Results in Mathematics

, 74:21 | Cite as

On m-Kropina Finsler Metrics of Scalar Flag Curvature

  • Guojun YangEmail author


In this paper, we consider a special class of singular Finsler metrics: m-Kropina metrics which are defined by a Riemannian metric and a 1-form. We show that an m-Kropina metric (\(m\ne -1\)) of scalar flag curvature must be locally Minkowskian in dimension \(n\ge 3\). We characterize by some PDEs a Kropina metric (\(m=-1\)) which is respectively of scalar flag curvature and locally projectively flat in dimension \(n\ge 3\), and obtain some principles and approaches of constructing non-trivial examples of Kropina metrics of scalar flag curvature.


m-Kropina metric flag curvature projective flatness 

Mathematics Subject Classification

53B40 53A20 



  1. 1.
    Bao, D., Robles, C., Shen, Z.: Zermelo navigation on Riemann manifolds. J. Diff. Geom. 66, 391–449 (2004)zbMATHGoogle Scholar
  2. 2.
    Chen, B., Zhao, L.: A note on Randers metrics of scalar flag curvature. Can. Math. Bull. 55(3), 474–486 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ishihara, S., Tashiro, Y.: On Riemann manifolds admitting a concircular transformation. Math. J. Okayama Univ. 9, 19–47 (1959)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Kropina, V.K.: On projective two-dimensional Finsler spaces with a special metric. Trudy Sem. Vektor. Tenzor. Anal. 11, 277–292 (1961). (in Russian)MathSciNetGoogle Scholar
  5. 5.
    Matsumoto, M.: Projective changes of Finsler metrics and projectively flat Finsler spaces. Tensor N. S. 34, 303–315 (1980)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Rafie-Rad, M.: Time-optimal solutions of parallel navigation and Finsler geodesics. Nonlinear Anal. RWA 11, 3809C3814 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Shen, Z., Yang, G.: On square metrics of scalar flag curvature. Israel J. Math. 224, 159–188 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Shen, Z., Yang, G.: On a class of weakly Einstein Finsler metrics. Israel J. Math. 199, 773–790 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Shen, Z., Yildirim, G.C.: On a class of projectively flat metrics with constant flag curvature. Can. J. Math. 60(2), 443–456 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Shen, Z., Yildirim, G.C.: A characterization of Randers metrics of scalar flag curvature. Surv. Geom. Anal. Relativ. ALM 23, 330–343 (2012)MathSciNetGoogle Scholar
  11. 11.
    Tashiro, Y.: Complete Riemannian manifolds and some vectors. Trans. Am. Math. Soc. 117, 251–275 (1965)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Yajima, T., Nagahama, H.: Zermelos condition and seismic ray path. Nonlinear Anal. RWA 8, 130C135 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Yang, G.: On a class of Finsler metrics of scalar flag curvature, preprintGoogle Scholar
  14. 14.
    Yang, G.: On a class of two-dimensional singular Douglas and projectively flat Finsler metrics, preprintGoogle Scholar
  15. 15.
    Yang, G.: On a class of singular Douglas and projectively flat Finsler metrics. Diff. Geom. Appl. 32, 113–129 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Yang, G.: On a class of singular projectively flat Finsler metrics with constant flag curvature. Int. J. Math. 24, 1350087 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yoshikawa R., Okubo K.: Kropina Spaces of Constant Curvature II. arXiv:1110.5128v1 [math.DG] 24 Oct 2011
  18. 18.
    Zhou, L.: A local classification of a class of (\(\alpha,\beta \))-metrics with constant flag curvature. Diff. Geom. Appl. 28, 170–193 (2010)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsSichuan UniversityChengduPeople’s Republic of China

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