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Generic Existence of Infinitely Many Non-contractible Closed Geodesics on Compact Space Forms

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Abstract

Let M = Sn /Γ and h be a nontrivial element of finite order p in π1(Μ), where the integers n, p ≥ 2, Γ 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 there are infinitely many non-contractible closed geodesics of class [h] on the compact space form with Cr-generic Finsler metrics, where 4 ≤ r ≤ ∞. The conclusion also holds for Cr-generic Riemannian metrics for 2 ≤ r ≤ ∞. The proof is based on the resonance identity of non-contractible closed geodesics on compact space forms.

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Authors and Affiliations

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

    Hui Liu

  2. School of Mathematics, Tianjin Normal University, Tianjin, 300384, P. R. China

    Yu Chen Wang

Authors
  1. Hui Liu
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  2. Yu Chen Wang
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Correspondence to Yu Chen Wang.

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Conflict of Interest The authors declare no conflict of interest.

Additional information

The first author was partially supported by NSFC (Grant Nos. 12371195, 12022111) and the Fundamental Research Funds for the Central Universities (Grant No. 2042023kf0207); the second author was partially supported by NSFC (Grant No. 11831009) and Fundings of Innovating Activities in Science and Technology of Hubei Province

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Liu, H., Wang, Y.C. Generic Existence of Infinitely Many Non-contractible Closed Geodesics on Compact Space Forms. Acta. Math. Sin.-English Ser. (2024). https://doi.org/10.1007/s10114-024-3009-1

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  • DOI: https://doi.org/10.1007/s10114-024-3009-1

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