Mathematische Annalen

, Volume 365, Issue 3–4, pp 1379–1408 | Cite as

Deformations and Hilbert’s fourth problem

Article
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Abstract

In this paper a special class of Finsler metrics defined by a Riemannian metric and an 1-form is studied. The projectively flat metrics in dimension \(n\ge 3\) are classified by a new class of metric deformations in Riemann geometry. The results show that the projective flatness of such Finsler metrics always arises from that of some Riemannian metric.

Mathematics Subject Classification

53B40 53C60 
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Notes

Acknowledgments

I’d like to thank my tutor Professor Xiaohuan Mo for the useful discussions on this topic when I studied in Peking University. I’d like to thank postgraduate student Xiaoyun Tang in Fudan University for helping me to build some fundamental formulas of \(\beta \)-deformations when she studied in South China Normal University. I should also thank the referees for careful reading and many useful comments.

References

  1. 1.
    Bácsó, S., Cheng, X., Shen, Z.: Curvature properties of \(\alpha , \beta \)-metrics, 73–110. In: Sabau, S., Shimada, H. (eds.) Finsler Geometry, Sapporo 2005-In Memory of Makoto Matsumoto. Advanced Studies in Pure Mathematics, vol. 48, Mathematical Society of Japan (2007)Google Scholar
  2. 2.
    Bácsó, S., Matsumoto, M.: On Finsler spaces of Douglas type. A generalization of the notion of Berwald space. Publ. Math. Debr. 51, 385–406 (1997)MathSciNetMATHGoogle Scholar
  3. 3.
    Bao, D., Robles, C., Shen, Z.: Zermelo navigation on Riemannian manifolds. J. Differ. Geom. 66, 391–449 (2004)MathSciNetMATHGoogle Scholar
  4. 4.
    Berwald, L.: Über die n-dimensionalen Geometrien konstanter Krümmung, in denen die Geradendie kürzesten sind. Math. Z. 30, 449–469 (1929)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bryant, R.: Projectively flat Finsler 2-sphere of constant curvature. Select. Math., New Ser. 3, 161–204 (1997)CrossRefMATHGoogle Scholar
  6. 6.
    Bryant, R.: Some remarks on Finsler manifolds with constant flag curvature. Houst. J. Math. 28(2), 221–262 (2002)MathSciNetMATHGoogle Scholar
  7. 7.
    Busemann, H.: Problem IV: Desarguesian spaces in Mathematical developments arising from Hilbert problems. In: Proc. Sympos. Pure Math., vol. XXVIII, AMS., Providence, R. I. (1976)Google Scholar
  8. 8.
    Chen, X., Li, M.: On a class of projectively flat \(\alpha,\beta \)-metrics. Publ. Math. Debr. 71, 195–205 (2007)MathSciNetGoogle Scholar
  9. 9.
    Chern, S.S., Shen, Z.: Riemann-Finsler Geometry. World Scientific, Singapore (2005)CrossRefMATHGoogle Scholar
  10. 10.
    Hamel, G.: Über die Geometrien in denen die Geraden die Kürzesten sind. Math. Ann. 57, 231–264 (1903)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hilbert, D.: Mathematical problems. Bull. Am. Math. Soc. 37, 407–436 (2001). Reprinted from Bull. Am. Math. Soc., 8, 437-479 (1902)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Li, B., Shen, Z.: On a class of projectively flat Finsler metrics with constant flag curvature. Int. J. Math. 18(7), 1–12 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Matsumoto, M.: On \(C\)-reducible Finsler spaces. Tensor N.S., 24, 29–37 (1972)Google Scholar
  14. 14.
    Mo, X., Shen, Z., Yang, C.: Some constructions of projectively flat Finser metric. Sci. China (Ser. A) 49, 703–714 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Álvarez Paiva, J.C.: Symplectic geometry and Hilbert’s fourth problem. J. Differ. Geom. 69, 353–378 (2005)MATHGoogle Scholar
  16. 16.
    Pogorelov, A.V.: Hilbert’s Fourth Problem. Scripta Series in Mathematics. V. H. Winston, Washington, NY (1979)MATHGoogle Scholar
  17. 17.
    Randers, G.: On an asymmetric in the four-space of general relativily. Phys. Rev. 59, 195–199 (1941)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Rapcsák, A.: Über die bahntreuen Abbildungen metrischer Räume. Publ. Math. Debr. 8, 285–290 (1961)MATHGoogle Scholar
  19. 19.
    Sevim, E., Shen, Z., Zhao, L.: On a class of Ricci-flat Douglas metrics. Int. J. Math. 23(6), 1250046 (2012). doi: 10.1142/S0129167X12500462 MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Shen, Z.: On projectively flat \(\alpha,\beta \)-metrics. Can. Math. Bull. 52(1), 132–144 (2009)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Shen, Z.: Projectively flat Finsler metrics of constant flag curvature. Trans. Am. Math. Soc. 355(4), 1713–1728 (2002)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Shen, Z., Xing, H.: On Randers metrics with isotropic \(S\)-curvature. Acta. Math. Sin. 24, 789–796 (2008)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Shen, Z., Yildirim, G.C.: On a class of projectively flat metrics with constant flag curvature. Can. J. Math. 60(2), 443–456 (2008)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Shen, Z., Yu, C.: On Einstein square metrics. Publ. Math. Debr. 85(3–4), 413–424 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Yang, G.: On a class of two-dimensional Douglas and projectively flat Finsler metrics. Sci. World J. (2013). doi: 10.1155/2013/291491 Google Scholar
  26. 26.
    Yu, C., Zhu, H.: On a new class of Finsler metrics. Differ. Geom. Appl. 29, 244–254 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesSouth China Normal UniversityGuangzhouPeople’s Republic of China

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