IJPAM: Volume 119, No. 4 (2018)


SPACE WITH A GENERAL $\left({\alpha ,\beta }\right)$-METRIC


Afsoon Goodarzian$^1$, Megerdich Toomanian$^2$,
and Mehdi Nadjafkhah$^3$
$^{1,2}$Department of Mathematics, Karaj Branch
Islamic Azad University
Karaj, I.R. IRAN
$^3$Department of Pure Mathematics
School of Mathematics
Iran University of Science and Technology, Narmak
Tehran, 16846-13114, I.R. IRAN


In this paper, we find the main scalars of a three dimensional Finsler space equipped with a general $ \left(\alpha,\beta\right)$-metric $ F=\alpha+\kappa\beta+{{\varepsilon\beta}^{2}}/{\alpha} $ (where $ \kappa $ and $ \varepsilon $ are constants which are not zero simultaneously). Consequently, we will obtain results that give the main scalars of special three dimensional $ \left(\alpha,\beta\right)$-metrics such as square metric $ F=(\alpha+\beta )^2/\alpha $ and the first approximate matsumoto metric $ F=\alpha+\beta+{{\beta}^{2}}/{\alpha} $. Moreover, we present a sufficient and necessary condition for Finsler spaces equipped with $ \left(\alpha,\beta\right)$-metrics to be Riemannian spaces. Some fundamental theorems on main scalars of three dimensional Finsler spaces has also been dealt.


Received: March 13, 2018
Revised: August 3, 2018
Published: August 10, 2018

AMS Classification, Key Words

AMS Subject Classification: 58B20, 58J60
Key Words and Phrases: three dimensional Finsler space, $(\alpha,\beta)$-metric, Moor's frame, main scalar, the first matsumoto metric, square metric, Cartan tensor vector

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M. Matsumoto, Theory of Finsler spaces with $(\alpha,\beta)$-Metrics, Rep. Math. Phys., 31, No. 1 (1992), 43-83, DOI: 10.1016/0034-4877(92)90005-L.

T. N. Pandey, B. N. Prasad, V. K. Chaubey, On three-dimensional Finsler spaces with $(\alpha,\beta)$-Metric, The Aigrah Bull. of Maths, 28, No. 1-2 (2009).

G. Randers, On an asymmetric metric in the four space of general relativity, Phys. Rev., 59, (1941), 195-199, DOI: https://doi.org/10.1103/PhysRev.59.195.

Z. Shen, C. Yu, On Einstein square metrics, preprint, arXiv: 1209.3876, (2012).

H. S. Park, I. Y. Lee, C. K. Park, Finsler space with the general approximate Matsumoto metric, Indian J. Pure. Appl. Math., 34, No. 1 (2002), 59-77.

P. L. Antonelli, I. Bucataru, S. F. Rutz, Computer algebra and two and three dimensional Finsler geometry, Publ. Math. DebrecenProof-sheets for paper Ref., 34, No. 2872 (2003), 1-24.

M. K. Gupta, P. N. Pandey, Relations between the main scalars of a four-dimensional Finsler space and its hypersurface, Differ. Geom. Dyn. Syst, (2008), 132-138.

S. K. Narasimhamurthy, D. M. Vasantha, Projective change between randers metricand special $ \left(\alpha,\beta\right)$-metric, Journal of Informatics and Mathematical Sciences, 4, No. 3 (2012), 293-303.

A. Goodarzian, M. Nadjafikhah, M.Toomanian, On Main Scalar of Two Dimentional Finsler Spaces with $ \left(\alpha,\beta\right)$-Metrics, International Journal of Applied Mathematics $\&$ Statistics, 57, No. 3 (2018).

How to Cite?

DOI: 10.12732/ijpam.v119i4.9 How to cite this paper?

International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 119
Issue: 4
Pages: 661 - 683

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