where H = H(x, y) is a positively homogeneous scalar function on T M_{0} = T M ∖ {0} [8].

Theorem 1.3. Let F = \sqrt[m]{A} (m > 2), be an m-th root Finsler metric on an open subset U ⊂ ℝ^{n}. Suppose that F̄ = F^{2}/(F − β) be the Matsumoto change of F. Then F̄ is a locally dually flat Finsler metric if and only if A_{x^{l}} = 0 and b_{i} = constant.

2. Proof of Theorem 1.1

In this section, we are going to prove Theorem 1.1. To this aim we first prove the following result.

ON MATSUMOTO CHANGE OF m-th ROOT FINSLER METRICS 185

Lemma 2.1 (Rapcsák [7]). Let F and F̄ be two Finsler metrics on a manifold M. Then F̄ is projectively related to F if and only if F̄ satisfies F̄_{|k,l} y^{k}−F̄_{|l} = 0, where | denotes the horizontal derivation with respect to the Berwald connection of F. In this case, the spray coefficients are related by Ḡ^{i} = G^{i} + P y^{i}, where

(2.1)  P = \frac{F̄_{|k}y^{k}}{2F̄} .

The P = P (x, y) is called the projective factor of F (x, y).

Throughout this paper, we use the Berwald connection and the h- and v- covariant derivatives of a Finsler tensor field are denoted by “ | " and “ , " respectively. Now, let (M, F) be a Finsler manifold and β = b_{i}(x)y^{i} a 1-form on M. Put

r_{ij} := \frac{1}{2}(b_{i|j} + b_{j|i} ),  r_{i0} := r_{ij} y^{j} , r_{00} := r_{ij} y^{i}y^{j} ,
s_{ij} := \frac{1}{2}(b_{i|j} − b_{j|i} ),  s_{i0} := s_{ij} y^{j} ,
R_{ij} := \frac{1}{2}( \frac{∂b_{i}}{∂x^{j}} + \frac{∂b_{j}}{∂x^{i}} ) , R_{0i} := R_{ji} y^{j} , R_{00} := R_{ij} y^{i}y^{j} .

Proof of Theorem 1.1. The following relations hold

b_{i|j} = \frac{∂b_{i}}{∂x^{l}} − Γ_{il}^{s} b_{s}, Γ_{il}^{s} = Γ_{li}^{s},

where Γ_{jk}^{i} = Γ_{jk}^{i} (x, y) is the Christoffel symbols of the Berwald connection of F. Then we have

s_{ij} := \frac{1}{2}(b_{i|j} − b_{j|i} ) = \frac{1}{2}( \frac{∂b_{i}}{∂x^{j}} − \frac{∂b_{j}}{∂x^{i}} ) , (2.2)

r_{ij} := \frac{1}{2}(b_{i|j} + b_{j|i} ) = \frac{1}{2}( \frac{∂b_{i}}{∂x^{j}} + \frac{∂b_{j}}{∂x^{i}} − 2b_{s}Γ_{ij}^{s} ) = R_{ij} − b_{s}Γ_{ij}^{s} . (2.3)

By (2.2) and (2.3), we get

β_{|l} = b_{i|l} y^{i} = ( \frac{∂b_{i}}{∂x^{l}} − Γ_{il}^{s} b_{s}) y^{i}, (2.4)

β_{|l}y^{l} = ( \frac{∂b_{i}}{∂x^{l}} − Γ_{il}^{s} b_{s}) y^{i}y^{l} = R_{00} − 2b_{s}G^{s} = r_{00}, (2.5)

β_{|k,l} y^{k} = ( \frac{∂b_{l}}{∂x^{k}} − b_{s}Γ_{lk}^{s}) y^{k}, (2.6)

whence (2.4) and (2.6) imply β_{|k,l} y^{k} − β_{|l} = s_{lk}y^{k} = s_{l0}. For F̄ = F^{2}/(F − β), we have

