Open Access Library Journal
Vol.06 No.03(2019), Article ID:91116,8 pages
10.4236/oalib.1105244
On Landsberg and Berwald Spaces of Two Dimensional Finslerian Space with Special (α, β)-Metric
Pradeep Kumar1, T. S. Madhu1, B. R. Sharath2
1Department of Mathematics, Acharya Institute of Technology, Soladevanahalli, Bengaluru, India
2Department of Mathematics, Vemana Institute of Technology, Bengaluru, India
Copyright © 2019 by author(s) and Open Access Library Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: February 13, 2019; Accepted: March 10, 2019; Published: March 13, 2019
ABSTRACT
In the present paper, we find a condition for the two-dimensional Finsler space with a special -metric to be a Berwald space. Also we have proved that, if the two-dimensional Finsler space with above metric is a Landsberg space, then it is a Berwald space.
Subject Areas:
Mathematical Analysis
Keywords:
Finsler Space, Berwald Space, Cartan Connection, Landsberg Space, Main Scalar
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.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
Kumar, P., Madhu, T.S. and Sharath, B.R. (2019) On Landsberg and Berwald Spaces of Two Dimensional Finslerian Space with Special (α, β)-Metric. Open Access Library Journal, 6: e5244. https://doi.org/10.4236/oalib.1105244
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