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Multiplicity of non-contractible closed geodesics on Finsler compact space forms

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Abstract

Let \(M=S^n/ \Gamma \) and h be a nontrivial element of finite order p in \(\pi _1(M)\), where the integer \(n, p\ge 2\), \(\Gamma \) is a finite abelian group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to a compact space form. In this paper, we prove that for every irreversible Finsler compact space form (M, F) with reversibility \(\lambda \) and flag curvature K satisfying

$$\begin{aligned} \frac{4p^2}{(p+1)^2} \left( \frac{\lambda }{\lambda +1} \right) ^2< K \le 1,\;\;\lambda < \frac{p+1}{p-1}, \end{aligned}$$

there exist at least \(n-1\) non-contractible closed geodesics of class [h]. In addition, if the metric F is bumpy and

$$\begin{aligned} \left( \frac{4p}{2p+1}\right) ^2 \left( \frac{\lambda }{\lambda +1}\right) ^2< K \le 1,\;\;\lambda <\frac{2p+1}{2p-1}, \end{aligned}$$

then there exist at least \(2\left[ \frac{n+1}{2}\right] \) non-contractible closed geodesics of class [h], which is the optimal lower bound due to Katok’s example. For \(C^4\)-generic Finsler metrics, there are infinitely many non-contractible closed geodesics of class [h] on (M, F) if \(\frac{\lambda ^2}{(\lambda +1)^2} < K \le 1\) with n being odd, or \(\frac{\lambda ^2}{(\lambda +1)^2}\frac{4}{(n-1)^2} < K \le 1\) with n being even.

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References

  1. Bangert, V.: On the existence of closed geodesics on two-spheres. Internat. J. Math. 4(1), 1–10 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bangert, V., Hingston, N.: Closed geodesics on manifolds with infinite abelian fundamental group. J. Differ. Geom. 19, 277–282 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bangert, V., Long, Y.: The existence of two closed geodesics on every Finsler 2-sphere. Math. Ann. 346, 335–366 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ballmann, W., Thorbergsson, G., Ziller, W.: Closed geodesics and the fundamental group. Duke Math. J. 48, 585–588 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ballmann, W., Thorbergsson, G., Ziller, W.: Existence of closed geodesics on positively curved manifolds. J. Differ. Geom. 18(2), 221–252 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. Birkhoff, G.: Dynamical Systems. Colloquium publications of American Mathematical Society (1927)

  7. Chang, K.C.: Infinite dimensional morse theory and multiple solution problems. Birkhäuser, Boston (1993)

    Book  MATH  Google Scholar 

  8. Duan, H., Long, Y.: Multiple closed geodesics on bumpy Finsler \(n\)-spheres. J. Differ. Equa. 233(1), 221–240 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Duan, H., Long, Y.: The index growth and multiplicity of closed geodesics. J. Funct. Anal. 259, 1850–1913 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Duan, H., Long, Y., Wang, W.: Two closed geodesics on compact simply-connected bumpy Finsler manifolds. J. Differ. Geom. 104(2), 275–289 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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. and PDEs. 55(6), 1–28 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Duan, H., Long, Y., Xiao, Y.: Two closed geodesics on \({\mathbb{R} }P^n\) with a bumpy Finsler metric. Calc. Var. and PDEs 54, 2883–2894 (2015)

    Article  MATH  Google Scholar 

  13. Ekeland, I.: Convexity methods in hamiltonian mechanics. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  14. Ekeland, I., Hofer, H.: Convex Hamiltonian energy surfaces and their closed trajectories. Comm. Math. Phys. 113, 419–467 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fadell, E., Rabinowitz, P.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45(2), 139–174 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  16. Franks, J.: Geodesics on \(S^2\) and periodic points of annulus homeomorphisms. Invent. Math. 108(2), 403–418 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gromoll, D., Meyer, W.: On differentiable functions with isolated critical points. Topology 8, 361–369 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gromoll, D., Meyer, W.: Periodic geodesics on compact Riemannian manifolds. J. Differ. Geom. 3, 493–510 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hingston, N.: On the growth of the number of closed geodesics on the two-sphere. Inter. Math. Resear. Notices 9, 253–262 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hingston, N., Rademacher, H.-B.: Resonance for loop homology of spheres. J. Differ. Geom. 93, 133–174 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Katok, A.B.: Ergodic properties of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk. SSSR 37,: Russian. Math. USSR-Izv. 7(1973), 535–571 (1973)

