On conformal -changes of more generalized
Abolfazl
1 2 Departmentof mathematics, Faculty of Mathematical Science, University of
ABSTARCT:
A change of Finsler metric
, , , where
and is conformal factor. The present paper is devoted mainly to studying the conditions for more
generalized -th root metrics conformal -change.
Keywords: -th root metric; more generalized
1. Introduction
Studying Finsler geometry one encounters substantial difficulties trying
sometimes even local, results of Riemannian geometry. These difficultiesarise mainly from the fact that in
Finsler geometry all geometric objects depend not only onpositional coordinates, as in Riemannian geometry,
but also on directional arguments.
Let ( , ) be an n-dimensional Finsler manifold. For a differential one
Randers [1], in 1941, introduced a special Finsler space defined by the change
where is Riemannian. M. Matsumoto [2], in 1974, studied Randers space and generalized Randers space in
which is Finslerian. On the other hand, in 1976, M. Hashiguchi [3] studied the conformal change of Finsler
metrics, namely, ,
named C-conformal. This change has been studied by many authors ([4], [5]). In 2008, S. Abed ([6], [7])
introduced the transformation
,
Moreover, he established the relationships between some important tensors associated with
corresponding tensors associated with
changes of more generalized m-throot metrics
Abolfazl Taleshian
1,Dordi Mohamad Saghali
2of mathematics, Faculty of Mathematical Science, University of Mazandaran, Babolsar, Iran
1Supervisor 2
M.SC.Thesis
, → , is called a conformal -change of
, is a one-form on an n-dimensional smooth manifold is conformal factor. The present paper is devoted mainly to studying the conditions for more
+ and + ,
th root metric; more generalized -th root metric;Randers "-change;conformal
Studying Finsler geometry one encounters substantial difficulties trying to seek analoguesof classical global, or
sometimes even local, results of Riemannian geometry. These difficultiesarise mainly from the fact that in
Finsler geometry all geometric objects depend not only onpositional coordinates, as in Riemannian geometry,
dimensional Finsler manifold. For a differential one-form " ,
Randers [1], in 1941, introduced a special Finsler space defined by the change ,
is Riemannian. M. Matsumoto [2], in 1974, studied Randers space and generalized Randers space in
is Finslerian. On the other hand, in 1976, M. Hashiguchi [3] studied the conformal change of Finsler
, . In particular, he also dealt with the special conformal transformation conformal. This change has been studied by many authors ([4], [5]). In 2008, S. Abed ([6], [7])
, " , . (1.1)
Moreover, he established the relationships between some important tensors associated with
corresponding tensors associated with , . He also studied some invariant and #-invariant properties and
42
root metrics
Mazandaran, Babolsar, Iran
change of , if , dimensional smooth manifold $ is conformal factor. The present paper is devoted mainly to studying the conditions for more
, When is established
conformal"-change.
to seek analoguesof classical global, or
sometimes even local, results of Riemannian geometry. These difficultiesarise mainly from the fact that in
Finsler geometry all geometric objects depend not only onpositional coordinates, as in Riemannian geometry,
, %& &on , G.
, " , , is Riemannian. M. Matsumoto [2], in 1974, studied Randers space and generalized Randers space in
is Finslerian. On the other hand, in 1976, M. Hashiguchi [3] studied the conformal change of Finsler
articular, he also dealt with the special conformal transformation
conformal. This change has been studied by many authors ([4], [5]). In 2008, S. Abed ([6], [7])
Moreover, he established the relationships between some important tensors associated with , and the
obtained a relationship between the Cartan connection associated with
connection associated with , .
In 1979, Shimada [8] introduced the
'
wherethe coefficients (&)&*…&' are the components of symmetric covariant tensor field of order
functions of positional co-ordinates only. Since then various geometers such as [
theory of -th root metric and studied its transformations.
There exist the following important one class of Finsler metric,
, -'* . / , (1.3)
where- =(&)&*…&' &) &*… &', .
generalized -th root metric and more general generalized
reversible Finsler metric and is Randers change of generalized
In this paper, we have considered a transformation of the more generalized
transforms to a similar metric as the conformal
is replaced with more generalized
two more generalized -th root metrics
change.
