On generalized 4-th root metrics of isotropic scalar curvature
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Akbar Tayebi
Abstract
By an interesting physical perspective and a suitable contraction of the Riemannian curvature tensor in Finsler geometry, Akbar-Zadeh introduced the notion of scalar curvature for the Finsler metrics. A Finsler metric is called of isotropic scalar curvature if the scalar curvature depends on the position only. In this paper, we study the class of generalized 4-th root metrics. These metrics generalize 4-th root metrics which are used in Biology as ecological metrics. We find the necessary and sufficient condition under which a generalized 4-th root metric is of isotropic scalar curvature. Then, we find the necessary and sufficient condition under which the conformal change of a generalized 4-th root metric is of isotropic scalar curvature. Finally, we characterize the Bryant metrics of isotropic scalar curvature.
Communicated by Július Korbaš
Acknowledgement
The author would like to thank Professors Hassan Akbar-Zadeh, Vladimir Balan and Robert Bryant for their valuable comments. Also, we would like to thank the referees for their careful reading of the manuscript and helpful suggestions.
References
[1] Akbar-Zadeh, H.: Sur les espaces de Finsler á courbures sectionnelles constantes, Bull. Acad. Roy. Belg. Cl. Sci. 74(5) (1988), 271–322.10.3406/barb.1988.57782Search in Google Scholar
[2] Akbar-Zadeh, H.: Generalized Einstein manifolds, J. Geom. Phys. 17 (1995), 342–380.10.1016/0393-0440(94)00052-2Search in Google Scholar
[3] Antonelli, P. L.—Ingarden, R.—Matsumoto, M.: The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Acad. Publ., Netherlands, 1993.10.1007/978-94-015-8194-3Search in Google Scholar
[4] Bácsó, S.—Cheng, X.: Finsler conformal transformations and the curvature invariant, Publ. Math. Debrecen 70(1-2) (2007), 221–231.10.5486/PMD.2007.3606Search in Google Scholar
[5] Balan, V.—Brinzei, N.: Einstein equations for (h, v)-Berwald-Moór relativistic models, Balkan. J. Geom. Appl. 11(2) (2006), 20–26.Search in Google Scholar
[6] Bryant, R.: Finsler structures on the 2-sphere satisfyingK = 1, Contemp. Math. 196 (1996), 27–42.10.1090/conm/196/02427Search in Google Scholar
[7] Bryant, R.: Projectively flat Finsler 2-spheres of constant curvature, Selecta Math. (N.S.) 3 (1997), 161–204.10.1007/s000290050009Search in Google Scholar
[8] Cheng, X.—Yuan, M.: On Randers metrics of isotropic scalar curvature, Publ. Math. Debrecen 84 (2014), 63–74.10.5486/PMD.2014.5833Search in Google Scholar
[9] Hashiguchi, H.—Ichijyo, Y.: Randers spaces with rectilinear geodetics, Rep. Fac. Sci. Kagoshima Univ. (Math. Phys. Chem) 13 (1980), 33–40.Search in Google Scholar
[10] A. Heydari, H.—Peyghan, E.—Tayebi, A.: Generalized P-reducible Finsler metrics, Acta Math. Hungar. 149(2) (2016), 286–296.10.1007/s10474-016-0615-0Search in Google Scholar
[11] Li, B.—Shen, Z.: Ricci curvature tensor and non-Riemannian quantities, Canad. Math. Bull. 58 (2015), 530–537.10.4153/CMB-2014-063-4Search in Google Scholar
[12] Li, B.—Shen, Z.: On projectively flat fourth root metrics, Canad. Math. Bull. 55 (2012), 138–145.10.4153/CMB-2011-056-5Search in Google Scholar
[13] Shen, Z.: Projectively flat Finsler metrics of constant flag curvature, Trans. Amer. Math. Soc. 355(4) (2003), 1713–1728.10.1090/S0002-9947-02-03216-6Search in Google Scholar
[14] Shibata, C.: On invariant tensors of β-changes of Finsler metrics, J. Math. Kyoto Univ. 24 (1984), 163–188.10.1215/kjm/1250521391Search in Google Scholar
[15] Shimada, H.: On Finsler spaces with metricL =
[16] Shimada, H.: On the Ricci tensors of particular Finsler spaces, J. Korean Math. Soc. 14 (1977), 41–63.Search in Google Scholar
[17] Tayebi, A.: On the class of generalized Landsbeg manifolds, Period. Math. Hungar. 72 (2016), 29–36.10.1007/s10998-015-0108-xSearch in Google Scholar
[18] Tayebi, A.—Alipour, A.: On distance functions induces by Finsler metrics, Publ. Math. Debrecen 90 (2017), 333–357.10.5486/PMD.2017.7505Search in Google Scholar
