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Weighted projective Ricci curvature in Finsler geometry

  • Tayebeh Tabatabaeifar , Behzad Najafi EMAIL logo and Akbar Tayebi
Published/Copyright: January 29, 2021
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Abstract

In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.

MSC 2010: 53B40; 53C60
  1. (Communicated by Július Korbaš)

References

[1] Akbar-Zadeh, H.: Champ de vecteurs projectifs sur le fibre unitaire, J. Math. Pures Appl. 65 (1986), 47–79.Search in Google Scholar

[2] Asanov, G. S.: Finsler Geometry, Relativity and Gauge Theories, D. Reidel Publishing Company, Dordrecht, Holland, 1985.10.1007/978-94-009-5329-1Search in Google Scholar

[3] Bacso, S.—Matsumoto, M.: On Finsler spaces of Douglas type. A generalization of the notion of Berwald space, Publ. Math. Debrecen 51 (1997), 385–406.10.5486/PMD.1997.1975Search in Google Scholar

[4] Bao, D.—Shen, Z.: Finsler metrics of constant positive curvature on the Lie group S3, J. Lond. Math. Soc. 66 (2002), 453–467.10.1112/S0024610702003344Search in Google Scholar

[5] Cheng, X.—Shen, Z.: A comparison theorem on the Ricci curvature in projective geometry, Ann. Global Anal. Geom. 23 (2003), 141–156.10.1023/A:1022446831429Search in Google Scholar

[6] Cheng, X.—Shen, Z.: Randers metrics of scalar flag curvature, J. Aust. Math. Soc. 87 (2009), 359–370.10.1017/S1446788709000408Search in Google Scholar

[7] Cheng, X.—Shen, Y.—Ma, X.: On a class of projective Ricci flat Finsler metrics, Publ. Math. Debrecen 7528 (2017), 1–12.10.5486/PMD.2017.7528Search in Google Scholar

[8] Ingarden, R. S.: Geometry of Thermodynamics.Diff. Geom. Methods in Theor. Phys., XV Intern. Conf. Clausthal 1986, World Scientific, Singapore, 1987.Search in Google Scholar

[9] Kropina, V. K.: On projective two-dimensional Finsler spaces with a special metric, Trudy Sem. Vektor. Tenzor. Anal. 11 (1961), 277–292.Search in Google Scholar

[10] Matsumoto, M.—HōJō, S.: A conclusive theorem on C-reducible Finsler spaces, Tensor N. S. 32 (1978), 225–230.Search in Google Scholar

[11] Najafi, B.—Tayebi, A.: Finsler metrics of scalar flag curvature and projective invariants, Balkan J. Geom. Appl. 15 (2010), 90–99.Search in Google Scholar

[12] Najafi, B.—Tayebi, A.: A new quantity in Finsler geometry, C. R. Acad. Sci. Paris, Ser. I. 349 (2011), 81–83.10.1016/j.crma.2010.11.015Search in Google Scholar

[13] Najafi, B.—Tayebi, A.: Some curvature properties of (α, β)-metrics, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 108(3) (2017), 277–291.Search in Google Scholar

[14] Ohta, S. I.: Weighted Ricci curvature estimates for Hilbert and Funk geometries, Pacific J. Math. 265 (2013), 185–197.10.2140/pjm.2013.265.185Search in Google Scholar

[15] Randers, G.: On an asymmetric metric in the four-space of general relativity, Phys. Rev. 59 (1941), 195–199.10.1103/PhysRev.59.195Search in Google Scholar

[16] Shen, Z.: Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001.10.1007/978-94-015-9727-2Search in Google Scholar

[17] Shen, Z.: Landsberg Curvature, S-Curvature and Riemann Curvature, MSRI Publication Series, Cambridge University Press, 2004.Search in Google Scholar

[18] Tayebi, A.—Barzegari, M.: Generalized Berwald spaces with (α, β)-metrics, Indag. Math. (N.S.) 27 (2016), 670–683.10.1016/j.indag.2016.01.002Search in Google Scholar

[19] Tayebi, A.—Nankali, A.: On generalized Einstein Randers metrics, Int. J. Geom. Meth. Modern. Phys. 12(9) (2015), #1550105.10.1142/S0219887815501054Search in Google Scholar

[20] Tayebi, A.—Nankali, A.—Najafi, B.: On the class of Einstein exponential-type Finsler metrics, Journal of Mathematical Physics, Analysis, Geometry 14(1) (2018), 100–11410.15407/mag14.01.100Search in Google Scholar

[21] Tayebi, A.—Rafie-Rad, M.: S-curvature of isotropic Berwald metrics, Science in China, Series A: Math. 51 (2008), 2198–2204.10.1007/s11425-008-0095-ySearch in Google Scholar

[22] Tayebi, A.—Razgordani, M.: Four families of projectively flat Finsler metrics with K = 1 and their non-Riemannian curvature properties, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 112 (2018), 1463–1485.10.1007/s13398-017-0443-2Search in Google Scholar

[23] Tayebi, A.—Sadeghi, H.: On generalized Douglas-Weyl (α, β)-metrics, Acta Math. Sin. (Engl. Ser.) 31(10) (2015), 1611–1620.10.1007/s10114-015-3418-2Search in Google Scholar

[24] Xia, Q.: On Kropina metrics of scalar flag curvature, Differ. Geom. Appl. 31 (2013), 393–404.10.1016/j.difgeo.2013.03.001Search in Google Scholar

[25] Zhao, W.—Shen, Y. B.: A universal volume comparison theorem for Finsler manifolds and related results, Canad. J. Math. 65 (2013), 1401–1435.10.4153/CJM-2012-053-4Search in Google Scholar

[26] Zhang, X.—Shen, Y.: On Einstein Kropina metrics, Differ. Geom. Appl. 31 (2013), 80–92.10.1016/j.difgeo.2012.10.011Search in Google Scholar

[27] Zhu, H.—Zhang, H.: Projective Ricci flat spherically symmetric Finsler metrics, Int. J. Math. 29 (2018), #1850078.10.1142/S0129167X18500787Search in Google Scholar

Received: 2020-01-13
Accepted: 2020-04-06
Published Online: 2021-01-29
Published in Print: 2021-02-23

© 2021 Mathematical Institute Slovak Academy of Sciences

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