The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form S2n+1/ G

The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form \(S^{2n+1}/ \Gamma \)



Let \(M=S^{2n+1}/ \Gamma \), \(\Gamma \) is a finite group which acts freely and isometrically on the \((2n+1)\)-sphere and therefore M is diffeomorphic to a compact space form. In this paper, we first investigate Katok’s famous example about irreversible Finsler metrics on the spheres to study the topological structure of the contractible component of the free loop space on the compact space form M, then we apply the result to establish the resonance identity for homologically visible contractible minimal closed geodesics on every Finsler compact space form (MF) when there exist only finitely many distinct contractible minimal closed geodesics on (MF). As its applications, using this identity and the enhanced common index jump theorem for symplectic paths proved by Duan et al. (Calc Var PDEs 55(6):55–145, 2016), we show that there exist at least \(2n+2\) distinct closed geodesics on every compact space form \(S^{2n+1}/ \Gamma \) with a bumpy irreversible Finsler metric F under some natural curvature condition, which is the optimal lower bound due to Katok’s example.

Mathematics Subject Classification

53C22 58E05 58E10 



I would like to thank sincerely the referee for his careful reading of the manuscript, valuable comments on improving the exposition, and for his deep insight on the main ideas of this paper. And also I would like to sincerely thank Professor Yiming Long, for introducing me to Hamiltonian dynamics, and for his constant helps and encouragements on my research.


