Journal of Modern Physics, 2013, 4, 983-990
http://dx.doi.org/10.4236/jmp.2013.47132 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Energy β-Conformal Change and Special Finsler Spaces
Amr Soleiman1, Amira A. Ishan2
1Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt
2Department of Mathematics, Taif University, Makkah, KSA
Email: amrsoleiman@yahoo.com, amiraishan@hotmail.com
Received February 18, 2013; revised March 21, 2013; accepted April 20, 2013
Copyright © 2013 Amr Soleiman, Amira A. Ishan. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The main aim of the present paper is to establish an intrinsic investigation of the energy β-conformal change of the most
important special Finsler spaces, namely, Ch-recurrent, Cv-recurrent, C0-recurrent, Sv-recurrent, quasi-C-reducible,
semi-C-reducible, C-reducible, P-reducible, C2-like, S3-like, P2-like and h-isotropic, ···, etc. Necessary and sufficient
conditions for such special Finsler manifolds to be invariant under an energy β-conformal change are obtained. It should
be pointed out that the present work is formulated in a prospective modern coordinate-free form.
Keywords: Energy β-Conformal Change; Ch-Recurrent; Cv-Recurrent; C0-Recurrent; C2-Like; Quasi-C-Reducible;
C-Reducible; Berwald Space; Sv-Recurrent; S3-Like; P2-Like; h-Isotropic
1. Introduction
An important aim of Finsler geometry is the construction
of a natural geometric framework of variational calculus
and the creation of geometric models that are appropriate
for dealing with different physical theories, such as gen-
eral relativity, relativistic optics, particle physics and
others. As opposed to Riemannian geometry, the extra
degrees of freedom offered by Finsler geometry, due to
the dependence of its geometric objects on the direc-
tional arguments, make this geometry potentially more
suitable for dealing with such physical theories at a deeper
level.
Studying Finsler geometry, however, one encounters
substantial difficulties trying to seek analogues of classi-
cal global, or sometimes even local, results of Rieman-
nian geometry. These difficulties arise mainly from the
fact that in Finsler geometry all geometric objects depend
not only on positional coordinates, as in Riemannian
geometry, but also on directional arguments.
In Riemannian geometry, there is a canonical linear
connection on the manifold M, whereas in Finsler ge-
ometry there is a corresponding canonical linear connec-
tion due to E. Cartan. However, this is not a connection
on M but is a connection on T (TM), the tangent bundle
of TM, or on π1 (TM), the pullback of the tangent bun-
dle TM by :TM M
. The infinitesimal transforma-
tions (changes) in Riemannian and Finsler geometry are
important, not only in differential geometry, but also in
application to other branches of science, especially in the
process of geometrization of physical theories [1].
In [2], we investigated intrinsically energy β-confor-
mal change of the fundamental linear connections on the
pullback bundle of a Finsler manifold, namely, the Car-
tan connection, the Berwald connection, the Chern con-
nection and the Hashiguchi connection. Moreover, the
change of their curvature tensors is obtained.
The present paper is a continuation of [2] where we
present an intrinsic investigation of energy β-conformal
change of the most important special Finsler spaces,
namely, Ch-recurrent, Cv-recurrent, C0-recurrent, Sv-re-
current, quasi-C-reducible, semi-C-reducible.
C-reducible, P-reducible, C2-like, S3-like, P2-like and
h-isotropic, ··· , etc. Moreover, we obtain necessary and
sufficient conditions for such special Finsler manifolds to
be invariant under an energy β-conformal change.
Finally, it should be pointed out that all results ob-
tained are formulated in a prospective modern coordi-
nate-free form.
2. Notation and Preliminaries
In this section, we give a brief account of the basic con-
cepts of the pullback approach to intrinsic Finsler ge-
ometry necessary for this work. For more details, we
refer to [3-6].
We assume, unless otherwise stated, that all geometric
objects treated are of class C. The following notation
will be used throughout this paper:
M: a real paracompact differentiable manifold of finite
C
opyright © 2013 SciRes. JMP
A. SOLEIMAN, A. A. ISHAN
984
dimension n and of class C,