[19] Tayebi, A.—Barzegari, B.: Generalized Berwald spaces with (α, β)-metrics, Indag. Math. 27 (2016), 670–683.10.1016/j.indag.2016.01.002Search in Google Scholar
[20] Tayebi, A.—Najafi, B.: On m-th root Finsler metrics, J. Geom. Phys. 61 (2011), 1479–1484.10.1016/j.geomphys.2011.03.012Search in Google Scholar
[21] Tayebi, A.—Najafi, B.: On m-th root metrics with special curvature properties, C. R. Math. Acad. Sci. Paris Ser. I. 349 (2011), 691–693.10.1016/j.crma.2011.06.004Search in Google Scholar
[22] Tayebi, A.—Nankali, A.: On generalized Einstein Randers metrics, Int. J. Geom. Methods Mod. Phys. 12(9) (2015), 1550105, 14 pages.10.1142/S0219887815501054Search in Google Scholar
[23] Tayebi, A.—Nankali, A.—Peyghan, E.: Some curvature properties of Cartan spaces with mth root metrics, Lith. Math. J. 54(1) (2014), 106–114.10.1007/s10986-014-9230-3Search in Google Scholar
[24] Tayebi, A.—Nankali, A.—Peyghan, E.: Some properties of m-th root Finsler metrics, J. Contemp. Math. Anal. 49(4) (2014), 157–166.10.3103/S1068362314040049Search in Google Scholar
[25] Tayebi, A.—Peyghan, E.: On Ricci tensors of Randers metrics, J. Geom. Phys. 60 (2010), 1665–1670.10.1016/j.geomphys.2010.05.016Search in Google Scholar
[26] Tayebi, A.—Peyghan, E.—Shahbazi Nia, M.: On generalized m-th root Finsler metrics, Linear Algebra Appl. 437 (2012), 675–683.10.1016/j.laa.2012.02.025Search in Google Scholar
[27] Tayebi, A.—Peyghan, E.—Shahbazi Nia, M.: On Randers change of m-th root Finsler metrics, Int. Electron. J. Geom. 8(1) (2015), 14–20.10.36890/iejg.592790Search in Google Scholar
[28] Tayebi, A.—Sadeghi, H.: On Cartan torsion of Finsler metrics, Publ. Math. Debrecen 82(2) (2013), 461–471.10.5486/PMD.2013.5379Search in Google Scholar
[29] Tayebi, A.—Sadeghi, H.—Peyghan, E.: On generalized Douglas-Weyl spaces, Bull. Malays. Math. Sci. Soc. (2) 36(3) (2013), 587–594.Search in Google Scholar
[30] Tayebi, A.—Shahbazi Nia, M.: A new class of projectively flat Finsler metrics with constant flag curvatureK = 1, Differ. Geom. Appl. 41 (2015), 123–133.10.1016/j.difgeo.2015.05.003Search in Google Scholar
[31] Tayebi, A.—Tabatabaeifar, T.—Peyghan, E.: On Kropina-change of m-th root Finsler metrics, Ukrainian J. Math. 66(1) (2014), 140–144.10.1007/s11253-014-0919-6Search in Google Scholar
[32] Xu, B.—Li, B.: On a class of projectively flat Finsler metrics with flag curvatureK = 1, Differ. Geom. Appl. 31 (2013), 524–532.10.1016/j.difgeo.2013.05.001Search in Google Scholar
[33] Yu, Y.—You, y.: On Einstein m-th root metrics, Differ. Geom. Appl. 28 (2010), 290–294.10.1016/j.difgeo.2009.10.011Search in Google Scholar
© 2018 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- A multi-parameter generalization of the symmetric algorithm
- Single identities forcing lattices to be Boolean
- Weighted uniform density ideals
- A generalized class of restricted Stirling and Lah numbers
- The Riemann hypothesis and universality of the Riemann zeta-function
- Distance functions on the sets of ordinary elliptic curves in short Weierstrass form over finite fields of characteristic three
- The drazin inverse of the sum of two matrices
- Refinements of the majorization-type inequalities via green and fink identities and related results
- Characterizations and properties of graphs of Baire functions
- Some improvements of the young mean inequality and its reverse
- On characteristic of bounded analytic functions involving hyperbolic derivative
- A uniqueness problem for entire functions related to Brück’s conjecture
- System of nonlocal resonant boundary value problems involving p-Laplacian
- Construction of a unique mild solution of one-dimensional Keller-Segel systems with uniformly elliptic operators having variable coefficients
- Infinitely many solutions for non-homogeneous Neumann problems in Orlicz-Sobolev spaces
- Characterization of rough weighted statistical limit set
- Approximation by Baskakov-Durrmeyer operators based on (p, q)-integers
- On generalized 4-th root metrics of isotropic scalar curvature
- Compactifications from generators and relations
- Diophantine quadruples with values in k-generalized Fibonacci numbers
