Research ArticleNo Access

On Einstein-reversible $m$ $m$ th root Finsler metrics

Jila Majidi

Department of Mathematics, Basic Sciences Faculty, University of Bonab, Bonab, Iran

Search for more papers by this author

Akbar Tayebi

Department of Mathematics, Faculty of Science, University of Qom, Qom Iran

Search for more papers by this author

, and

Ali Haji-Badali

https://orcid.org/0000-0001-5309-5902

Department of Mathematics, Basic Sciences Faculty, University of Bonab, Bonab, Iran

E-mail Address: haji.badali@ubonab.ac.ir

E-mail Address: ahajibadali@gmail.com

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0219887823500998Cited by:1 (Source: Crossref)

Abstract

The theory of mth root Finsler metrics has been applied to Biology, Ecology, Gravitation, Seismic ray theory, etc. It is regarded as a direct generalization of Riemannian metric in a sense, namely, the second root metric is a Riemannian metric. On the other hand, the Riemannian curvature faithfully reveals the local geometric properties of a Riemann–Finsler metric. The reversibility of Riemannian and Ricci curvatures of Finsler metrics is an essential concept in Finsler geometry. Here, we study the Riemannian curvature of the class of third and fourth root $(α, β)$ $(α, β)$ -metrics. Then, we find the necessary and sufficient condition under which a cubic and fourth root $(α, β)$ $(α, β)$ -metric be Einstein-reversible.

Keywords:

AMSC: 53C60, 53B40

References

1. N. Abazari and T. R. Khoshdani, Characterization of weakly Berwald fourth-root metrics, Ukr. Math. J. 71(7) (2019) 1–23. Web of Science, Google Scholar
2. G. S. Asanov, Finslerian Extension of General Relativity (Reidel, Dordrecht, 1984). Google Scholar
3. S. Bácsó, X. Cheng and Z. Shen, Curvature properties of $(α, β)$ $(α, β)$ -metrics, Adv. Stud. Pure Math. 48 (2007) 73–110. Crossref, Google Scholar
4. V. Balan, Notable submanifolds in Berwald-Moór spaces, in BSG Proc., Vol. 17 (Geometry Balkan Press, 2010), pp. 21–30. Google Scholar
5. V. Balan and S. Lebedev, On the Legendre transform and Hamiltonian formalism in Berwald-Moór geometry, Differ. Geom. Dyn. Syst. 12 (2010) 4–11. Google Scholar
6. V. Balan and N. Brinzei, Einstein equations for $(h, v)$ $(h, v)$ -Berwald-Moór relativistic models, Balkan J. Geom. Appl. 11(2) (2006) 20–26. Web of Science, Google Scholar
7. X. Cheng, Z. Shen and Y. Tian, A class of Einstein $(α, β)$ $(α, β)$ -metrics, Israel J. Math. 192 (2012) 221–249. Crossref, Web of Science, Google Scholar
8. S. S. Chern and Z. Shen, Riemann–Finsler Geometry (World Scientific, 2005). Link, Google Scholar
9. M. Crampin, Randers spaces with reversible geodesics, Publ. Math. (Debrecen) 67 (2005) 401–409. Crossref, Web of Science, Google Scholar
10. V. K. Kropina, On projective two-dimensional Finsler spaces with special metric, Trudy Sem. Vektor. Tenzorn. Anal. 11 (1961) 277–292. Google Scholar
11. I.-Y. Lee and H.-S. Park, Finsler spaces with infinite series $(α, β)$ $(α, β)$ -metric, J. Korean Math. Soc. 41(3) (2004) 567–589. Crossref, Web of Science, Google Scholar
12. B. Li and Z. Shen, On projectively flat fourth root metrics, Canad. Math. Bull. 55 (2012) 138–145. Crossref, Web of Science, Google Scholar
13. I. M. Masca, S. V. Sabau and H. Shimada, Reversible geodesics for $(α, β)$ $(α, β)$ -metrics, Int. J. Math. 21(8) (2010) 1071–1094. Link, Web of Science, Google Scholar
14. I. M. Masca, S. V. Sabau and H. Shimada, Necessary and sufficient conditions for two-dimensional $(α, β)$ $(α, β)$ -metrics with reversible geodesics, preprint (2012), arXiv:1203.1377 . Google Scholar
15. M. Matsumoto, Theory of Finsler spaces with m-th root metric. II, Publ. Math. (Debrecen) 49 (1996) 135–155. Crossref, Web of Science, Google Scholar
16. M. Matsumoto, Theory of Finsler spaces with $(α, β)$ $(α, β)$ -metric, Rep. Math. Phys. 31 (1992) 43–84. Crossref, Google Scholar
17. M. Matsumoto and S. Numata, On Finsler spaces with a cubic metric, Rep. Math. Phys. 32(2) (1979) 153–162. Google Scholar
18. M. Matsumoto and H. Shimada, On Finsler spaces with 1-form metric. II. Berwald-Moór’s metric $L = {(y^{1} y^{2} \dots y^{n})}^{1 / n}$ $L = {(y^{1} y^{2} \dots y^{n})}^{1 / n}$ , Tensor (N.S.) 32 (1978) 275–278. Google Scholar
19. D. G. Pavlov, Gh. Atanasiu and V. Balan, Space-Time Structure, Algebra and Geometry (Russian Hypercomplex Society, Lilia Print, Moscow, 2007). Google Scholar
20. D. G. Pavlov, Four-dimensional time, Hypercomplex Numbers Geom. Phys. 1 (2004) 31–39. Google Scholar
21. Z. Shen and G. Yang, Randers metrics of reversible curvature, Int. J. Math. 24(1) (2013) 1350006. Link, Web of Science, Google Scholar
22. H. Shimada, On Finsler spaces with metric $L = \sqrt[m]{a_{i_{1} \dots i_{m}} (x) y_{1}^{i_{1}} y_{2}^{i_{2}} \dots y_{m}^{i_{m}}}$ $L = \sqrt[m]{a_{i_{1} \dots i_{m}} (x) y^{i_{1}} y^{i_{2}} \dots y^{i_{m}}}$ , Tensor (N.S.) 33 (1979) 365–372. Google Scholar
23. A. Tayebi, On generalized 4-th root metrics of isotropic scalar curvature, Math. Slovaca 68 (2018) 907–928. Crossref, Web of Science, Google Scholar
24. A. Tayebi, On the theory of 4-th root Finsler metrics, Tbil. Math. J. 12(1) (2019) 83–92. Web of Science, Google Scholar
25. A. Tayebi, On 4-th root Finsler metrics of isotropic scalar curvature, Math. Slovaca 70 (2020) 161–172. Crossref, Web of Science, Google Scholar
26. A. Tayebi, M. Amini and B. Najafi, On conformally Berwald m-th root $(α, β)$ $(α, β)$ -metrics, Facta Univ. Ser. Math. Inform. 35(4) (2020) 963–981. Web of Science, Google Scholar
27. A. Tayebi and B. Najafi, On m-th root Finsler metrics, J. Geom. Phys. 61 (2011) 1479–1484. Crossref, Web of Science, Google Scholar
28. A. Tayebi and B. Najafi, On m-th root metrics with special curvature properties, C. R. Acad. Sci. Paris, Ser. I 349 (2011) 691–693. Crossref, Google Scholar
29. A. Tayebi, A. Nankali and E. Peyghan, Some curvature properties of Cartan spaces with m-th root metrics, Lithuanian Math. J. 54(1) (2014) 106–114. Crossref, Web of Science, Google Scholar
30. A. Tayebi, A. Nankali and E. Peyghan, Some properties of m-th root Finsler metrics, J. Contemp. Math. Anal. 49(4) (2014) 157–166. Google Scholar
31. A. Tayebi, E. Peyghan and M. Shahbazi Nia, On generalized m-th root Finsler metrics, Linear Algebra Appl. 437 (2012) 675–683. Crossref, Web of Science, Google Scholar
32. A. Tayebi and M. Razgordani, Four families of projectively flat Finsler metrics with $K = 1$ $K = 1$ and their non-Riemannian curvature properties, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 112 (2018) 1463–1485. Crossref, Web of Science, Google Scholar
33. A. Tayebi and M. Razgordani, On conformally flat fourth root $(α, β)$ $(α, β)$ -metrics, Differ. Geom. Appl. 62 (2019) 253–266. Crossref, Web of Science, Google Scholar
34. A. Tayebi and H. Sadeghi, On generalized Douglas–Weyl $(α, β)$ $(α, β)$ -metrics, Acta Math. Sin. (Engl. Ser.) 31(10) (2015) 1611–1620. Crossref, Web of Science, Google Scholar
35. A. Tayebi and T. Tabatabaeifar, Matsumoto metric of reversible curvatures, Acta Math. Acad. Paedagog. Nyiregyhaziensis 32 (2016) 165–200. Google Scholar
36. A. Tayebi, T. Tabatabaeifar and E. Peyghan, On Kropina-change of m-th root Finsler metrics, Ukrain. Math. J. 66(1) (2014) 140–144. Crossref, Web of Science, Google Scholar
37. B. Tiwari, M. Kumar and A. Tayebi, On generalized Kropina change of generalized m-th root Finsler metrics, Proc. Nat. Acad. Sci. India, Sec. A Phys. Sci. 91 (2021) 443–450. Crossref, Web of Science, Google Scholar
38. J. M. Wegener, Untersuchungen der zwei- und dreidimensionalen Finslerschen Räume mit der Grundform $L = \sqrt[3]{a_{i k ℓ} {x^{'}}^{i} {x^{'}}^{k} {x^{'}}^{ℓ}}$ , Proc. K. Akad. Wet. (Amsterdam) 38 (1935) 949–955. Google Scholar
39. G. Yang, On a class of Einstein-reversible Finsler metrics, Differ. Geom. Appl. 60 (2018) 80–103. Crossref, Web of Science, Google Scholar
40. Y. Yu and Y. You, On Einstein m-th root metrics, Differ. Geom. Appl. 28 (2010) 290–294. Crossref, Web of Science, Google Scholar