M

: the R-algebra of differentiable functions on M,
M
: the

M
M
: module of vector fields on M,
:TM
M
:TM M

: the tangent bundle of M,
M
:
: the cotangent bundle of M,
M
M
: the subbundle of nonzero vectors tan-
gent to M,
VTM
1
:PT
M
M
: the vertical subbundle of the bundle TTM,

MT: the pullback of the tangent bun-
dle TM by π,

1
:PTMT
 
TM
: the pullback of the cotangent
bundle by π,

:M

1

:M

the : module of differentiable
sections of ,
TM

TM
TM
the : module of differentiable
sections of ,

M
1
T
iX: the interior product with respect to
X
M

,
df: the exterior derivative of
f
M,
d: ,
L
Lid, iL being the interior derivative with re-
spect to a vector form L.
Elements of


M
will be called π-vector fields
and will be denoted by barred letters. Tensor fields on
will be called π-tensor fields. The fundamen-
tal π-vector field is the π-vector field
1TM
defined by
 
,uuu
for all uM.
We have the following short exact sequence of vector
bundles, relating the tangent bundle T (TM) and the pull-
back bundle :

1TM

1
00MT TMTM

 1
,
where the bundle morphisms ρ and γ are defined respec-
tively by

:,d
TM

and , where ju
is the natural isomorphism
 
,:u
uvj v
 

:
uu
Mv Mv
jT MTTM

.
The vector 1-form J on TM defined by :J
is
called the natural almost tangent structure of TM. The ver-
tical vector field C on TM defined by :C
is
called the fundamental or the canonical (Liouville) ve-
ctor field.
Let D be a linear connection (or simply a connection)
on the pullback bundle . We associate with D
the map
1TM

1
::
X
KTMTMXD

called the connection (or the deflection) map of D. A
tangent vector
u
X
TTM

X TTM
is said to be horizontal if
. The vector space

0KX

HTM


:
uu
KX
uTM
0
of the hori-
zontal vectors at is called the horizontal space
to M at u. The connection D is said to be regular if
 
.
uu u
TTMVTMH TMuTM 
(1)
If M is endowed with a regular connection, then the
vector bundle maps


 

 
1
1
1
:,
:,
:.
HTM
VTM
TMV TM
H
TM TM
KVTM TM


are vector bundle isomorphisms. Let us denote

1
:HTM

, this map will be called the horizontal
map of the connection D. According to the direct sum
decomposition (1), a regular connection D gives rise to a
horizontal projector hD and a vertical projector vD, given
by
,
DD
hvI,
 
 (2)
where I is the identity endomorphism on T (TM).
The (classical) torsion tensor T of the connection D is
defined by

,,,
,.
YX
X
YDYDXXY
XY TM



T
The horizontal ((h)h-) and mixed ((h)hv-) torsion ten-
sors, denoted by Q and T respectively, are defined by



,,,, ,
,.
QXYTXY TXYTXX
XY M
 



,
The (classical) curvature tensor K of the connection D
is defined by


,
,,
,, .
XY YXXY
K
XYZDDZ DDZ DZ
XYZ TM

 

The horizontal (h-), mixed (hv-) and vertical (v-) cur-
vature tensors, denoted by R, P and S respectively, are
defined by

 
 
,,
,,
,,
RXYZ KX YZ
PXY
S
,
,
.
Z
KXYZ
X
YZKX Y Z



The contracted curvature tensors, denoted by , R
̂, P
̂
and respectively, are also known as the (v)h-, (v)hv-
and (v)v-torsion tensors and are defined by
ˆ
S



ˆ,,
ˆ,,
,,
ˆ
RXYRXY
PXY PXY
XYSSXY
,
,
.
If M is endowed with a metric g on
1
πTM
, we
write

:,,,, ,RXYZWgRXYZW
(3)
On a Finsler manifold
,
M
L, there are canonically
Copyright © 2013 SciRes. JMP
A. SOLEIMAN, A. A. ISHAN 985
associated four linear connections on [7]: the
Cartan connection , the Chern (Rund) connection Dc,
the Hashiguchi connection
1
πTM
D
and the Berwald con-
nection D. Each of these connections is regular with (h)
hv-torsion T satisfying

