A NEW PROOF OF FRANKS' LEMMA FOR GEODESIC FLOWS

Given a Riemannian manifold (M, g) and a geodesic 7, the perpendicular part of the derivative of the geodesic flow phi(t)(g) : SM -> SM along 7 is a linear symplectic map. The present paper gives a new proof of the following Franks' lemma, originally found in [7] and [61: this map can be perturbed freely within a neighborhood in Sp(n) by a C-2-small perturbation of the metric g that keeps 7 a geodesic for the new metric. Moreover, the size of these perturbations is uniform over fixed length geodesics on the manifold. When dim M >= 3, the original metric must belong to a C-2 open and dense subset of metrics.