On the minimal number of closed geodesics on positively curved Finsler spheres
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916297539846144 |
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| author | Duan, Huagui Xie, Dong |
| author_facet | Duan, Huagui Xie, Dong |
| contents | In this paper, we proved that for every Finsler metric on $S^n$ $(n\ge 4)$ with reversibility $λ$ and flag curvature $K$ satisfying $(\frac{2n-3}{n-1})^2 (\fracλ{λ+1})^2<K\le 1$ and $ λ<\frac{n-1}{n-2} $, there exist at least $n$ prime closed geodesics on $(S^n,F)$, which solved a conjecture of Katok and Anosov for such positivley curved spheres when $n$ is even. Furthermore, if the number of closed geodesics on such positively curved Finsler $S^n$ is finite, then there exist at least $2\left[\frac{n}{2}\right]-1$ non-hyperbolic closed geodesics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15705 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the minimal number of closed geodesics on positively curved Finsler spheres Duan, Huagui Xie, Dong Differential Geometry Dynamical Systems Symplectic Geometry In this paper, we proved that for every Finsler metric on $S^n$ $(n\ge 4)$ with reversibility $λ$ and flag curvature $K$ satisfying $(\frac{2n-3}{n-1})^2 (\fracλ{λ+1})^2<K\le 1$ and $ λ<\frac{n-1}{n-2} $, there exist at least $n$ prime closed geodesics on $(S^n,F)$, which solved a conjecture of Katok and Anosov for such positivley curved spheres when $n$ is even. Furthermore, if the number of closed geodesics on such positively curved Finsler $S^n$ is finite, then there exist at least $2\left[\frac{n}{2}\right]-1$ non-hyperbolic closed geodesics. |
| title | On the minimal number of closed geodesics on positively curved Finsler spheres |
| topic | Differential Geometry Dynamical Systems Symplectic Geometry |
| url | https://arxiv.org/abs/2406.15705 |