,0TX
. The following
theorem guarantees the existence and uniqueness of the
Cartan connection on the pullback bundle.
Theorem 2.1. [8] Let
,
M
L
be a Finsler manifold
and g the Finsler metric defined by L. There exists a
unique regular connection on π1 (TM) such that
(a) is metric : ,
0g
(b) The (h) h-torsion of vanishes: ,
0Q
(c) The (h) hv-torsion T of
satisfies:



,,,,
g
TXY ZgTX
Z Y
Definition 2.2. Let
,
M
L be a Finsler manifold and
g the Finsler metric dened by L. We define:

1
:,XLgX,
:g
ħ
: the angular metric tensor,

,,:, ,TXYZgTXY Z
: the Cartan tensor,
 
:CXTrY TXY
,: the contracted torsion,


,:
g
CXC X: C
̅ is the π-vector field associated
with the π-form C,
 
,: ,
v
RicX YTrZSXZY
: the vertical Ricci
tensor,


0,: ,
vv
g
Ric XYRic XY: the vertical Ricci map,

0
:
v
ScTr XRicX
v
: the vertical scalar curva-
ture.
Deicke theorem [9] can be formulated globally as fol-
lows:
Lemma 2.3. Let
,
M
L be a Finsler manifold. The
following assertions are equivalent:
(a)
,
M
L is Riemannian,
(b) The (h)hv-torsion tensor T vanishes,
(c) The π-form C vanishes.
Concerning the Berwald connection on the pullback
bundle, we have
Theorem 2.4. [8] Let
,
M
L be a Finsler manifold.
There exists a unique regular connection D on
such that
1TM
D
(a) = 0,
hX
(b) D is torsion-free: T= 0,
L
(c) The (v)hv-torsion P of D vanishes : P
̂(X
̅,Y
̅) = 0.
Such a connection is called the Berwald connection
associated with the Finsler manifold

,
M
L.
We terminate this section by some concepts and re-
sults concerning the Klein-Grifone approach to ic
Finsler geometry. For more details, we refer to [103].
ntrinsi
-1
A semispray is a vector field X on TM, C on M, C1
on TM, such that X
.
A semispray X which is homogeneous of degree 2 in
the directional argument
,CX X

,
is called a spray.
Proposition 2.5. [12] Let
M
L be a Finsler mani-
fold. The vector field G on TM defined by d
G
iE

is a spray, where
2
2L:1E is the energy function
and :d
J
dE
. Such a spray is called the canonical
spray.
A nonlinear connection on M is a vector 1-form Γ on
TM, C on M, C0 on TM, such that
,.
J
JJ J

The horizontal and vertical projectors hΓ and vΓ asso-
ciated with
are defined by
 
12:2:,1hIvI

.

To each nonlinear connection Γ there is associated a
semispray S defined by , where S' is an arbi-
trary semispray. A nonlinear connection Γis homoge-
neous if
ShS
,C0
. The torsion of a nonlinear connec-
tion Γ is the vector 2-form t on TM defined by
:12t,J
. The curvature of Γ is the vector 2-form
on TM defined by
R

:12,hh

R. A nonlinear
connection Γ is said to be conservative if d0hE
.
Theorem 2.6. [11] On a Finsler manifold
,
M
L,
there exists a unique conservative homogenous nonlinear
connection with zero torsion. It is given by:
,,
J
G
where G is the canonical spray. Such a nonlinear connec-
tion is called the canonical connection, the Barthel con-
nection or the Cartan nonlinear connection associated
with (M,L).
It should be noted that the semispray associated with
the Barthel connection is a spray, which is the canonical
spray.
3. Energy β-Conformal Change and Special
Finsler Spaces
In [2], we investigated intrinsically a particular β-change,
called an energy β-conformal change:


 
2222
,e ,,
x
xy LxL,
x
yB y

(4)
where
,
M
L is a Finsler manifold admitting a con-
current π-vector field
,

:,Bg

;
being the
fundamental π-vector field and σ(x) is a function on M.
Moreover, the relation between the two Barthel connec-
tions Γ and
, corresponding to this change, is obtained.
The energy β-conformal change of the fundamental lin-
ear connections on the pullback bundle of a Finsler
manifold is studied.
In this section, we introduce the effect of energy
β-conformal change on some important special Finsler
spaces. The intrinsic definitions of the special Finsler
spaces treated here are quoted from [14].
Copyright © 2013 SciRes. JMP
A. SOLEIMAN, A. A. ISHAN
986
The following definition and three lemmas are useful
for subsequence use.
Definition 3.1. [15] Let
,
M
L be a Finsler mani-
fold. A π-vector field

M

is called a concur-
rent π-vector field if it satisfies the following condi-
tions
,0
XX
X


. (5)
In other words,
is a concurrent π-vector field if
X
X

 for all

X
TM.
Lemma 3.2. [15] Let

M

be a concurrent
π-vector field. For every

,
X
YM
, we have
(a)

0,X,,TX T

(b)

ˆˆ
,,PXP X


0,
(c)
 
,,PXY PXY

0.
Lemma 3.3. [15] Let
,
M
L be a Finsler manifold
which admits a concurrent π-vector
. Then, we have:
(a) The concurrent π-vector eld
is everywhere non
-zero.
(b) The scalar function
:,Bg

is everywhere
non-zero.
(c) The π-vector field

2
:mBL

 is every-
where non-zero and is orthogonal to
.
(d) The π-vector fields m and
satisfy


,,gm gmm
0
.
(e) The angular metric tensor ħ satisfies

,0X
ħ
for all X
.
Lemma 3.4. [2] Under the energy β-conformal change
(4), we have
,,hhLvvL 
or equivalently
,,LK KKL
 
 
where





1
22
21
:,
1e 1d;
J
x
JJ
LE JdEJdC
pdEBdB
 
 






2
1
2
,:d d
:d, :,
and: d.
,
g
XX
Gp g
 
 



hX
Theorem 3.5. [2] Let
,
M
L and
,
M
L be two
Finsler manifolds related by the energy β-conformal
change (4). Then the associated Cartan connections
and are related by:
,
XX
YYXY
 
,
where
(6)











22
:d d,
,,
1e
d,d ,
,, ,
x
hX YYXgXY
TYXTLXY
p
hXYY gX
g
L
YgXY
g
gX
 
 

 
,XY
 

(7)
where T
is a 2-form on TM, with values in π1 (T
by
M),
defined


,, ,,
g
T.LXhYZgTLZhYX

Moreover,
v-torsion T and the π-form C are invariant. (a) The (h) h
(b)
,,.SXYZ SXYZ
(c)

,,,PXYZPXYZ VXYZ
,
wV is the vector π-form defined by here


 

,,
,,
Y
SX
YZB XZBTXYZ
  (8)
(d)
,VXYZ

,, ,RXYZRXYZ HXYZ
,
wH is the vector π-form defined by here
,,HXYZ SNXNYZ





 



,
1
,,
,,
, ,,
,
X
XY
NX
UP
XNYZ BYZ
BYZ BXBYBTZY
Z
X

 
(9)
where
,: ,BXY XY

and :LN
 is
given b
Definiti on 3.6. A Fi
y Lemma 3.4.
nsler manifold

,
M
L
n tenso
is :
hori-
zo
(a) a Berwald manifold if the torsior T is
ntally parallel: 0
XT
.
(b) a Landsberga mnifold if

ˆ,0PXY , or equiva-
lently, if 0T

.
(c) a gerg manifold if t
lin
neral Landsbehe trace of the
ear map
ˆ,YPXY is identically zero for all
X
M
tly, if , or equivalen0C


.
Now, we have
Theorem 3.7. Let
,
M
L
ent π-v
be a Finsler manifold
which admits a concurrector field
. Under the
energy β-conformal change (4), we have:
(a) if
,
M
L is Berwald, then

,
M
L and
,
M
L
are Riem
(b) if
annian.
,
M
L is Landsberg, then

,
M
L and
,
M
L
annian. are Riem
1

,,: ,,
XY
UAXYAXYAYX
Copyright © 2013 SciRes. JMP
A. SOLEIMAN, A. A. ISHAN 987
Proo
f.
(a) If
,
M
L is a Berwald manifold, then 0
XT
.
(Definition 3.6(a)). Hence ˆ0TP

 [16]-
quently, the hv-curvature P va
Hence,
. Conse
nishes [15].


