Honam Mathematical Journal (호남수학학술지)

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WEAKLY BERWALD SPACE WITH A SPECIAL (α, β)-METRIC

  • PRADEEP KUMAR (Department of Mathematics, Presidency University) ;
  • AJAYKUMAR AR (Department of Mathematics, KNS Institute of Technology)
  • Received : 2022.12.06
  • Accepted : 2023.02.09
  • Published : 2023.09.14
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Abstract

As a generalization of Berwald spaces, we have the ideas of Douglas spaces and Landsberg spaces. S. Bacso defined a weakly-Berwald space as another generalization of Berwald spaces. In 1972, Matsumoto proposed the (α, β) metric, which is a Finsler metric derived from a Riemannian metric α and a differential 1-form β. In this paper, we investigated an important class of (α, β)-metrics of the form $F={\mu}_1\alpha+{\mu}_2\beta+{\mu}_3\frac{\beta^2}{\alpha}$, which is recognized as a special form of the first approximate Matsumoto metric on an n-dimensional manifold, and we obtain the criteria for such metrics to be weakly-Berwald metrics. A Finsler space with a special (α, β)-metric is a weakly Berwald space if and only if Bmm is a 1-form. We have shown that under certain geometric and algebraic circumstances, it transforms into a weakly Berwald space.

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References

  1. AR AjayKumar and Pradeep Kumar, On the conformal transformation of Douglas space of second kind with generalized (α, β)-metric, Advances in Dynamical Systems and Applications 17 (2022), no. 1, 299-310.
  2. A. V. Arhangel'skii, Bicompact sets and the topology of spaces, Soviet Mathematics Doklady 4 (1963), 561-564.
  3. S. Bacso and B. Szilagyi, On a weakly Berwald Finsler space of Kropina type, Mathematica Pannonica 13 (2002), no. 1, 91-95.
  4. M. Hashiguchi, On some special (α, β) metrics, Rep. Fac. Sci. Kagoshima University 8 (1975), 39-46.
  5. T. R. Khoshdani and N. Abazari, Charactarization of weakly Berwald fourth-root metrics, Ukrainian Mathematical Journal 71 (2019), no. 7, 1115-1138.
  6. 6] Pradeep Kumar, S. K. Narasimhamurthy, H. G. Nagaraja, and S. T. Aveesh, On a special hypersurface of a Finsler space with (α, β)-metric, Tbilisi Mathematical Journal 2 (2009), 51-60. https://doi.org/10.32513/tbilisi/1528768841
  7. Pradeep Kumar, T. S. Madhu, and B. R. Sharath, On Ricci tensors of a Finsler space with special (α, β)-metric, J. T. S. 13 (2019), 73-80.
  8. Pradeep Kumar, T. S. Madhu, and B. R. Sharath, Projective changes between generalized (α, β)-metric and Randers metric, Advances in Pure Mathematics 10 (2020), no. 5, 312-321.
  9. Pradeep Kumar and AR Ajaykumar, Locally dually flat and reversible geodesics of a Finsler space with generalized (α, β)-metric, Stochastic Modeling and Applications 26 (2022), no. 3, 640-652.
  10. Pradeep Kumar and AR Ajaykumar, New class of weakly Berwald space with a special (α, β)-metric, Ganita 72 (2022), no. 1, 335-344.
  11. Pradeep Kumar and AR Ajaykumar, Reversible geodesics of a Finsler space with generalized (α, β)-metric, Gulf Journal of Mathematics 14 (2023), no. 1, 125-138.
  12. I. Y. Lee and M. H. Lee, On weakly-Berwald spaces of special (α, β)-metrics, Bulletin of the Korean Mathematical Society 43 (2006), no. 2, 425-441.
  13. M. Matsumoto, On c-reducible Finsler spaces, Tensor NS 24 (1972), 29-37.
  14. M. Matsumoto, Theory of Finsler spaces with (α, β)-metric, Reports on Mathematical Physics 31 (1992), no. 1, 43-83.
  15. S. K. Narasimhamurthy and M. Ramesha, Projectively flat Finsler space of Douglas type with weakly-Berwald (α, β)-metric, International Journal of Pure Mathematical Sciences 18 (2017), 1-12. https://doi.org/10.18052/www.scipress.com/IJPMS.18.1
  16. G. Shanker and D. Choudhary, On a new class of weakly Berwald spaces with (α, β)-metric, British Journal of Mathematics and Computer Science 14 (2016), no. 1, 1-13.
  17. C. Shibata, On Finsler space with an (α, β)-metric, J. Hokkaido Univ. Ed. 35 (1984), 1-16.
  18. A. Tayebi, A class of weakly Berwald Finsler metrics, Mathematica Pannonica 2015 (2014), 63-74.
  19. R. Yoshikawa, K. Okubo, and M. Matsumoto, The conditions for some (α, β)-metric spaces to be weakly-Berwald spaces, Tensor NS 65 (2004), no. 3, 277-290.