Advertisement

Science China Mathematics

, Volume 61, Issue 4, pp 769–782 | Cite as

Dually flat general spherically symmetric Finsler metrics

  • Xiaoming Wang
  • Wangfu Liu
  • Benling Li
Articles
  • 63 Downloads

Abstract

Dually flat Finsler metrics arise from information geometry which has attracted some geometers and statisticians. In this paper, we study dually flat general spherically symmetric Finsler metrics which are defined by a Euclidean metric and two related 1-forms. We give the equivalent conditions for those metrics to be locally dually flat. By solving the equivalent equations, a group of new locally dually flat Finsler metrics is constructed.

Keywords

Finsler metric general spherically symmetric dually flat Euclidean metric 

MSC(2010)

53B40 53C60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11371209) and K. C. Wong Magna Fund in Ningbo University.

References

  1. 1.
    Amari S-I, Nagaoka H. Methods of Information Geometry. Oxford: Oxford University Press, 2000zbMATHGoogle Scholar
  2. 2.
    Berwald L. Über eine characteristic Eigenschaft der allgemeinen Räume konstanter Krümmung mit gradlinigen Extremalen. Monatsh Math Phys, 1929, 36: 315–330MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berwald L. Über die n-dimensionalen Geometrien konstanter Krümmung, in denen die Geraden die Kürzesten sind. Math Z, 1929, 30: 449–469zbMATHGoogle Scholar
  4. 4.
    Bryant R. Projectively flat Finsler 2-spheres of constant curvature. Selecta Math (NS), 1997, 3: 161–204MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cheng X Y, Shen Z M, Zhou Y S. On locally dually flat Finsler metrics. Internat J Math, 2010, 21: 1531–1543MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Funk P. Über Geometrien bei denen die Geraden die Kürtzesten sind. Math Ann, 1929, 101: 226–237MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Funk P. Über zweidimensionale Finslersche Räume insbesondere über solche mit geradlinigen Extremalen und positiver konstanter Krümmung. Math Z, 1936, 40: 86–93MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Huang L B, Mo X H. On spherically symmetric Finsler metrics of scalar curvature. J Geom Phys, 2012, 62: 2279–2287MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Huang L B, Mo X H. On some explicit constructions of dually flat Finsler metrics. J Math Anal Appl, 2013, 405: 565–573MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Huang L B, Mo X H. On some dually flat Finsler metrics with orthogonal invariance. Nonlinear Anal, 2014, 108: 214–222MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jiang J N, Cheng X Y, Tian Y F. On locally dually flat Douglas (α,β)-metrics. Adv Math (China), 2013, 42: 723–730MathSciNetzbMATHGoogle Scholar
  12. 12.
    Li B L. On dually flat Finsler metrics. Differential Geom Appl, 2013, 31: 718–724MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liu W F, Li B L. Projectively flat Finsler metrics defined by the Euclidean metric and related 1-forms. Differential Geom Appl, 2016, 46: 14–24MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mo X H, Solórzano N M, Tenenblat K. On spherically symmetric Finsler metrics with vanishing Douglas curvature. Differential Geom Appl, 2013, 31: 746–758MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mo X H, Wang X Y. On projectively related spherically symmetric Finsler metric in ℝn. Publ Math Debrecen, 2016, 88: 249–259MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rutz S F. Symmetry in Finsler Spaces. Providence: Amer Math Soc, 1996Google Scholar
  17. 17.
    Sevim E S, Shen Z M, Ulgen S. Spherically symmetric Finsler metrics with constant Ricci and flag curvature. Publ Math Debrecen, 2015, 87: 463–472MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Shen Z M. Riemann-Finsler geometry with applications to information geometry. Chinese Ann Math Ser B, 2006, 27: 73–94MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tayebi A, Peyghan E, Sadeghi H. On locally dually flat (α,β)-metrics with isotropic S-curvature. Indian J Pure Appl Math, 2012, 43: 521–534MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tayebi A, Sadeghi H, Peyghan E. On a claßs of locally dually flat (α,β)-metrics. Math Slovaca, 2015, 65: 191–198MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Xia Q L. On locally dually flat (α,β)-metrics. Differential Geom Appl, 2011, 29: 233–243MathSciNetCrossRefGoogle Scholar
  22. 22.
    Yu C T. On dually flat Randers metrics. Nonlinear Anal, 2014, 95: 146–155MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yu C T. On dually flat (α,β)-metrics. J Math Anal Appl, 2014, 412: 664–675MathSciNetCrossRefGoogle Scholar
  24. 24.
    Yu C T. On dually flat general (α,β)-metrics. Differential Geom Appl, 2015, 40: 111–122MathSciNetCrossRefGoogle Scholar
  25. 25.
    Yu C T, Zhu H. On a new class of Finsler metrics. Differential Geom Appl, 2011, 29: 244–254MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhou L F. Spherically symmetric Finsler metric in ℝn. Publ Math Debrecen, 2012, 80: 67–77MathSciNetGoogle Scholar
  27. 27.
    Zhou L F. Projective spherically symmetric Finsler metrics with constant flag curvature in ℝn. Geom Dedicata, 2012, 158: 353–364MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zhou L F. The spherically symmetric Finsler metrics with isotropic S-curvature. J Math Anal Appl, 2015, 431: 1008–1021MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsNingbo UniversityNingboChina

Personalised recommendations