Non-contractible closed geodesics on compact Finsler space forms without self-intersections
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911758831058944 |
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| author | Wang, Yuchen |
| author_facet | Wang, Yuchen |
| contents | Let $M=S^n/ Γ$ and $h \in π_1(M)$ be a non-trivial element of finite order $p$, where the integers $n, p\geq2$ and $Γ$ is a finite abelian group which acts on the sphere freely and isometrically, therefore $M$ is diffeomorphic to a compact space form which is typical a non-simply connected manifold. We prove there exist at least two non-contractible closed geodesics on $\mathbb{R}P^2$ and obtain the upper bounds on their lengths. Moreover, we prove there exist at least $n$ prime non-contractible simple closed geodesics on $(M,F)$ of prescribed class $[h]$, provided \[ F^2 <(\frac{λ+1}λ)^2 g_0 \;\; \text{ and } \;\; (\fracλ{λ+1})^2 < K \leq 1 \text{ for $n$ is odd or }\; 0<K \leq 1 \text{ for $n$ is even}, \] where $λ$ is the reversibility, $K$ is the flag curvature and $g_0$ is standard Riemannian metric. Stability of these non-contractible closed geodesics is also studied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00946 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-contractible closed geodesics on compact Finsler space forms without self-intersections Wang, Yuchen Differential Geometry Dynamical Systems Let $M=S^n/ Γ$ and $h \in π_1(M)$ be a non-trivial element of finite order $p$, where the integers $n, p\geq2$ and $Γ$ is a finite abelian group which acts on the sphere freely and isometrically, therefore $M$ is diffeomorphic to a compact space form which is typical a non-simply connected manifold. We prove there exist at least two non-contractible closed geodesics on $\mathbb{R}P^2$ and obtain the upper bounds on their lengths. Moreover, we prove there exist at least $n$ prime non-contractible simple closed geodesics on $(M,F)$ of prescribed class $[h]$, provided \[ F^2 <(\frac{λ+1}λ)^2 g_0 \;\; \text{ and } \;\; (\fracλ{λ+1})^2 < K \leq 1 \text{ for $n$ is odd or }\; 0<K \leq 1 \text{ for $n$ is even}, \] where $λ$ is the reversibility, $K$ is the flag curvature and $g_0$ is standard Riemannian metric. Stability of these non-contractible closed geodesics is also studied. |
| title | Non-contractible closed geodesics on compact Finsler space forms without self-intersections |
| topic | Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2401.00946 |