0,,, ,,PXYZ TXYZ
 by Proposi-
tio of [15]. The result follows then
n 3.4(d) from Deicke
theorem (Lemma 2.3) and noting that under β-conformal
change the (h)hv-torsion T is invariant (Theorem 3.5(a)).
(b) The proof is similar to that of (a).
Definition 3.8. A Finsler manifold
,
M
L is said to
be
) Ch-recurrent if the (h)hv-torsion tensor T satisfies
th
:
(a
e condition

o
XTXT
 , where o
is a π-form
of order one.
(b) Cv-recurrent if the (h)hv-torsion tensor T satisfies
the condition

 
,,TYZ XTYZ
 .
o
X
rent if the (h)hv-torsion tensor T satisfies
th
(c) C0-recur
e condition

 
,,DT YZXTYZ
.
o
X
.9. Under the energy β-conform l change
(4
Theorem 3a
), we have:
(a) if
,
M
L is Ch-recurrent, then

,
M
L and

,
M
L
arnnian.
f

,
e Riema
(b) i
M
L is Cv-recurrent, then

,
M
L and

,
M
L
arennian.
f

,
Riema
(c) i
M
L is C0-recurrent, then

,
M
L and

,
M
L
arnnian.
f. e Riema
Proo
(a) We have [16]











,, ,,
ˆˆ
,, ,,.,,
ZW
P
gTYXWg TYXZ
gT XWPZYgT XZPWY


(10)
Setting
,,,XYZW
W
in Equation (10), making use of
 
ˆ0PX T
,,X

(Lemma 3.2),


,Z [15] and ,,, ,PXYZT XY






,, ,
ZZ
,
g
TXYW gY

 (Proposi-
tion 3.3 of [16]), we get
TXW
0.T

On the other hand, Definition 3.8(a) for X
,
yields
.
o
TT


The above two equations and Theorem 3.5(a) imply
th
ma 2.3, the result follows.
4], together
wDefinition 3.10. A Finsler manifold (M,L) is said to
r T has the from
at 0.TT
Hence, by Lem
(b) and (c) follow from Theorem 4.7 of [1
ith Theorem 3.5(a).
be:
(a) quasi-C-reducible if

dim 3M and the Cartan
tenso
 

,,
Z AY
AZXCY (11)
where A is a symmetric π-tensor field satisfying
,, ,,T XYZAXYCZCX
,0AX
(b) semi-C-reducible if and the Cartan
tensor T has the form
dim 3M
,,TXYZ


  



2
1
,, ,
,
n
X
YCZYZCXZXCY
CCXCYCZ


ħħħ
(12)
where
2:0CCC
,
g 1
μ and τ are scalar functions sat-
isfyin
.
(c) Cm 3M and the Cartan tensor T
has the f
-reducible if
orm
di
,,TXYZ


 

11
,, ,
n
X
.YCZ YZCXZXCY

ħħ
(13)
(d) C2-like if and the Cartan tensor T has
the form
dim 2M

, .TXYCXCYCZ
2
, 1ZC
Theorem 3.11. Under the energy β-conformal change
(4), we have:
(a) if
,
M
L is quasi-C-reducible provided that
and

,
M
L
,0A

, th
,
M
Len are Rieman-
ni
(b) if
an.
,
M
L is Cible, the
,
M
L and -reducn
,
M
L
are Riemannian.
(c) if
,
M
L is semi-C-reducible,, then

M
L and
,
M
L
are C2-like.
Proof.
(M,L) is qua
(a) Ifsi-C-reducible, then the Cartan ten-
so fies Relation (11). Setting r T satis
XY
into Equation (11) and using the fact that