  1. 1.
    Anosov, D.V.: Geodesics in Finsler geometry. In: Proceedings of I.C.M. (Vancouver, B.C. 1974), vol. 2, pp. 293–297 Montreal (1975) (Russian), Am. Math. Soc. Transl. 109, 81–85 (1977)Google Scholar
  2. 2.
    Ballmann, W., Thorbergsson, G., Ziller, W.: Closed geodesics and the fundamental group. Duke Math. J. 48, 585–588 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bangert, V.: On the existence of closed geodesics on two-spheres. Int. J. Math. 4(1), 1–10 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bangert, V., Hingston, N.: Closed geodesics on manifolds with infinite abelian fundamental group. J. Differ. Geom. 19, 277–282 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bangert, V., Klingenberg, W.: Homology generated by iterated closed geodesics. Topology 22, 379–388 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bangert, V., Long, Y.: The existence of two closed geodesics on every Finsler 2-sphere. Math. Ann. 346, 335–366 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bott, R.: On the iteration of closed geodesics and the Sturm intersection theory. Commun. Pure Appl. Math. 9, 171–206 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Burns, K., Matveev, S.: Open problems and questions about closed geodesics. arXiv:1308.5417v2 (2014)
  9. 9.
    Chang, K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston (1993)zbMATHCrossRefGoogle Scholar
  10. 10.
    Duan, H., Long, Y.: Multiple closed geodesics on bumpy Finsler \(n\)-spheres. J. Differ. Equ. 233(1), 221–240 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Duan, H., Long, Y.: The index growth and multiplicity of closed geodesics. J. Funct. Anal. 259, 1850–1913 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Duan, H., Long, Y., Wang, W.: Two closed geodesics on compact simply-connected bumpy Finsler manifolds. J. Differ. Geom. 104(2), 275–289 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Duan, H., Long, Y., Wang, W.: The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds. Calc. Var. PDEs 55(6), 55–145 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Duan, H., Long, Y., Xiao, Y.: Two closed geodesics on \({\mathbb{R}P^{n}}\) with a bumpy Finsler metric. Calc. Var. PDEs 54, 2883–2894 (2015)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Franks, J.: Geodesics on \(S^2\) and periodic points of annulus homeomorphisms. Invent. Math. 108(2), 403–418 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Ginzburg, V., Gurel, B., Macarini, L.: Multiplicity of closed Reeb orbits on prequantization bundles. Israel J. Math. 228(1), 407–453 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Gromoll, D., Meyer, W.: Periodic geodesics on compact Riemannian manifolds. J. Differ. Geom. 3, 493–510 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Hingston, N.: Equivariant Morse theory and closed geodesics. J. Differ. Geom. 19, 85–116 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Hingston, N.: On the growth of the number of closed geodesics on the two-sphere. Int. Math. Res. Not. 9, 253–262 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Hingston, N., Rademacher, H.-B.: Resonance for loop homology of spheres. J. Differ. Geom. 93, 133–174 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Katok, A.B.: Ergodic properties of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk. SSSR 37 (1973), [Russian]; Math. USSR-Izv. 7, 535–571 (1973)Google Scholar
  22. 22.
    Klingenberg, W.: Lectures on Closed Geodesics. Springer, Berlin (1978)zbMATHCrossRefGoogle Scholar
  23. 23.
    Klingenberg, W.: Riemannian Geometry, 2 Rev. edn. de Gruyter, Berlin (1995)zbMATHCrossRefGoogle Scholar
  24. 24.
    Liu, C.: The relation of the Morse index of closed geodesics with the Maslov-type index of symplectic paths. Acta Math. Sin. 21, 237–248 (2005)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Liu, C., Long, Y.: Iterated index formulae for closed geodesics with applications. Sci. China 45, 9–28 (2002)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Liu, H.: The Fadell–Rabinowitz index and multiplicity of non-contractible closed geodesics on Finsler \({\mathbb{R}}P^{n}\). J. Differ. Equa. 262, 2540–2553 (2017)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Liu, H., Long, Y., Xiao, Y.: The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form. Discrete Contin. Dyn. Syst. 38, 3803–3829 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Liu, H., Xiao, Y.: Resonance identity and multiplicity of non-contractible closed geodesics on Finsler \({\mathbb{R}}P^{n}\). Adv. Math. 318, 158–190 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Long, Y.: Bott formula of the Maslov-type index theory. Pac. J. Math. 187, 113–149 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Long, Y.: Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics. Adv. Math. 154, 76–131 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Long, Y.: Index Theory for Symplectic Paths with Applications 207. Progress in Mathematics. Birkhäuser, Berlin (2002)CrossRefGoogle Scholar
  32. 32.
    Long, Y.: Multiplicity and stability of closed geodesics on Finsler 2-spheres. J. Eur. Math. Soc. 8, 341–353 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Long, Y., Duan, H.: Multiple closed geodesics on 3-spheres. Adv. Math. 221, 1757–1803 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Long, Y., Wang, W.: Multiple closed geodesics on Riemannian 3-spheres. Calc. Var. PDEs 30, 183–214 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Long, Y., Zhu, C.: Closed characteristics on compact convex hypersurfaces in \(\mathbb{R}^{2n}\). Ann. Math. 155, 317–368 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Oancea, A.: Morse theory, closed geodesics, and the homology of free loop spaces, With an appendix by Umberto Hryniewicz. IRMA Lect. Math. Theor. Phys., 24, Free loop spaces in geometry and topology, pp. 67–109, Eur. Math. Soc., Zürich, 2015. arXiv:1406.3107 (2014)
  37. 37.
    Rademacher, H.-B.: On the average indices of closed geodesics. J. Differ. Geom. 29, 65–83 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Rademacher, H.-B.: Morse-Theorie und geschlossene Geodätische, Bonner Mathematis-che Schriften, vol. 229. Universität Bonn Mathematisches Institut, Bonn (1992)Google Scholar
  39. 39.
    Rademacher, H.-B.: A Sphere Theorem for non-reversible Finsler metrics. Math. Ann. 328, 373–387 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Rademacher, H.-B.: Existence of closed geodesics on positively curved Finsler manifolds. Ergod. Theory Dyn. Syst. 27(3), 957–969 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Rademacher, H.-B.: The second closed geodesic on Finsler spheres of dimension \(n>2\). Trans. Am. Math. Soc. 362(3), 1413–1421 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Shen, Z.: Lectures on Finsler Geometry. World Scientific, Singapore (2001)zbMATHCrossRefGoogle Scholar
  43. 43.
    Taimanov, I.A.: The type numbers of closed geodesics. Regul. Chaotic Dyn. 15(1), 84–100 (2010). arXiv:0912.5226 MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Taimanov, I.A.: The spaces of non-contractible closed curves in compact space forms. Mat. Sb. 207(10), 105–118 (2016). arxiv:1604.05237 MathSciNetCrossRefGoogle Scholar
  45. 45.
    Wang, W.: Closed geodesics on positively curved Finsler spheres. Adv. Math. 218, 1566–1603 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Wang, W.: On a conjecture of Anosov. Adv. Math. 230, 1597–1617 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Westerland, C.: Dyer-Lashof operations in the string topology of spheres and projective spaces. Math. Z. 250(3), 711–727 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Westerland, C.: String Homology of Spheres and Projective Spaces. Algebr. Geom. Topol. 7, 309–325 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Xiao, Y., Long, Y.: Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions. Adv. Math. 279, 159–200 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Vigué-Poirrier, M., Sullivan, D.: The homology theory of the closed geodesic problem. J. Differ. Geom. 11, 633–644 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Ziller, W.: The free loop space of globally symmetric spaces. Invent. Math. 41, 1–22 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Ziller, W.: Geometry of the Katok examples. Ergod. Theory Dyn. Syst. 3, 135–157 (1982)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

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