Science China Mathematics

, Volume 62, Issue 5, pp 935–960 | Cite as

On a class of almost regular Landsberg metrics

  • Shasha Zhou
  • Jiayue Wang
  • Benling LiEmail author


There is a long existing ıicorn" problem in Finsler geometry: whether or not any Landsberg metric is a Berwald metric? Some classes of metrics were studied in the past and no regular non-Berwaldian Landsberg metric was found. However, if the metric is almost regular (allowed to be singular in some directions), some non-Berwaldian Landsberg metrics were found in the past years. All of them are composed by Riemannian metrics and 1-forms. This motivates us to find more almost regular non-Berwaldian Landsberg metrics in the class of general (α,β)-metrics. In this paper, we first classify almost regular Landsberg general (α,β)-metrics into three cases and prove that those regular metrics must be Berwald metrics. By solving some nonlinear PDEs, some new almost regular Landsberg metrics are constructed which have not been described before.


Finsler metric Landsberg metric Berwald metric (α,β)-metric 


53B40 53C60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was supported by Zhejiang Provincial Natural Science Foundation of China (ZPNSFC) (Grant No. R18A010002), National Natural Science Foundation of China (Grant No. 11371209) and K.C. Wong Magna Fund in Ningbo University. The authors thank the referees for their helpful comments.


  1. 1.
    Asanov G S. Finsleroid-Finsler space with Berwald and Landsberg conditions. Rep Math Phys, 2006, 58: 275–300MathSciNetCrossRefGoogle Scholar
  2. 2.
    Asanov G S. Finsleroid-Finsler space and geodesic spray coefficients. Publ Math Debrecen, 2007, 71: 397–412MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bácsó S. Reduction theorems of certain Landsberg spaces to Berwald spaces. Publ Math Debrecen, 1996, 48: 357–366MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bao D. On two curvature-driven problems in Finsler geometry. Adv Stud Pure Math, 2007, 48: 19–71MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cheng X, Li H, Zou Y. On conformally flat (a,ß)-metrics with relatively isotropic mean Landsberg curvature. Publ Math Debrecen, 2014, 85: 131–144MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Crampin M. On Landsberg spaces and the Landsberg-Berwald problem. Houston J Math, 2011, 37: 1103–1124MathSciNetzbMATHGoogle Scholar
  7. 7.
    He Y, Zhong C P. Strongly convex weakly complex Berwald metrics and real Landsberg metrics. Sci China Math, 2018, 61: 535–544MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Li B L, Shen Z M. On a class of weak Landsberg metrics. Sci China Ser A, 2007, 50: 573–589MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Li B L, Shen Z M. On a class of locally projectively flat Finsler metrics. Internat J Math, 2016, 27: 1650052MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Matveev V S. On ⋚gular Landsberg metrics are always Berwald" by Z. I. Szabó. Balkan J Geom Appl, 2009, 14: 50–52MathSciNetzbMATHGoogle Scholar
  11. 11.
    Mo X, Zhou L. The curvatures of spherically symmetric Finsler metrics in Rn. ArXiv:1202.4543v4, 2014Google Scholar
  12. 12.
    Najafi B, Saberali S. On a class of isotropic mean Landsberg metrics. Differ Geom Dyn Syst, 2016, 18: 72–80MathSciNetzbMATHGoogle Scholar
  13. 13.
    Randers G. On an asymmetric metric in the four-space of general relativity. Phys Rev, 1941, 59: 195–199MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shen Z. On a class of Landsberg metrics in Finsler geometry. Canad J Math, 2009, 61: 1357–1374MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shibata C, Shimada H, Azuma M, et al. On Finsler spaces with Randers' metric. Tensor (NS), 1977, 31: 219–226MathSciNetzbMATHGoogle Scholar
  16. 16.
    Szabó Z I. Positive definite Berwald spaces (structure theorem on Berwald spaces). Tensor (NS), 1981, 35: 25–39MathSciNetzbMATHGoogle Scholar
  17. 17.
    Szabó Z I. All regular Landsberg metrics are Berwald. Ann Global Anal Geom, 2008, 34: 381–386MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Szabó Z I. Correction to "All regular Landsberg metrics are Berwald". Ann Global Anal Geom, 2009, 35: 227–230MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tayebi A. On the class of generalized Landsberg manifolds. Period Math Hungar, 2016, 72: 29–36MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tayebi A, Tabatabaeifar T. Unicorn metrics with almost vanishing H- and E-curvatures. Turkish J Math, 2017, 41: 998–1008MathSciNetCrossRefGoogle Scholar
  21. 21.
    Vincze C. An observation on Asanov's unicorn metrics. Publ Math Debrecen, 2017, 90: 251–268MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wang X, Li B. On Douglas general (a,ß)-metrics. Acta Math Sin Engl Ser, 2017, 33: 951–968MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yu C, Zhu H. On a new class of Finsler metrics. Differential Geom Appl, 2011, 29: 244–254MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zhou L. Projective spherically symmetric Finsler metrics with constant flag curvature in Rn. Geom Dedicata, 2012, 158: 353–364MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhou S, Li B. On Landsberg general (a,β)-metrics with a conformal 1-form. ArXiv:1706.00533, 2017Google Scholar
  26. 26.
    Zhu H. On a class of Finsler metrics with relatively isotropic mean Landsberg curvature. Publ Math Debrecen, 2016, 89: 483–498MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zhu H. On general (a,β)-metrics with isotropic Berwald curvature. Bull Korean Math Soc, 2017, 54: 399–416MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zohrehvand M, Maleki H. On general (a,β)-metrics of Landsberg type. Int J Geom Methods Mod Phys, 2016, 13: 1650085MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNingbo UniversityNingboChina

Personalised recommendations