Locally projectively flatness and locally dually flatness of generalized Kropina conformal change of m-th root metric

Document Type : Original Article

Authors

1 Department of Mathematics, Institute of Applied Sciences and Humanities, GLA University Mathura-281406, India

2 Department of Mathematics and Statistics, Dr. Rammanohar Lohiya Avadh University, Ayodhya-224001

Abstract

In this paper, we consider the generalized Kropina conformal change of m-th root metric and for this, prove a necessary and sufficient condition of locally projectively flatness. Also we proved a necessary and sufficient condition for the generalized Kropina conformal change of m-th root metric is locally dually flat.

Keywords

Main Subjects

References

[1] S.-i. Amari and H. Nagaoka, Methods of information geometry, vol. 191 of Translations of Mathemat[1]ical Monographs, American Mathematical Society, Providence, RI; Oxford University Press, Oxford, 2000. Translated from the 1993 Japanese original by Daishi Harada.
[2] P. L. Antonelli, R. S. Ingarden, and M. Matsumoto, The theory of sprays and Finsler spaces with applications in physics and biology, vol. 58 of Fundamental Theories of Physics, Kluwer Academic Publishers Group, Dordrecht, 1993.
[3] G. S. Asanov, A Finslerian extension of general relativity, Found. Phys., 11 (1981), pp. 137–154.
[4] V. Balan, Notable submanifolds in Berwald-Mo´or spaces, in Proceedings—the International Conference of Differential Geometry and Dynamical Systems (DGDS-2009), vol. 17 of BSG Proc., Geom. Balkan Press, Bucharest, 2010, pp. 21–30.
[5] V. Balan and N. Brinzei, Einstein equations for (h, v)-Berwald-Moor relativistic models, Balkan J. Geom. Appl., 11 (2006), pp. 20–27.
[6] N. Brinzei, Projective relations for m-th root metric spaces, J. Calcutta Math. Soc., 5 (2009), pp. 21–35.
[7] M. Hashiguchi, On conformal transformations of Finsler metrics, J. Math. Kyoto Univ., 16 (1976), pp. 25–50.
[8] M. Knebelman, Conformal geometry of generalized metric spaces, Proceedings of the National Academy of Sciences, 15 (1929), pp. 376–379.
[9] B. Li and Z. Shen, Projectively flat fourth root Finsler metrics, Canad. Math. Bull., 55 (2012), pp. 138–145.
[10] M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, Kaiseisha Press, Shigaken, 1986.
[11] B. N. Prasad and A. K. Dwivedi, On conformal transformation of Finsler spaces with mth root metric, Indian J. Pure Appl. Math., 33 (2002), pp. 789–796.
[12] H. Rund, The  differential geometry of Finsler spaces, Die Grundlehren der mathematischen Wissenschaften, Band 101, Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1959.
[13] Z. Shen, Riemann-Finsler geometry with applications to information geometry, Chinese Ann. Math. Ser. B, 27 (2006), pp. 73–94.
[14] C. Shibata, On invariant tensors of β-changes of Finsler metrics, J. Math. Kyoto Univ., 24 (1984), pp. 163– 188.
[15] H. Shimada, On Finsler spaces with the metric L = mp (ai1i2···im(x)y i1 y i2 · · · y im, Tensor (N.S.), 33 (1979), pp. 365–372. [16] A. Tayebi, E. Peyghan, and M. Shahbazi Nia, On generalized m-th root Finsler metrics, Linear Algebra Appl., 437 (2012), pp. 675–683.
[17] A. Tayebi, T. Tabatabaeifar, and E. Peyghan, On Kropina change for mth root Finsler metrics, Ukrainian Math. J., 66 (2014), pp. 160–164. Reprint of Ukra¨ın. Mat. Zh. 66 (2014), no. 1, 140–144.
[18] B. Tiwari and M. Kumar, On Randers change of a Finsler space with mth-root metric, Int. J. Geom. Methods Mod. Phys., 11 (2014), pp. 1450087, 13.
[19] Y. Yu and Y. You, On Einstein m-th root metrics, Differential Geom. Appl., 28 (2010), pp. 290–294.
AUT Journal of Mathematics and Computing
Volume 3, Issue 1
February 2022
Pages 27-33

Files

Share

How to cite

Statistics