A Characterisation for Finsler Metrics of Constant Curvature and a Finslerian Version of Beltrami Theorem

  • Ioan BucataruEmail author
  • Georgeta Creţu


We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classical projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from the Weyl projective curvature, it is not a projective invariant, and hence Beltrami Theorem does not work in Finsler geometry. We provide the relation between the Weyl-type curvature tensors of two projectively related Finsler metrics. Using this formula we show that a projective deformation preserves the property of having constant flag curvature if and only if the projective factor is a Hamel function. This way we provide a Finslerian version of Beltrami Theorem.


Finsler metrics Constant flag curvature Weyl-type curvature Beltrami Theorem 

Mathematics Subject Classification

53C60 53B40 58E30 49N45 



We express our thanks to Vladimir Matveev, Zhongmin Shen and József Szilasi for their comments and suggestions on this work. This work is supported by Ministry of Research and Innovation within Program 1—Development of the national RD system, Subprogram 1.2— Institutional Performance—RDI excellence funding projects, Contract no. 34PFE/19.10.2018.


  1. 1.
    Akbar-Zadeh, H.: Initiation to Global Finslerian Geometry. Elsevier, North-Holland (2006)zbMATHGoogle Scholar
  2. 2.
    Bao, D., Chern, S.-S., Shen, Z.: An Introduction to Riemann–Finsler Geometry. Springer, GMT, New York (2000)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bucataru, I., Muzsnay, Z.: Projective and Finsler metrizability: parameterization rigidity of the geodesics. Int. J. Math. 23(9), 1250099-1–1250099-15 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bucataru, I., Muzsnay, Z.: Sprays metrizable by Finsler functions of constant flag curvature. Differ. Geom. Appl. 31(3), 405–415 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Grifone, J., Muzsnay, Z.: Variational Principles for Second-Order Differential Equations. World Scientific, Singapore (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    Li, B., Shen, Z.: Sprays of isotropic curvature. Int. J. Math. 29(1), 1850003-1–1850003-12 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Matsumoto, M.: Foundations of Finsler Geometry and Special Finsler Spaces. Kaiseisha Press, Otsu (1986)zbMATHGoogle Scholar
  8. 8.
    Matveev, V.: Projectively invariant objects and the index of the group of affine transformations in the group of projective transformations. Bull. Iran. Math. Soc. 44(2), 341–375 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Najafi, B., Tayebi, A.: Finsler metrics of scalar flag curvature and projective invariants. Balk. J. Geom. Appl. 15(2), 82–91 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Shen, Z.: Differential Geometry of Spray and Finsler Spaces. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Shen, Z.: Projectively flat Finsler metrics of constant flag curvature. Trans. Am. Math. Soc. 355(4), 1713–1728 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sinha, B.B., Matharoo, A.S.: On Finsler spaces of constant curvature. Indian J. Pure Appl. Math. 17(1), 66–73 (1986)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Szilasi, J., Lovas, R., Kertész, D.: Connections, Sprays and Finsler Structures. World Scientific, Singapore (2014)zbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsAlexandru Ioan Cuza UniversityIasiRomania

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