0as:, 0CTrXTX
 
 C
and
,,, ,0TXYgTX Y

, we get

,0. ACX

From which together with the given assumption,
,0A

, it follows that the π-orm C vanishes.
Hence, by Lemma 2.3 and Theorem 3.5(a) imply that
f
Copyright © 2013 SciRes. JMP
A. SOLEIMAN, A. A. ISHAN
988
0C. Consequently, again by Lemma 2.3, C
,
M
L
and
,
M
L
bility by se
are Riemannian manifolds.
(b) Follows from the defining property of C-reduci-
tting XY
, taking into account Lemma
3.3(ea 2.3, Theorem 3.5(a) and ), Lemm

0C
.
(c) Let

,
M
L be semi-C-reducible. Setting
XY
 and
Z
C in Equation (12), taking into
account Lemma 3.2(a) and
0C
, we get

,0.CC
From which, since
ħ
ħand
,0

CC
0
, it
follows that 0
. Consequently, again by E
(12)
quation
, we get



2
,, 1.TXYZC CXCYCZ (14)
Now, unde
r the energy β-conformal change (4
Theorem 3.5(a), Lemma 2.3 and the Relation [
), from
2]


2
,e,, ,,
x
g
XYg XYg XgY


it follows that

2
,, e,,
x
TXYZ TXYZ
and
. Hence, from Equation (14), we get

2
22
ex
CC



2
1 .C CXCYCZ 15)
Therefore, aefini-
tion 3.10(d), the result follows.
Definitio n 3 .12. The condition
,,TXYZ
(
gain from Equations (14), (15) and D


 


2
,,,
0,,
XYZW
YZWTX
(1
will be called the T-condition.
The more relaxed condition
,,,TXYZW
:,
,
X
LT
YZW
 ,
6)





,,,
0.
XYZW
CY
(17)
will be called the To-condition.
Theorem 3.13. Under the energy β-conformal change
(4), we have:
,:
oX
TXY L

XCY
(a) if

,
M
L satisfies the T-condition, then
,
M
L
and

,
M
L
are Riemannian.
(b) if
,
M
Lsatisfies the To-condition, then
,
M
L
and

,
M
L
are Riemannian.
Proof.
(a) If
,
M
L satisfies the T-condition, then - by set
ting W
into Equation (16), taking into account that

,0TT X
, we get ,X

,, 0.TXYZ
r with


0BL

This equation, togethe (Lem-
ma 3.3(b)) and Theorem 3.5(a), impl thaty 0TT
.
Consequently, by Lemma 2.3,

,
M
L and
,
M
L
are
Riemannian manifolds.
(b) Follows from Equation (17) by setting X
,
taking into account that


CX (by 0, X
CCY
 Y
,
XY
TYZT X

,Z [16]),
0BL
, together with 3.5(a) and
Lemma 3.3.
Definition 3.14. A Finsler m
Theore m
anifold

,
M
Lis said to
be S3-like if
4M and the v-curvature tensor S
has the form:
dim




12
,, , ,.
n n
,,,
v
SXYZW
Sc
X
ZYWXWYZ

ħ-ħħħ
(18)
Theorem 3.15. Under the energy β-conforma
(4), if
l change
,
M
L
curvature te
is S3-like with , then, the v-
nsors S andvanis
dim 4M
h. S
Proof. Setting Z
in Equation (18), taking into
accounct that
t the fa
,0SXY
position 3.4(a)
of [13]) and
(Pro

,,:
X
YgX
ħY, we immediately
t ge
 


12 ,,0.Sc nnYX
vXY

ħħ
Taking the trace of the above equation with respect to
, noting that
1Tr n
Y[14], we have

–1, 0
v
Sc nX
ħ
From which, s
0
(Lemma 3.
ince ,Xħ3(e)), the
vertical scalar curvature Scv vanishes.
Now, again from Equation (18) togeth with Theorem
3.s.
ld
er
5(b), the result follow
Definition 3.16. A Finsler manifo
,
M
L, where
dim 3M, is said to be:
(a) P2-like if the hv-curvature tensor P has the form:

,, ,,,
,,,PXYZW
Z
TXYW WTXYZ

 (19)
where ω is a (1) π-form (positively homogeneous of de-
gree 0).
(b) P-reducible if the π-tensor field
ˆˆ
,, :,,PXYZgPXYZ has the form

ˆ,,PXYZ
,,,,
X
YZYXXY

ħħ
(20)
2




,,
,, ,,,:,
XYZ
XYZXYZ YZXA AAA,,
Z
XY
Z Zħ
Copyright © 2013 SciRes. JMP
A. SOLEIMAN, A. A. ISHAN 989
where δ is the π-form defined by




11.
X
nC

 X
gy β-conformal change
(4), we have:
(a) if
Theorem 3.17. Under the ener

,
M
L is P-like, provided that
2
1

,
then

,
M
L and

,
M
L
are Riemannian.
(b) if

,
M
L is P-reducible, provided that

,0XY
(defined by Theorem 3.5(c)),
,
V then
M
L
, and
M
L
Landsberg. are
(a) Setting
Proof.
Z
in Equation (19), taking into
mma 3.2, we im
ac-
count Pion 3.4-
m
roposit(a) of [15] and Le
get ediately


1,0.TXY

This, together
with the given assumption and Theorem
3.5(a), follow that and T vanish. He by Lemma
2.3,
T
nce,

,
M
L and

,
M
L
are Riemannian.
(b) If

,
M
L is P-reducible, then, by Definition
3.16(b), the (v)hv-tosion ˆ
P satisfy Relation (20). Set-
ting
r
XY
to Equ
co
ination (20) and taking into ac-
unt that


0
, we get C




,0.CZ


ħ
From which, noting that
,

ħ0 (Lemma 3.3(e)),
implies that 0C


Theorem 3.
. Hence, again, from Definition
3.16(b) and5(c) (under the given assumption),
th H
f
e (v)hv-torsion tensors ence, by Defini-
t
ion 3.6, the resultollows.
Definition 3.18. A Finsler manifold

,
ˆ0PP
.
M
L, where
dim 3M, is said to be h-otropic if there exists a sca-
lar ko such that the horizontal curvature tensor R has the
fo
is
rm
  

,,,.
o
XYZk gXZYgYZX
The
R
orem 3.19. Under the energy β-conformal change
(4), we have:
(a) if

,
M
L is h-isotropic, provided that

,0XYZ (defined by Theorem 3.5(d)), then the
h-curvature tensors R and R
of the Cartan connection
va
H
) if
nish.
(b

,
M
L is an h-isotropic Berwald manifold,
provided that

,0HXYZ, then

,
M
L and

,
M
L
are Riemannian.
Proof.
(a) From De we havfinition 3.18(a),e


,, ,
o
kg
XZYWgYZgg
,.
R
WX
(2
Setting
,,,XYZW
1)
Z


0,,,,
o
kgmYW ggWgYm

.
Taking the trace of this equation, we get

,0.
o
kn
1gm
, since
and
X
m and noting that

,0RXY
(Proposition 3.4(g) of [15]), we have
From which

,,gm gmm
 (Lemma 0
3.3) and , the scalar ko vanishes. Now, ain,
fromder the given
as nd of the
Cartan connection vanish
(b) Follows from (a), taking Theorem 3.7 into account.
n
better
plications of indices.
of the present work, where
dim 3Mag
Equation (21) and Theorem 3.5(c) (un
sumption), the h-curvature tensors R a R
4. Concluding Remark
It should be pointed out that a global formulatio of dif-
ferent aspects of Finsler geometry may give more insight
into the infrastructure of physical theories and make a
understanding on the essence of such theories
without being trapped into the com
This is one of the motivations
all results obtained are formulated in a prospective mod-
ern coordinate-free form. Moreover, it should be noted
that the outcome of this work is twofold. Firstly, the local
expressions of the obtained results, when calculated, co-
incide with the existing local results. Secondly, new glo-
bal proofs have been established.
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