  22. Klingenberg, W., Takens, F.: Generic properties of geodesic flows. Math. Ann. 197, 323–334 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  23. Klingenberg, W.: Lectures on closed geodesics. Springer, Berlin and Heidelberg (1978)

    Book  MATH  Google Scholar 

  24. 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)

    Article  MATH  Google Scholar 

  25. 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)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, H.: The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form \({S}^{2n+1}/ \Gamma \). Calc. Var. and PDEs 58, 107 (2019)

    Article  MATH  Google Scholar 

  28. Long., Y.: Index Theory for Symplectic Paths with Applications. Progress in Math. 207, Birkhäuser. (2002)

  29. Long, Y., Duan, H.: Multiple closed geodesics on 3-spheres. Adv. Math. 221, 1757–1803 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Long, Y., Zhu, C.: Closed characteristics on compact convex hypersurfaces in \(^{2n}\). Ann. of Math. 155, 317–368 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lyusternik, L. A., Fet, A. I.: Variational problems on closed manifolds. Dokl. Akad. Nauk SSSR (N.S.) 81 (1951) 17-18 (in Russian)

  32. Rademacher, H.-B.: On the average indices of closed geodesics. J. Diff. Geom. 29, 65–83 (1989)

    MathSciNet  MATH  Google Scholar 

  33. Rademacher, H.-B., Theorie, Morse, Geodatische, geschlossene: Bonner Math. Schr. 229 (1992)

  34. Rademacher, H.-B.: The Fadell-Rabinowitz index and closed geodesics. J. London. Math. Soc. 50, 609–624 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rademacher, H.-B.: A Sphere Theorem for non-reversible Finsler metrics. Math. Ann. 328, 373–387 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rademacher, H.-B.: Existence of closed geodesics on positively curved Finsler manifolds. Ergod. Th. Dyn. Sys. 27(3), 957–969 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rademacher, H.-B.: The second closed geodesic on Finsler spheres of dimension \(n>2\). Trans. Amer. Math. Soc. 362(3), 1413–1421 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rademacher, H.-B., Taimanov, I.A.: The second closed geodesic, the fundamental group, and generic Finsler metrics. Math. Z. 302, 629–640 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  39. Shen, Z.: Lectures on Finsler Geometry. World Scientific, Singapore (2001)

    Book  MATH  Google Scholar 

  40. Taimanov, I.A.: The type numbers of closed geodesics. Regul. Chaotic Dyn. 15(1), 84–100 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Taimanov, I.A.: The spaces of non-contractible closed curves in compact space forms. Mat. Sb. 207(10), 105–118 (2016)

    MathSciNet  Google Scholar 

  42. Vigué-Poirrier, M., Sullivan, D.: The homology theory of the closed geodesic problem. J. Differ. Geom. 11, 633–644 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wang, W.: Closed geodesics on positively curved Finsler spheres. Adv. Math. 218, 1566–1603 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wang, W.: On a conjecture of Anosov. Adv. Math. 230, 1597–1617 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wang, W.: On the average indices of closed geodesics on positively curved Finsler spheres. Math. Ann. 355, 1049–1065 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  46. 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)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ziller, W.: Geometry of the Katok examples. Ergod. Th. Dyn. Sys. 3, 135–157 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to sincerely thank our advisor, Professor Yiming Long, for introducing us to the theory of closed geodesics.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, China

    Hui Liu

  2. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, Hubei, China

    Yuchen Wang

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  1. Hui Liu
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  2. Yuchen Wang
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Correspondence to Yuchen Wang.

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Partially supported by NSFC (Nos. 12022111, 11771341) and the Fundamental Research Funds for the Central Universities (No. 2042021kf1059)

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Liu, H., Wang, Y. Multiplicity of non-contractible closed geodesics on Finsler compact space forms. Calc. Var. 61, 224 (2022). https://doi.org/10.1007/s00526-022-02323-3

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