# On a Special Form of(

Authors: Brijesh Kumar Tripathi; K. B. Pandey
Journal: Journal of Mathematics
 (2016)

---

## Abstract

A special form of (h) h
ν-torsion tensor was introduced which may be considered generalization of P
⁎-Finsler space and P-reducible Finsler space and then some properties of this space were studied. We also introduce connection and give some case and condition of torsion tensor T
j
k
i
.


---

## Body

## 1. Introduction

LetM
n be an n-dimensional differentiable manifold and T
M
n be its tangent bundle. The manifold M
n is covered by neighborhoods (U), in each U of which we have a local coordinate system (
x
i
). A tangent vector at a point x
=
(
x
i
) of U is written as y
i
∂
/
∂
x
i
x, and we have a local coordinate system (
x
i
,
y
i
) of T
M
n over U.In paper [1] let F
n
n
≥
1 be an n-dimensional Finsler space with metric function L
x
,
y. There are five kinds of function L
x
,
y. There are five kinds of torsion tensors in the theory of Finsler space based on Carton’s connection, out of which(1)
P
i
j
k
=
y
h
P
h
i
j
k
,
C
i
j
k
=
1
4
∂
3
L
3
∂
y
i
∂
y
j
∂
y
kas (ν) h
ν-torsion tensor and (h) h-Torsion tensor are of great important tensors for the present study, where P
h
i
j
k ish
ν-curvature tensor. In Finsler geometry based on Cartan’s connection, there are three kinds of covariant differentiations denoted as |
i and v-covariant differentiation denoted as |
i.Ann
(
n
≥
3
)-dimensional Finsler space F
n is said to be a semi-C-reducible Finsler space, whose Cartan’s tensor C
i
j
k is written as (2)
C
i
j
k
=
p
n
+
1
C
i
h
j
k
+
C
j
h
k
j
+
C
k
h
i
j
+
q
C
2
C
i
C
j
C
k
,where C
2
=
g
i
j
C
i
C
j and scalars satisfy p
+
q
=
1. Moreover if scalars p and q are constants, F
n is said to be C-reducible Finsler space with constants coefficients.A Special semi-C-reducible Finsler space has been introduced by Ikeda [2] as follows.Ann
(
n
≥
3
)-dimensional Finsler space F
n is said to be a Special semi-C-reducible Finsler space (in short we call SSR-Finsler space) [1, 2] whose h (h
ν)-torsion tensor C
i
j
k is written as (3)
C
i
j
k
=
1
n
-
2
C
i
h
j
k
+
C
j
h
k
j
+
C
k
h
i
j
-
3
n
-
2
C
2
C
i
C
j
C
k
.Various interesting forms of these tensors have been studied by many ([3–7], …), two of them are C-reducible Finsler space and a Special semi-C-reducible Finsler space ([1, 2]) in which the torsion tensor C
i
j
k, respectively, is in the forms(4)
C
i
j
k
=
1
n
+
1
C
i
h
j
k
+
C
j
h
k
j
+
C
k
h
i
j
,
(5)
C
i
j
k
=
1
n
-
2
C
i
h
j
k
+
C
j
h
k
j
+
C
k
h
i
j
-
3
n
-
2
C
2
C
i
C
j
C
k
,where h
i
j is angular metric tenser and C
i
=
C
i
j
k
g
j
k, where g
j
k is reciprocal of the metric tensor g
j
k
.Izumi ([4, 5]) introduced P
∗-Finsler space in which P
i
j
k is of the form (6)
P
i
j
k
=
λ
C
i
j
k
,where λ is the scalar homogeneous function on T
M of zero degree in y
i. In P-reducible Finsler space the tensor P
i
j
k is the form [8](7)
P
i
j
k
=
1
n
+
1
J
i
h
j
k
+
J
j
h
k
i
+
J
k
h
i
j
,where J
i
=
C
i
∣
0
=
C
i
∣
j
y
i. A Finsler space with P
i
j
k
=
0 is called a Landsbergs space [9]. If C
