An elementary proof of Franks' lemma for geodesic flows

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.

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