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The Non-contractibility of Closed Geodesics on Finsler ℝPn

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Abstract

Let (ℝPn, F) be a bumpy and irreversible Finsler n-dimensional real projective space with reversibility λ and flag curvature K satisfying \({({\lambda \over {1 + \lambda}})^2} < K \le 1\) when n is odd, and K ≥ 0 when n is even. We show that if there exist exactly \(2[{{n + 1} \over 2}]\) prime closed geodesics on such (ℝPn, F), then all of them are non-contractible, which coincides with the Katok’s examples.

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, P. R. China

    Hua Gui Duan

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

    Hui Liu

Authors
  1. Hua Gui Duan
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  2. Hui Liu
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Correspondence to Hui Liu.

Additional information

The first author was partially supported by National Key R&D Program of China (Grant No. 2020YFA0713300) and NSFC (Grant Nos. 11671215 and 11790271), LPMC of MOE of China and Nankai University; the second author was partially supported by NSFC (Grant Nos. 11771341 and 12022111)

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Duan, H.G., Liu, H. The Non-contractibility of Closed Geodesics on Finsler ℝPn. Acta. Math. Sin.-English Ser. 38, 1–21 (2022). https://doi.org/10.1007/s10114-021-0023-4

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  • DOI: https://doi.org/10.1007/s10114-021-0023-4

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