i
j
k
∣
h
=
0 then F
n is called Berwald’s affinely connected space ([10, 11]).Rund [11] introduced a special form of torsion tensor P
i
j
k as follows:(8)
P
i
j
k
=
λ
C
i
j
k
+
a
i
h
j
k
+
a
j
h
k
i
+
a
k
h
i
j
,where λ
=
λ
x
,
y is a scalar homogeneous function on T
M of degree 1 and a
i
=
a
i
x is a homogeneous function of degree 0 with respect to y
i. He then studied some properties of F
n satisfying (8). The present author introduced a more general form of (8) and studies some properties of F
n satisfying it [12].We quote the following lemmas, which will be used in the present paper.Lemma 1 (see [6]).
If the curvature tensorP
i
j
k of a C-reducible Finsler space vanishes then the space vanishes and then the space is Berwald’s affinely connected space.Lemma 2 (see [13]).
A Finsler spaceF
n is locally Minkowskian if h-curvature tensor R
h
i
j
k
=
0 and C
i
j
k
∣
h
=
0.Definition 3 (see [1]).
A Finsler connectionF
Γ is defined as tried F
j
k
i
x
,
y
,
V
j
k
i
(
x
,
y
) as h-connection and v-connection which are components of a tensor field of (1, 2)-type. The tensor D of component D
j
i is called deflection tensor of F
Γ. Therefore, D
j
i
=
N
j
i and y
r
V
r
j
i
=
0 are desirable conditions for a Finsler connection.
LetM be an n-dimensional C
∞ modified by T
x
M (we mean the tangent space at x
∈
M) and by T
M
∖
0 (we mean the slit tangent bundle of M).A Finsler metric onM is a function L
:
T
M
→
[
0
,
∞
) which has the following properties:(i)
L is C
∞
o
n
T
M
∖
0.
(ii)
L is positively homogeneous function of degree 1 on T
M.
(iii)
For eachy
∈
T
x
M, the metric tensor g
i
j and the angular metric tensorh
i
j are, respectively, given by (9)
g
i
j
=
1
2
∂
2
L
2
∂
y
i
∂
y
j
,
h
i
j
=
L
∂
2
L
∂
y
i
∂
y
j
.The angular metric tensor h
i
j can also be written in terms of the normalized element of support(10)
l
i
=
1
L
g
i
j
y
i
y
j
,
h
i
j
=
g
i
j
-
l
i
l
j(see [14]). Fory
∈
T
x
M
∖
0, Cartan’s tensor vector is defined as(11)
C
i
=
g
j
k
C
i
j
k
.According to Deicke’s theorem C
i
=
0 is the necessary and sufficient condition for F
n to be Riemannian. Let F
n
=
M
n
,
L be a Finsler space for y
∈
T
x
M
∖
0. We define Matsumoto torsions of C-reducible and Special semi-C-reducible Finsler space, respectively, as follows:(12)
M
i
j
k
=
C
i
j
k
-
1
n
+
1
C
i
h
j
k
+
C
j
h
i
k
+
C
k
h
i
j
,
M
-
i
j
k
=
C
i
j
k
-
1
n
-
2
C
i
h
j
k
+
C
j
h
k
j
+
C
k
h
i
+
3
n
-
2
C
2
C
i
C
j
C
k
.A Finsler space F
n is said to be C-reducible if M
i
j
k
=
0 and is Special semi-C-reducibleif(13)
M
-
i
j
k
=
0
.
Next, we define a tensor(14)
L
i
j
k
=
C
i
j
k
∣
l
y
l
,
where “∣” means h-covariant differentiation with respect to Cartan’s connection.A Finsler spaceF
n is called a Landsberg space if P
i
j
k
=
0, or equivalently L
i
j
k
=
C
i
j
k
∣
h
=
0.Define(15)
J
i
=
g
j
k
L
i
j
k
.A Finsler space is said to be weakly Landsberg space if J
i
=
0 [15].It is obvious that everyC-reducible Finsler space is P-reducible, but the converse is not true.In paper [1] define(16)
M
̿
i
j
k
=
P
i
j
k
-
1
n
+
1
J
i
h
i
j
+