(2.7) \begin{matrix}F̄_{|l} = \frac{β_{|l}F^{2}}{(F − β)^{2}} , F̄_{|k,l} = [ \frac{(2F_{l}β_{|k} + F β_{|k,l} )(F − β) − 2(F_{l} − b_{l})F β_{|k}}{(F − β)^{3}} ]F, \\ F̄_{|k,l} y^{k} = [ \frac{(2F_{l}r_{00} + F β_{|k,l} y^{k})(F − β) − 2(F_{l} − b_{l})F r_{00}}{(F − β)^{3}} ]F. \end{matrix}

Then

F̄_{|k,l} y^{k} − F̄_{|l} = [ \frac{(2F_{l}r_{00} + F β_{|k,l} y^{k} − F^{2}β_{|l})(F − β) − 2(F_{l} − b_{l})F r_{00}}{(F − β)^{3}} ]F

= [ \frac{(2F_{l}r_{00} + F s_{l0})(F − β) − 2(F_{l} − b_{l})F r_{00}}{(F − β)^{3}} ]F.

Put

A_{i} := \frac{2(F_{i} − b_{i})F − 2F_{i}(F − β)}{F (F − β)} .

Then by Lemma 2.1, F̄ is projectively related to F if and only if s_{l0} = A_{l}r_{00}, which, taking a vertical derivation, yields

(2.8)  s_{li} = A_{li}r_{00} + 2A_{l}r_{0i}.

Since s_{li} = −s_{il}, then by (2.8) we get A_{li}r_{00} + 2A_{l}r_{0i} = −A_{il} r_{00} − 2A_{i}r_{0l} or

(2.9)  A_{li}r_{00} = −A_{l}r_{0i} − A_{i}r_{0l}.

By (2.8) and (2.9), we get s_{ij} = A_{i}r_{0j} − A_{j} r_{0i}. Now, by (2.5) and (2.7), we have

F̄_{|y}y^{k} = \frac{β_{|k}y^{k}F^{2}}{(F − β)^{2}} = \frac{F^{2}r_{00}}{(F − β)^{2}} .

By (2.1), it follows that

P = \frac{F̄_{|k}y^{k}}{2F̄} = \frac{r_{00}}{2(F − β)} .  □

3. Proof of Theorem 1.2

It is known that a Finsler metric F (x, y) on U ⊂ R^{n} is projective if and only if its geodesic coefficients G^{i} are of the form G^{i}(x, y) = P (x, y)y^{i}, where P: T U = U × R^{n} → R is positively homogeneous of degree one with respect to y. In [4], Hamel showed that a Finsler metric F on U ⊂ R^{n} is projectively flat if and only if it satisfies F_{x^{k} y^{l}} y^{k} = F_{x^{l}} .

Lemma 3.1. Let F = \sqrt[m]{A} (m > 2), be an m-th root Finsler metric on an open subset U ⊂ R^{n}. Suppose that the equation

ΨA^{\frac{2}{m} −1} + ΞA^{\frac{2}{m}} + ΦA^{\frac{1}{m}} + ΘA^{\frac{1}{m} +1} + ΥA^{\frac{1}{m} −1} + A^{\frac{1}{m} +2} Ω + Γ = 0

holds, where Φ, Ψ, Θ, Υ, Ω, Ξ are homogeneous polynomials in y. Then Ψ = Ξ = Φ = Θ = Υ = Ω = Γ = 0.

For an m-th root metric F = \sqrt[m]{A}, put

A_{i} = \frac{∂A}{∂y^{i}} , A_{ij} = \frac{∂^{2}A}{∂y^{j} ∂y^{j}} , A_{x^{i}} = \frac{∂A}{∂x^{i}} , A_{0} = A_{x^{i}} y^{i}, A_{0l} = A_{x^{k} y^{l}} y^{k} = \frac{∂^{2}A}{∂x^{i}∂y^{l}} y^{k}.

Then we have the following.

