Intrinsic Properties of Finsler Space with a cubic change in the Matsumoto Metric

Krishnamoni  Medhi; Maphisha  Nongsiej; V. K. Chaubey

doi:10.5644/SJM.21.01.12

Authors

Krishnamoni Medhi North Eastern Hill University
Maphisha Nongsiej North Eastern Hill University
V. K. Chaubey North Eastern Hill University

DOI:

https://doi.org/10.5644/SJM.21.01.12

Keywords:

Matsumoto Metric, Cubic Metric, one form metric, Cartan Tensor, C-reducible

Abstract

In this paper, we investigate Finsler space with a cubic modification of the Matsumoto metric, given by $F=\frac{\gamma^{2}}{\gamma-\beta}$, where $\gamma$ is a cubic metric and $\beta$ is one form metric. We identify the fundamental characteristics of this modified metric. The reducibility of the Cartan torsion tensor is a key factor, as it measures how closely a Finsler metric approximates a Riemannian metric. Specifically, if $C_{ijk}$ vanishes, the Finsler metric becomes Riemannian. Accordingly, we analyze various forms of the Cartan torsion tensor's reducibility within the context of this cubic-changed Matsumoto metric. We also establish conditions for determining whether the Finsler space is quasi C-reducible, semi C-reducible, C-reducible and C2-like.

Author Biographies

Krishnamoni Medhi, North Eastern Hill University

Student

Maphisha Nongsiej, North Eastern Hill University

Student

V. K. Chaubey, North Eastern Hill University

Mathematics, Associate Professor

Statistics

Abstract: 179 / PDF: 34

References

Aikou, T., Hashiguchi, M. and Yamaguchi, K.: On Matsumoto’s Finsler space with time measure, Rep. Fac. Sci. Kagoshima Univ., Math. Phy. Chem., (1990), 1-12.

Antonelli, P. L.: Handbook of Finsler geometry, Kluwer Academic Publishers, Netherlands (2003).

Bao, D., Chern, S. S., Shen, Z.: An introduction to Riemann-Finsler geometry, Springer, New York (2000).

Chaubey, V. K. and Tripathi, B. K.: Finslerian hypersurface and Finsler spaces with $(gamma, beta)$-Metric, Journal of Dynamical Systems in Geometric Theories, 12(1),(2014), 19-27.

Heydari, A., Peyghan, E. and Tayebi, A.: Generalized P-reducible Finsler Metrics, Acta Math. Hungar., 149(2),(2016), 286-296, DOI: 10.1007/s10474-016-0615-0.

Matsumoto, M.: A slope of Mountain is a Finsler surface with respect to time measure, J. Math. Kyoto Univ., 29(1), (1989), 17-25.

Matsumoto, M.: On C-reducible Finsler spaces, Tensor, N.S., 24, (1972), 29-37.

Matsumoto, M. and Hojo, S.: A conclusive theorem on C-reducible Finsler spaces, Tensor, N.S., 32, (1978), 225-230.

Matsumoto, M. and Numata, S.: On Finsler spaces with a cubic metric, Tensor, N.S., 33, (1979), 153-162.

Matsumoto, M.:Theory of Finsler spaces with $ (alpha, beta) $-metric, Rep. on Math, Phys., 31, (1992), 43-83.

Okada, T. and Numata, S.: On generalised C-reducible Finsler spaces, Tensor, N.S., 35, (1981), 313-318.

Pandey, T. N. and Chaubey, V. K.: On Finsler space with $(gamma, beta)$-metric: geodesic, connection and scalar curvature, Tensor, N.S., (74)2, (2013), 126-134.

Pandey, T. N. and Chaubey, V. K.:Theory of Finsler spaces with $(gamma, beta)$-metric, Bulletin of the Transilvania University of Brasov, 4(53)2, (2011), 43-56.

Pandey, T.N., Chaubey, V.K.: Main scalar of of two-dimensional Finsler space with $(gamma,beta)$, J. Rajasthan Acad. Phy. Sci., 11(1), (2012), 1-10.

Park, Hong-Suh, Lee, Il-Yong, Park, Ha-Yong and Kim, Byung-Doo: Projectively flat Finsler space with an approximate Matsumoto metric, Commun. Korean Math. Soc., (18)3, (2003), 501-513.

Shimada, H.: On Finsler spaces with $ L=sqrt[m]{a_{i_1i_{2}...i_{m}}y^{i_{1}}y^{i_{2}}...y^{i_{m}}} $, Tensor, N. S., 3, (1979), 365-372.

Shukla, H. S. and Mishra, Arunima: On Finsler spaces with $ (gamma, beta) $-metric, Matehamtical Fourm, 25, (2013), 63-76.

Shen, Z.: Differential geometry of Spray and Finsler Spaces, Springer-Verlag, (2001).

Tiwari, B., Gangopadhyay, R. and Prajapati, G. K.: On Semi C-reducibility of general $(alpha, beta)$-Finsler metrics, Kyungpook Math. J., 59(2), (2019), 353-362, DOI: https://doi.org/10.5666/KMJ.2019.59.2.353.