An elementary proof of Franks' lemma for geodesic flows

2013/07/24 by Daniel Visscher, Visscher, Daniel
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #math.DS

paper · pdf · doi:10.48550/arxiv.1307.6573

arxiv created 2013/12/03 · arxiv updated 2013/12/04

Abstract

Given a Riemannian manifold (M,g) and a geodesic γ, the perpendicular part of the derivative of the geodesic flow ϕ_g^t: SM → 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 C²-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 ≥ 3, the original metric must belong to a C²--open and dense subset of metrics.

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