In overall this paper,
-0 (&)&*…&'* &) &*… &'*,
B2 %&3 & 3 ,
lationship between the Cartan connection associated with , and the transformed Cartan
] introduced the -th root metric on the differentiable manifold defined as:
(&)&*…&' &) &*… &' '
(1.2)
are the components of symmetric covariant tensor field of order
ordinates only. Since then various geometers such as [9], [10],
th root metric and studied its transformations.
There exist the following important one class of Finsler metric,
-'* . ,
. = %&3 & 3 and/ 45 5 , that is one 1-form. This forms are called a root metric and more general generalized -th root metric, respectively.
reversible Finsler metric and is Randers change of generalized -th root metric .
we have considered a transformation of the more generalized -th root metricsuch that it
transforms to a similar metric as the conformal"-change one de_ned in (1.1) ina way that the Riemannian metric
ized -th root metrics ,defined in (1.3). Then, we obtain the conditions among
th root metrics ,2 -2
*
') .2+/2and,0 -0
*
'* .0+ /
-2 (&)&*…&') &) &*… &') , (1.
,
B0 %&3 & 3 ,
C2 45 5,
43 and the transformed Cartan
defined as:
are the components of symmetric covariant tensor field of order 0, being the
, etc. have explored the
form. This forms are called a
root metric, respectively. Obviously, , is not
root metricsuch that it
change one de_ned in (1.1) ina way that the Riemannian metric
defined in (1.3). Then, we obtain the conditions among
/0due to conformal "
and 2, 0 are belongs to natural numbers
2. Main results
case 1:
2, 0 are even numbers andTheorem 2.1Let ,2 -2
* ') .2+
on an open subset 8 ⊂ :;, where
numbers with 2 0and.2 0
< =*-0)and /2 /0 ".
Proof.For simplicity, we put 2
-2
* ' .2+
By putting (-y) instead of (y) in (2.1), we have
-2
*
' .2
Summing sides the two equations (2
-2
* '
Consequently, we get the proof.∎
We have the following.
Corollary 2.1 Let ,2 -2
* ')
metrics on an open subset 8 ⊂ :;where
even numbers with 2 0 and
,0 if and only if -2 < 0
-case2:
2, 0 are odd numbers andTheorem 2.2Let ,2 -2
* ') .2+
on an open subset 8 ⊂ :;, where are belongs to natural numbers.
are even numbers and 2 0.
+/2 and ,0 -0
*
'* .0+/0 are two more generalized
where -2,.2,/2, -0,.0 and /0 are given by (1.4). Suppose that
0 B0. If ,2 is conformal "-changeof,0, then -2
0 . Under the assumption, we have
+/2= -0'* .0+/0 ". (
.1), we have
? /2= -0'* .0? /0 ? ".
2.1) and (2.2), we have
.2 0 -0
*
' 0 .0. (
.2+/2 and ,0 -0
*
'* .0+/0 are two more generalized
where -2,.2,/2, -0,.0 and /0 are given by (1.4). Suppose that
and .2 0 B0. If '@-2 and '@-0are Riemannian metrics, then
-0and/2 /0.
numbers and 2 0.
+/2 and ,0 -0
*
'* .0+/0 are two more generalized
where -2,.2,/2, -0,.0 and /0 are given by (1.4). Suppose that
44
are two more generalized -th root metrics
. Suppose that 2, 0 are even
< =)-0 (or or-2
. (2.1)
(2.2)
. (2.3)
are two more generalized -th root
. Suppose that 2, 0 are
are Riemannian metrics, then ,2
are two more generalized -th root metrics
numbers with 2 0and .2
<A =)-0(or -2 < =*-0, -2
Proof.For simplicity, we put 2
-2
* '
By putting (-y) instead of (y) in (2.4
?-2
*
' .2?
Summing sides the two equations (2
-2
*
' .2 ?-2'*
Thus
.2 .2
Therefore,
.2 .2 0?