Proof of Theorem 1.2. For F̄ = \frac{\sqrt[m]{A^{2}}}{\sqrt[m]{A}−β} , we infer
(3.1)  [F̄]_{x^{l}} = \frac{1}{m( \sqrt[m]{A} − β)^{2}} [A^{\frac{3}{m} −2} A_{x^{l}} − 2A^{\frac{2}{m} −1} A_{x^{l}} β + mA^{\frac{2}{m}} β_{x^{l}} ],

[F̄]_{x^{k} y^{l}} y^{k} = \frac{1}{m}[ \frac{(\frac{1}{m} − 2)A_{0}A_{l}A^{\frac{4}{m} −3} +A_{0l}A^{\frac{4}{m} −2} +(2β_{l}A_{0} − A_{0l}β + 2A_{0}A_{l}β)A^{\frac{3}{m} −2}}{(A^{\frac{1}{m}} − β)^{3}}

+\frac{(−2βA_{0l} − 2A_{0}β_{l})A^{\frac{3}{m} −1} + (2 − \frac{3}{m})βA_{0}A_{l}A^{\frac{3}{m} −3} + mβ_{0l}A^{\frac{3}{m}}}{(A^{\frac{1}{m}} − β)^{3}}

ON MATSUMOTO CHANGE OF m-th ROOT FINSLER METRICS 187

+\frac{(2β^{2}A_{0l} − 2βA_{l}β_{0} − 6βA_{0}β_{l})A^{\frac{2}{m} −1} + (\frac{4}{m} − 2)β^{2}A_{0}A_{l}A^{\frac{2}{m} −2}}{(A^{\frac{1}{m}} − β)^{3}} (3.2)

+\frac{(2mβ_{0}β_{l} − mββ_{0l})A^{\frac{2}{m}}}{(A^{\frac{1}{m}} − β)^{3}} ],

where

β_{x^{i}} := \frac{∂β}{∂x^{i}} , β_{i} := \frac{∂β}{∂y^{i}} = b_{i}, β_{0} := β_{x^{i}} y^{i}, β_{0l} := β_{x^{i} l}y^{i}.

Since F̄ is locally projectively flat metric, we have [F̄]_{x^{k} y^{l}} y^{k} − [F̄]_{x^{l}} = 0. By substituting (3.1) and (3.2) into this, we get (3.3) ( \frac{1}{m} − 2)A_{0}A_{l}A^{\frac{4}{m} −3} + (A_{0l} − A_{x^{l}} )A^{\frac{4}{m} −2}

+ [2β_{l}A_{0} − β(A_{0l} − A_{x^{l}} ) + 2A_{0}A_{l}β]A^{\frac{3}{m} −2} + [ − 2β(A_{0l} − A_{x^{l}} ) − 2A_{0}β_{l}]A^{\frac{3}{m} −1} + (2 − \frac{3}{m} )βA_{0}A_{l}A^{\frac{3}{m} −3} + m(β_{0l} − β_{x^{l}} )A^{\frac{3}{m}} + [2β^{2}(A_{0l} − A_{x^{l}} ) − 2βA_{l}β_{0} − 6βA_{0}β_{l}]A^{\frac{2}{m} −1}

+ ( \frac{4}{m} − 2)β^{2}A_{0}A_{l}A^{\frac{2}{m} −2} + [2mβ_{0}β_{l} − mβ(β_{0l} − β_{x^{l}} )]A^{\frac{2}{m}} = 0.

Simplifying (3.3), it results that ( \frac{1}{m} − 2)A_{0}A_{l}A^{\frac{2}{m} −1} + (A_{0l} − A_{x^{l}} )A^{\frac{2}{m}} (3.4)

+ [2β_{l}A_{0} − β(A_{0l} − A_{x^{l}} ) + 2A_{0}A_{l}β]A^{\frac{1}{m}}
− [2β(A_{0l} − A_{x^{l}} ) + 2A_{0}β_{l}]A^{\frac{1}{m} +1} + (2 − \frac{3}{m} )βA_{0}A_{l}A^{\frac{1}{m} −1}
+ m(β_{0l} − β_{x^{l}} )A^{\frac{1}{m} +2} + m[2β_{0}β_{l} − β(β_{0l} − β_{x^{l}} )]A^{2}
+ 2β[β(A_{0l} − A_{x^{l}} ) − A_{l}β_{0} − 3A_{0}β_{l}]A + ( \frac{4}{m} − 2)β^{2}A_{0}A_{l} = 0.

