论文标题
在Finsler紧凑型空间形式上多样性的非合同封闭的大地测量学
Multiplicity of non-contractible closed geodesics on Finsler compact space forms
论文作者
论文摘要
Let $M=S^n/ Γ$ and $h$ be a nontrivial element of finite order $p$ in $π_1(M)$, where the integer $n, p\geq2$, $Γ$ is a finite abelian group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form.在本文中，我们证明，对于每个不可逆的Finsler紧凑型空间$（m，f）$，具有可逆性$λ$和Flag Curvature $ k $满足\ [\ frac {4p^2} {（p+1）^2} \ frac {p+1} {p-1}，\ \]至少存在$ n-1 $ nontractible of Cloctible of clable offibles $ [h] $。此外，如果度量$ f $是颠簸的，并且\ [（\ frac {4p} {2p+1}） $ 2 [\ frac {n+1} {2}] $ class $ [h] $的非合同的封闭地球学，这是由于Katok的示例而导致的最佳下限。对于$ c^4 $ - 类芬斯勒指标，如果$ \ frac {λ^2} {λ^2} {（λ+1）^2} <k \ leq 1 $奇怪，或$ \ frac {λ^2} {（λ+1）^2} \ frac {4} {（n-1）^2} <k \ leq 1 $，$ n $均匀。
Let $M=S^n/ Γ$ and $h$ be a nontrivial element of finite order $p$ in $π_1(M)$, where the integer $n, p\geq2$, $Γ$ 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 for every irreversible Finsler compact space form $(M,F)$ with reversibility $λ$ and flag curvature $K$ satisfying \[ \frac{4p^2}{(p+1)^2} \big(\fracλ{λ+1} \big)^2 < K \leq 1,\;\;λ< \frac{p+1}{p-1}, \] there exist at least $n-1$ non-contractible closed geodesics of class $[h]$. In addition, if the metric $F$ is bumpy and \[ (\frac{4p}{2p+1})^2 (\fracλ{λ+1})^2 < K \leq 1,\;\;λ<\frac{2p+1}{2p-1}, \] then there exist at least $2[\frac{n+1}{2}]$ 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 $\frac{λ^2}{(λ+1)^2} < K \leq 1$ with $n$ being odd, or $\frac{λ^2}{(λ+1)^2}\frac{4}{(n-1)^2} < K \leq 1$ with $n$ being even.
