测地流的Franks引理的初等证明
An elementary proof of Franks' lemma for geodesic flows
摘要：
给定一个黎曼流形$(M,g)$和一个测地线$γ$，测测地流$ϕ_g^t: SM \rightarrow SM$沿$γ$的导数的垂直部分是一个线性辛映射。本文给出了最初在[G. Contreras和G. Paternain，2002]和[G.孔特雷拉斯，2010年]：该映射可以在$Sp(n)$中的邻域内被度量$g$的$C^2$ --保持$γ$为新度量的测地线的小扰动自由地扰动。此外，这些扰动的大小是均匀的固定长度测地线上的流形。当使用$\dim M \geq 3$时，原始度量必须属于$C^2$ -度量的开放和密集子集。
英文摘要：
Given a Riemannian manifold $(M,g)$ and a geodesic $γ$, the perpendicular part of the derivative of the geodesic flow $ϕ_g^t: SM \rightarrow SM$ along $γ$ is a linear symplectic map. We give an elementary proof of the following Franks' lemma, originally found in [G. Contreras and G. Paternain, 2002] and [G. Contreras, 2010]: this map can be perturbed freely within a neighborhood in $Sp(n)$ by a $C^2$-small perturbation of the metric $g$ that keeps $γ$ a geodesic for the new metric. Moreover, the size of these perturbations is uniform over fixed length geodesics on the manifold. When $\dim M \geq 3$, the original metric must belong to a $C^2$--open and dense subset of metrics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