J
j
h
j
k
+
J
k
h
i
j
,where (17)
P
i
j
k
=
λ
C
i
j
k
+
A
a
i
h
j
k
+
a
j
h
k
i
+
a
k
h
i
j
-
b
∗
C
i
C
j
C
k
.Let B
=
-
b
∗; hence (18)
P
i
j
k
=
λ
C
i
j
k
+
A
a
i
h
j
k
+
a
j
h
k
i
+
a
k
h
i
j
+
B
C
i
C
j
C
k
,
(19)
J
j
=
g
i
k
P
i
j
k
,where λ, A, and B are some scalar function homogeneous of degree 1 and a
i’s are homogeneous of degree zero. It is obvious that F
n is a P-reducible Finsler space if M
̿
i
j
k
=
0.The purpose of the present paper is to studyF
n satisfying (18).IfF
n is a Landsberg space then P
i
j
k
=
0; hence from (18)(20)
C
i
j
k
=
-
A
λ
a
i
h
j
k
+
a
j
h
i
k
+
a
k
h
i
j
-
B
λ
C
i
C
j
C
k
,where (21)
a
i
=
-
λ
A
n
-
2
C
i
,
B
C
2
λ
=
3
n
-
2
.Corollary 4.
A Landsbergs space satisfying (18) is a Special semi-C-reducible Finsler space.
Since forF
n to be Landsberg space P
i
j
k
h
=
0, therefore from Lemma 1 and Corollary 4.Corollary 5.
A Landsbergs space satisfying (18) is Berwald’s affinely connected space, if B
=
0
.
In view of Lemma2 and Corollary 5 one has the following.Corollary 6.
If Landsbergs space satisfying (18) has vanishing h-curvature tensor, that is, R
i
j
k
h
=
0, then it is locally Minkowskian.Special Forms of
P
i
j
k. Let F
n be a Finsler space satisfying (18). A Finsler space with P
i
j
k of given form reduces to P
∗-Finsler space when A
=
0 and B
=
0, while it reduces to P-reducible Finsler space when λ
=
0 and B
=
0 and A
a
i
=
1
/
(
n
+
1
)
J
i.By definition from (18) we can write (22)
L
i
j
k
=
λ
C
i
j
k
+
A
a
i
h
j
k
+
a
j
h
i
k
+
a
k
h
i
j
+
B
C
i
C
j
C
k
.Contracting by g
i
j we get (23)
J
k
=
λ
+
B
C
2
C
k
+
A
n
+
1
a
k
,
a
k
=
1
n
+
1
A
J
k
-
λ
+
B
C
2
n
+
1
A
C
k
.By replacing (23) into (22)(24)
L
i
j
k
=
λ
C
i
j
k
+
1
n
+
1
J
i
h
j
k
+
J
j
h
k
i
+
J
k
h
i
j
-
λ
+
B
C
2
n
+
1
C
i
h
j
k
+
C
j
h
k
i
+
C
k
h
i
j
+
B
C
i
C
j
C
kor(25)
L
i
j
k
-
1
n
+
1
J
i
h
j
k
+
J
j
h
k
i
+
J
k
h
i
j
=
λ
C
i
j
k
-
1
n
-
2
C
i
h
j
k
+
C
j
h
k
i
+
C
k
h
i
j
+
3
n
-
2
C
2
C
i
C
j
C
k
,where (26)
n
+
1
n
-
2
=
1
+
B
λ
C
2
,
3
n
-
2
=
B
λ
C
2
.Hence we have the following.Theorem 7.
The Matsumoto torsion ofP-reducible Finsler space M
̿
i
j
k and Matsumoto torsion of Special semi-C-reducible Finsler space M
-
i
j
k are related by(27)
M
̿
i
j
k
=
λ
M
-
i
j
k
.Corollary 8.
A Finsler spaceF
n satisfying (18) is a weakly Landsberg space if(28)
a
i
=
-
λ
+
B
C
2
n
+
1
A
C
i
.The notation of stretch curvature denoted by Σ
h
i
j
k was introduced by Berwald as generalization of Landsberg curvature [10] in which (29)
Σ
h
i
j
k
≔
2
L
h
i
j
∣
k
-
L
h
i
k
∣
j
.A Finsler space is said to be stretch space if Σ
h
i
j
k
=
0.Again taking h-covariant derivative of (22) and then contracting by y
h, we get (30)
L
i
j
k
∣
h
y
h
=
λ
-
+
λ
2
C
i
j
k
+
λ
A
a
i
+
A
-
a
i
+
A
a
i
-
h
j
k
+
λ
A
a
j
+
A
-
a
j
+
A
a
j
-
h
i
k
+
λ
A
a
k
+
A
-
a
k
+
A
a
k
-
h
i
j
+
B
-
C
k
+
B
L
k
C
i
C
j
+
J
i
C
j
+
J
J
C
i
B
C
k
,where we put λ
-
=
λ
|
h, A
-
=
A
|
h, and B
-
=
B
|
h
.Suppose thatF
n is stretch space; then (31)
L
i
j
k
∣
h
-
L
i
j
h
∣
k
=
0
.By contacting (30) with y
k, we obtain (32)
L
i
j
k
∣
h
y
h
=