According to Lemma 3.1, (3.4) reduces to the following

(3.5)  A_{0}A_{l} = 0,
(3.6)  A_{0l} − A_{x^{l}} = 0,
2β_{l}A_{0} − β(A_{0l} − A_{x^{l}} ) + 2A_{0}A_{l}β = 0,
β(A_{0l} − A_{x^{l}} ) + A_{0}β_{l} = 0,
(3.7)  β_{0l} − β_{x^{l}} = 0,
2β_{0}β_{l} − β(β_{0l} − β_{x^{l}} ) = 0,
(3.8)  β(A_{0l} − A_{x^{l}} ) − A_{l}β_{0} − 3A_{0}β_{l} = 0.

By (3.6), we have

(3.9)  A_{x^{l}} − A_{0l} = 0.

The relations (3.5), A_{l} ≠ 0 and β ≠ 0 imply that

(3.10)  A_{0} = 0.

Taking a vertical derivation of (3.10) yields

(3.11)  A_{x^{l}} + A_{0l} = 0.

By (3.9) and (3.11), we get A_{x^{l}} = 0. On the other hand, by substituting (3.9) and (3.10) in (3.8), we have β_{0} = 0. Taking a vertical derivation of it implies that β_{0l} + β_{x^{l}} = 0. By considering (3.7), we get β_{x^{l}} = 0, which means that b_{i} are constants. □

4. Proof of Theorem 1.3

In [8], Shen proved that the Finsler metric F on an open subset U ⊂ R^{n} is dually flat if and only if it satisfies (F^{2})_{x^{k} y^{l}} y^{k} = 2(F^{2})_{x^{l}} . Now, we are going to characterize locally dually flat Finsler metrics which is obtained by a Matsumoto change of m-th root metrics. First, we remark the following.

Lemma 4.1. Let F = \sqrt[m]{A} (m > 2), be an m-th root Finsler metric on an open subset U ⊂ ℝ^{n}. Suppose that the equation

ΨA^{\frac{2}{m} +1} + ΞA^{\frac{2}{m}} + ΦA^{\frac{1}{m}} + ΘA^{\frac{1}{m} +1} + ΥA^{\frac{1}{m} +2} + Ω = 0

holds, where Φ, Ψ, Θ, Υ, Ω, Ξ are homogeneous polynomials in y. Then Ψ = Ξ = Φ = Θ = Υ = Ω = 0.

Proof of Theorem 1.3. The following holds

(4.1)  [F̄^{2}]_{x^{k}} = \frac{2A^{\frac{4}{m} −1} [A^{\frac{1}{m}} A_{x^{l}} − 2A_{x^{l}} β + mAβ_{x^{l}} ]}{m(A^{\frac{1}{m}} − β)^{3}} ,

[F̄^{2}]_{x^{k} y^{l}} y^{k} = \frac{2}{m}[ \frac{A_{0l}A^{\frac{6}{m} −1} + (β_{l}A_{0} − 3A_{0l}β + A_{l}β_{0})A^{\frac{5}{m} −1} + (\frac{2}{m} − 1)A_{0}A_{l}A^{\frac{6}{m} −2}}{(A^{\frac{1}{m}} − β)^{4}}

+\frac{(\frac{3}{m} − 1)βA_{0}A_{l}A^{\frac{5}{m} −2} + mβ_{0l}A^{\frac{5}{m}} + (\frac{8}{m} − 2)β^{2}A_{0}A_{l}A^{\frac{4}{m} −2}}{(A^{\frac{1}{m}} − β)^{4}} (4.2)

+\frac{(2β^{2}A_{0l} − 4βA_{0}β_{l} + 4βA_{l}β_{0})A^{\frac{4}{m} −1} + mββ_{0l}A^{\frac{4}{m}}}{(A^{\frac{1}{m}} − β)^{4}}  ].

Since F̄ is a locally dually flat metric, then

(4.3)  [F̄^{2}]_{x^{k} y^{l}} y^{k} − 2[F̄^{2}]_{x^{l}} = 0.