Because of .2 0 .0, -2 <
Theorem 2.3Let ,2 -2
* ') .2+
on an open subset 8 ⊂ :;, where
numbers with 2 0 and.2
Proof.From (2.8), we have
-2
B
' = -0 'B 2 0
By (1.4), one can see that 1.∎
case3:
2, 0 are even numbers and0 .
0. If ,2 is conformal "-changeof ,0, then
<A =*-0 )and /2 /0 ".
0 . Under the assumption, we have
.2+/2= -0'* .0+/0 ". (
4), we have
/2= ?-0'* .0? /0 ? ".
2.4) and (2.5), we have
' .2 -0'* .0 ?-0'* .0 .
2 0? -2
B
' 0 .0 .0 0? -0'B . (
? -2
B
' 0 .0 0 .0 0? = -0 'B(2.
< =)-0, -2 <A =)-0 and then /2 /0 ".∎
+/2 and ,0 -0
*
'* .0+/0 are two more generalized
where-2,.2,/2, -0,.0 and /0 are given by (1.4). Suppose that
E 0 .
0. If ,2 is conformal"-changeof ,0, then
.2.0 0? 0 .0 0-2
B
'? .2 0 = -0 'B
numbers and 2E 0.
45
, then -2 < =)-0, -2
. (2.4)
(2.5)
. (2.6)
. (2.7)
8)
∎
are two more generalized -th root metrics
. Suppose that 2, 0 are odd
1.
Theorem 2.4Let ,2 -2
* ') .2+
on an open subset 8 ⊂ :;, where
-numbers with 2E 0, 2F
< =* -0 ')'*and /2 /0
Proof.Under the assumption, we have
-2
*
') .
By putting (-y) instead of (y) in (2.10
-2
Summing sides the two equations (2
-2
* ')
Consequently, we get the proof.∎
In above theorem, if 2? 0 G, where
(H ):If 5
=*F 1, then
Case 1:5
=* 2I. Therefore, from theorem 2.4,
Case 2:5
=* 2I 1. Therefore, from theorem 2.4,
Case 3: 0∤ G. Because ofG
(H ):If 5
=* 1, then, from theorem 2.4,
(HL):If =5
*M 1, then
Case 1: =*
5 2I. Therefore, from theorem 2.4,
Case 2: =*
5 2I 1. Therefore, from theorem 2.4,
Case 3: G ∤ 0. Because of
case4:
2, 0 are odd numbers and+/2 and ,0 -0
*
'* .0+/0 are two more generalized
-2,.2,/2, -0,.0 and /0 are given by (1.4). Suppose that
0and.2 0 .0. If ,2 is conformal "-change
".
Under the assumption, we have
.2+/2= -0
*
'* .0+/0 ". (
10), we have
2
*
') .2? /2= -0
*
'* .0? /0 ? ".
2.10) and (2.11), we have
) .2 0 -0
*
'* .0 . (
, where G is even number, then by (2.12), we get
. Therefore, from theorem 2.4, -2 < *ON -0 2P0Q.
. Therefore, from theorem 2.4, -2 < *OR)N -0 0 2PQ.
Because ofG 0S T, from theorem 2.4, -2 <
NUV
W -0
, then, from theorem 2.4, -2 < 5 -0 0.
. Therefore, from theorem 2.4, -2 < 05Q -0 )R*O*O .
. Therefore, from theorem 2.4, -2 < 0QP2 5 -0 *R*O)R*O.
Because of 0 GŚ T́, from theorem 2.4, -2 < 5ÝPŹ
numbers and 2E 0.
46
are two more generalized -th root metrics
. Suppose that 2, 0 are even
changeof,0, then -2
. (2.10)
(2.11)
. (2.12)
2PYP'*V .
-0 2P
Theorem 2.5Let ,2 -2
* ') .2+
on an open subset 8 ⊂ :;, where
numbers with 2E 0, 2F
< =* -0 ')'*, -2 <A =*
-Proof.Under the assumption, we have
-2
*
') .
