An elementary proof of Franks' lemma for geodesic flows

TL;DR

This paper provides an elementary proof that the linear symplectic map associated with a geodesic's derivative can be freely perturbed via small metric changes, with uniform control over fixed-length geodesics.

Contribution

It offers a simplified, elementary proof of Franks' lemma for geodesic flows, showing perturbations can be made freely within a neighborhood in the symplectic group.

Findings

Perturbations of the derivative map are possible via small metric changes.

The size of perturbations is uniform over fixed-length geodesics.

The result holds for a generic set of metrics when dimension is at least 3.

Abstract

Given a Riemannian manifold $(M,g)$ and a geodesic $\gamma$, the perpendicular part of the derivative of the geodesic flow $\phi_g^t: SM \rightarrow SM$ along $\gamma$ 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 $\gamma$ 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.