By substituting (4.1) and (4.2) in (4.3), we infer:

(4.4) ( A_{0l} − 2A_{x^{l}} ) A^{\frac{6}{m} −1} + [β_{l}A_{0} − 3(A_{0l} − 2A_{x^{l}} )β + A_{l}β_{0}]A^{\frac{5}{m} −1}

+ ( \frac{2}{m} − 1) A_{0}A_{l}A^{\frac{6}{m} −2} + ( \frac{3}{m} − 1) βA_{0}A_{l}A^{\frac{5}{m} −2} + m(β_{0l} − 2β_{x^{l}} )A^{\frac{5}{m}}

+ 2β[β(A_{0l} − 2A_{x^{l}} ) − 2A_{0}β_{l} + 2A_{l}β_{0}]A^{\frac{4}{m} −1} + ( \frac{8}{m} − 2) β^{2}A_{0}A_{l}A^{\frac{4}{m} −2}

+ mβ(β_{0l} − 2β_{x^{l}} )A^{\frac{4}{m}} = 0.

Simplifying (4.4) implies that

(4.5) (A_{0l} − 2A_{x^{l}} )A^{\frac{2}{m} +1} + [β_{l}A_{0} − 3(A_{0l} − 2A_{x^{l}} )β + A_{l}β_{0}]A^{\frac{1}{m} +1}

+ ( \frac{2}{m} − 1) A_{0}A_{l}A^{\frac{2}{m}} + ( \frac{3}{m} − 1) βA_{0}A_{l}A^{\frac{1}{m}} + m( β_{0l} − 2β_{x^{l}} ) A^{\frac{1}{m} +2}

+ mβ(β_{0l} − 2β_{x^{l}} )A^{2} + 2β[β(A_{0l} − 2A_{x^{l}} ) − 2A_{0}β_{l} + 2A_{l}β_{0}]A

+ ( \frac{8}{m} − 2) β^{2}A_{0}A_{l} = 0.

By Lemma 4.1, (4.5) reduces to

(4.6)  A_{0}A_{l} = 0,
(4.7)  A_{0l} − 2A_{x^{l}} = 0,
2β(β_{0l} − 2β_{x^{l}} ) = 0,
(4.8)  3β(A_{0l} − 2A_{x^{l}} ) − A_{0}β_{l} − A_{l}β_{0} = 0,
(4.9)  β_{0l} − 2β_{x^{l}} = 0,
β(A_{0l} − 2A_{x^{l}} ) + 2(β_{l}A_{0} + β_{0}A_{l}) = 0.

By (4.6), we have A_{0} = 0. Taking a vertical derivation of it implies that A_{x^{l}} + A_{0l} = 0. Then by (4.7), it follows that A_{x^{l}} = 0. In this case, (4.8) reduces to β_{0} = 0. Taking a vertical derivation of it implies that β_{0l} + β_{x^{l}} = 0. Then by (4.9), we get β_{x^{l}} = 0 which means that b_{i} are constants. This completes the proof. □

Corollary 4.1. Let F = \sqrt[m]{A} (m > 2) be an m-th root Finsler metric on an open subset U ⊂ ℝ^{n}. Suppose that F̄ = F^{2}/(F − β) be the Matsumoto change of F. Then F̄ is locally projectively flat if and only if it is locally dually flat. In this case, F and F̄ are Berwald–Moór metrics.

Proof. By Theorem 1.2 and 1.3, F̄ is locally projectively flat if and only if it is locally dually flat. Since A_{x^{i}} = 0, then a_{i_{1} ...i_{m}} (x) = c is a constant. In this case, we get

F = \sqrt[m]{c y^{1}y^{2} …y^{m}}

which is a locally Minkowskian metric. Since b_{i} = constant and F̄ = F^{2}/(F − β), then F̄ is locally Minkowskian, too. □

5. Conclusion

Every locally Minkowskian metric is locally projectively flat and locally dually flat metric. In this paper, we study the Matsumoto change of a Finsler metric and prove that the Matsumoto change of an m-th root metric is locally projectively flat if and only if it is locally dually flat if and only if it is locally Minkowskian. The study of this Finslerian change on an m-th root metric will enhance our understanding of the geometric meaning of the class of m-th root metrics.

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Department of Mathematics (Received 18 08 2014) Faculty of Science (Revised 13 06 2016 and 27 07 2016) University of Qom Qom, Iran

  akbar.tayebi@gmail.com
  m.shahbazinia@gmail.com