By putting (-y) instead of (y) in (2.1
?-2
* ') .2
Summing sides the two equations (2
-2
*
') .2 ?-2')*
Thus
.2
Therefore,
.2 .2 0? -2
B ')
Because of .2 0 .0, we get
".∎
case5:
2, 0 are even and odd numberTheorem 2.6Let ,2 -2
* ') .2+
on an open subset 8 ⊂ :;, where
-odd numbers, respectivelyand
-2 < [20 ?.2< .2 0? =
') *
+/2 and ,0 -0
*
'* .0+/0 are two more generalized
where-2,.2,/2, -0,.0 and /0 are given by (1.4). Suppose that
0and.2 0 .0. If ,2 is conformal "-change
-0
')
'*and /2 /0 ".
Under the assumption, we have
.2+/2= -0
*
'* .0+/0 ".
13), we have
? /2= ?-0
*
'* .0? /0 ? ".
2.13) and (2.14), we have
.2 -0
*
'* .0 ?-0'** .0 .
.2 0? -2
B
') 0 .0 .0 0? -0'*B . (
0 .
0 0 .0 0? =* -0
B
'* .
, we get -2 < =* -0 ')'*, -2 <A =* -0 ')'* and then
numbers, respectively.
+/2 and ,0 -0
*
'* .0+/0 are two more generalized
-2,.2,/2, -0,.0 and /0 are given by (1.4). Suppose that
and.2 0 .0. If ,2 is conformal
=* -0 '*B and /2 /0 ".
47
are two more generalized -th root metrics
. Suppose that 2, 0 are odd
changeof,0, then -2
(2.13)
(2.14)
. (2.15)
. (2.16)
. (2.17)
and then /2 /0
are two more generalized -th root metrics
. Suppose that 2, 0are even and
Proof.Under the assumption, we have
-2
*
') .
By putting (-y) instead of (y) in (2.18
-2
* ') .2
Summing sides the two equations (2
2 -2
* ') .2
Thus
4-2
B
') 4 .2 0 ] .0 0
Because of .2 0 .0, we have
4-2
B
') 4 .2 -2')*
Consequently,-2 < [2 0 ?.2<
') *
3. Generalized Conformal
^
We investigated what we call a conformal
where, # is a function of only, and
generalizes various types of changes: conformal changes, generalized Randers changes, Randers change. Under
this change, we obtain the relationshi
tensors associated with , [11].
In this paper, we introduce a general transformation or changeof Finsler metrics, which is referred to as a
generalized conformal _-vector-change in
,
Under the assumption, we have
.2+/2= -0
*
'* .0+/0 ".
18), we have
? /2= ?-0
*
'* .0? /0 ? ".
2.18) and (2.19), we have
-0
*
'* .0 ?-0'** .0 .
? 4 0 .
2.0 4-2
*
')`2.2? 0 .0a ` 0 .
, we have
) =* -0 '*B 0 .
.2 0? =* -0
B
'* and then /2 /0
^
-vector-change in Finsler spaces
what we call a conformal _-vector-change in Finsler spaces, namely
, → , , ",
only, and " , : %& , &, where %&≔ %& , is an
generalizes various types of changes: conformal changes, generalized Randers changes, Randers change. Under
this change, we obtain the relationships between some tensors associated with , .
, we introduce a general transformation or changeof Finsler metrics, which is referred to as a
change in Finsler spaces:
→ , d , " , 48 (2.18)
(2.19)
. (2.20)
.0a0? =* -0
B '*. (2.21)
. (2.22)
".∎
(3.1)
is an _-vector. This change generalizes various types of changes: conformal changes, generalized Randers changes, Randers change. Under
and the corresponding
, we introduce a general transformation or changeof Finsler metrics, which is referred to as a
to investigate the characteristics of this change For those interested.
change and conformal change in a general setting.
spaces, as fundamental Finsler connections,together with their associated
The main results of this section are being
Acknowledgments
The authors would like to express their grateful thanks to
suggestions. The authors wishes to express here his sincere gratitude to Dr. M. Rafie comments and encouragements.
References
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49 This transformation combines both_
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