On Landsberg and Berwald Spaces of Two Dimensional Finslerian Space with Special (α, β)-Metric ()
1. Introduction
In the carton connection
, if the covariant derivative
satisfies
then the Finsler space is known as Landsberg space. L. Berwald introduced a class of Finslerian spaces which are known as Berwald spaces in which local coefficients of the Berwald connection depend only on position coordinates. If Landsberg space satisfy some conditions, then it is Berwald space [1] . On the other hand, in two-dimensional case, the main scalar of a general Finsler space
satisfies
if and only if general Finslerian space is a Landsberg space [2] .
The purpose of the present paper is to find a two-dimensional Landsberg space
with a special
-metric
satisfying some conditions. First
we find the condition for a Finsler space with a special
-metric to be a Berwald space. Next, we determine the difference vector and the main scalar of
with the aforesaid metric.
Finally, we derive the condition for a two-dimensional Finsler space
with
a special
-metric
to be a Landsberg space and we
have shown that if
with the mentioned metric is a Landsberg space, then it is a Berwald space.
2. Preliminaries
Let an n-dimensional Finsler space
with
-metric and the associated Riemannian space
where
. We put
since
is invertible. In the following, we restrict our discussions to a domain of
where
does not vanish by taking Riemannian metric
is not supposed to be positive definite. The semi-colon denotes the covariant differentiation in the Levi-Civita connection
of
.
We have the following symbols
Here
of
plays an important role. Denote by
the difference tensor of Matsumoto [3] of
from
:
(2.1)
Transvecting above by
and then by
, we have
(2.2)
Then
and
.
On account Matsumoto [3] , the components of
is determined by
(2.3)
According to Matsumoto [3] ,
is called the difference vector if
where
.
Then
is written as follows
(2.4)
where
Further, by means of M. Hashiguchi, S. Hojo and M. Matsumoto [4] , we have
(2.5)
We have the following lemmas
Lemma 2.1. [2] [5] . If
contains
as a factor i.e.
, then the dimension
and
vanishes. In this case we have 1-form
satisfying
and
.
Lemma 2.2. [4] . We consider the two dimension case.
1) If
, then
a sign
and
and
.
2) If
, then
and
.
If two functions
and
satisfies
, then it is cleared that
because
gives a contradiction
.
Throughout the chapter, for brevity we shall say “homogeneous polynomial (s) in
of degree r”. Hence
are
.
3. Berwald Space
In this section, Let us consider an n-dimensional Finslerian space
with the following special
-metric
(3.1)
First we shall assume
.
Suppose if
, then from lemma (2.2), we have
, then
, which is a Randers metric. So the assumption
is reasonable.
Then from the above, we have
(3.2)
Substituting (3.2) into (3.3), we obtain
(3.3)
Assume that the Finsler space with metric (3.1) be a Berwald space, i.e.,
.
Then we have
, so LHS of (3.3) has a form
where P and Q are polynomials in
while
is irrational in
. Hence the Equation (3.3) shows
.
By Lemma (2.1), we have
The former yields
, so we have
. Then the latter leads to
directly.
Conversely, if
, by well known Okada’s axioms
becomes the Berwald connection of
. Thus
is a Berwald space.
Hence we have the following result
Theorem 3.1. The Finsler space
with special
-metric (3.1) satisfying
is a Berwald space if and only if
, then Berwald connection is essentially Riemannian
.
4. Two-Dimensional Landsberg Space
In this section, Let us consider an n-dimensional Finslerian space
with the following special
-metric
(4.1)
By means of (2.4) and (3.2), the difference vector
[6] of the Finsler space becomes
(4.2)
where
It is trivial that
,
and
, because
is irrational in
. From (4.2) it follows that
In two-dimensional case, the main scalar of a general Finsler space
satisfies
if and only if general Finsler space is a Landsberg space [7] . If
with (4.1), then the main scalar I is obtained as follows
(4.3)
where
The covariant differentiation of (4.3) leads to
(4.4)
Transvecting (4.4) by
, we have
(4.5)
where
Hence (4.5) can be put in the form
where
Consequently, the Finslerian space
with special
-metric (4.1) is Landsberg space if and only if
since
.
If
, then
which is a contradiction.
In view of (2.5), the above equation written as
(4.6)
Substituting the values of
and
in (4.6), we obtain
(4.7)
Separating (4.7) as rational and irrational terms with respect to (
), we obtain
(4.8)
where
The Equation (4.8) yields two equations as follows
(4.9)
(4.10)
From (4.10), we obtain
Then
a function
Thus, we have
(4.11)
Transvecting above by
leads to
Eliminating
from (4.9) and (4.10), from (4.11), we have
(4.12)
From
it follows that
a function
.
Hence (4.12) is reduces to
(4.13)
Since only the term
of
seemingly does not contain
, we must have
such that
. Thus it is a contradiction because of
, that is,
does not contain
as a factor.
Thus, from (4.13) we have
, which leads to
and
. Hence
(4.14)
which implies
, which leads to
and
. From (4.11), we get
.
Summarizing up, we obtain
and
, that is,
Therefore
is the so-called killing vector field with a constant length.
According to Hashiguchi, Hojo and Matsumoto [4] , the condition (4.14) is equivalent to
.
Hence, we have the following result
Theorem 4.2. If a two dimensional Finsler space
with a special
-metric (4.1) satisfying
, is a Landsberg space then
is a Berwald space.
5. Conclusion
In this paper, first we found a condition for a Finslerian space with special
-metric
to be a Berwald space. Further we have proved that two-dimensional Finslerian space with a special
-metric
is a Landsberg space, then it is a Berwald space.