0
.From (32) and (30) we have(33)
C
i
j
k
=
-
1
λ
-
+
λ
2
λ
A
a
i
+
A
-
a
i
+
A
a
i
-
h
j
k
+
λ
A
a
j
+
A
-
a
j
+
A
a
j
-
h
i
k
+
λ
A
a
k
+
A
-
a
k
+
A
a
k
-
h
i
j
+
B
-
C
k
+
B
J
k
C
i
C
j
+
J
i
C
j
+
J
J
C
i
B
C
k
.Contacting by (33) by g
j
h
(34)
C
k
=
-
1
λ
-
+
λ
2
n
+
1
λ
A
a
k
+
A
-
a
k
+
A
a
k
-
+
B
-
C
k
+
B
L
k
C
2
+
2
B
J
C
C
kwhence (35)
λ
A
a
k
+
A
-
a
k
+
A
a
k
-
=
-
λ
-
+
λ
2
+
B
C
2
+
2
B
J
C
n
+
1
C
k
-
B
C
2
n
+
1
J
k
.Substituting (35) into (33), we get(36)
C
i
j
k
=
-
λ
-
+
λ
2
+
B
C
2
+
2
B
J
C
n
+
1
λ
-
+
λ
2
C
i
h
j
k
+
C
j
h
k
j
+
C
k
h
i
J
+
B
C
2
n
+
1
λ
-
+
λ
2
J
i
h
j
k
+
J
j
h
k
i
+
J
k
h
i
j
+
-
B
-
λ
-
+
λ
2
C
i
C
j
C
k
+
-
B
λ
-
+
λ
2
J
i
C
j
C
k
+
J
j
C
i
C
k
+
J
k
C
j
C
i
.From (36), it follows that F
n is a semi-C-reducible Finsler space if it is a weakly Landsberg space.Therefore we have the following.Theorem 9.
Let a Finsler spaceF
n satisfying (18) be a stretch space; then it is a Special semi-C-reducible Finsler space, if it is a weakly Landsberg space.2-Connection. A connection connects with tengent spaces of two points of manifold. The n
3 quantities L
j
k
i are connection coefficients if(37)
W
j
i
=
L
j
k
i
d
x
k
d
t (see [16]).Connection. L
j
k
i is uniquely expressible as the sum of the symmetric connections and the torsion tensor [12] (38)
L
j
k
i
=
S
j
k
i
+
T
j
k
i
,where(39a)
S
j
k
i
=
1
2
L
j
k
i
+
L
k
j
iis symmetric connection (39b)
L
j
k
i
=
P
j
k
i
+
T
j
k
i
s
c
i
e
n
c
e
P
j
k
i
=
S
j
k
i
.A connection L
j
k
i is called symmetric connection if L
j
k
i
=
L
k
j
i.Torsion tensor for symmetric connection science is(40)
T
j
k
i
=
L
j
k
i
-
L
k
j
i
2
,
(41)
T
k
j
i
=
-
T
j
k
i
;that is, T
j
k
i is a skew-symmetric tensor.Five kinds of torsion tensors [17] are as follows:(42)
T
=
T
j
k
i
h
h
-
t
o
r
s
i
o
n
,
V
=
V
j
k
i
h
h
ν
-
t
o
r
s
i
o
n
,
R
1
=
R
j
k
i
ν
h
-
t
o
r
s
i
o
n
,
P
1
=
P
j
k
i
ν
h
ν
-
t
o
r
s
i
o
n
,
S
1
=
S
j
k
i
ν
ν
-
t
o
r
s
i
o
n
.It is noted that ν-connection V
j
k
i also plays a role of torsion tensor and(43)
T
j
k
i
=
F
j
k
i
-
F
k
j
i(see [9]).(44)
P
j
k
i
=
∂
˙
k
N
J
i
-
F
k
j
i
,
(45)
S
j
k
i
=
V
j
k
i
-
V
k
j
i
.From (40) and (43) we have(46)
L
j
k
i
-
L
k
j
i
=
2
F
j
k
i
-
F
k
j
i
.From (38) and (46), we have(47)
S
j
k
i
-
T
j
k
i
=
2
F
j
k
i
-
F
k
j
i
.From (39b) and (43) we have(48)
L
j
k
i
-
P
j
k
i
=
F
j
k
i
-
F
k
j
i
,
P
j
k
i
=
L
j
k
i
-
F
j
k
i
+
F
k
j
i
.For Carton’s connection (h) torsion T
j
k
i
=
0.Hence from (47) we have(49)
S
j
k
i
=
2
F
j
k
i
-
F
k
j
i
.Using (45)(50)
V
j
k
i
-
V
k
j
i
=
2
F
j
k
i
-
F
k
j
i
.Also for (h) h-torsion S
j
k
i
=
0 from (47)(51)
T
i
j
i
=
2
F
j
k
i
-
F
k
j
i
(52)
V
j
k
i
=
V
k
j
i
.Then tensor D of component D
j
i is called the deflection tensor F.Therefore(53)
D
j
i
=
N
j
i
,
y
r
V
r
j
i
=
0
.Put r
=
k; we have(54)
y
k
V
k
j
j
=
0
.Using (52) we have(55)
y
k
V
j
k
i
=
0
⟹
y
r
V
k
r
j
=
0
.Theorem 10.
For Cartan’s connection (h) h tensorS
j
k
i and deflection FT V
j
k
i is symmetric as (52).



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*Source: 3694017-2016-09-29.xml*