{
  "id": "1307.6573",
  "version": "v2",
  "published": "2013-07-24T20:25:43.000Z",
  "updated": "2013-12-03T15:03:40.000Z",
  "title": "An elementary proof of Franks' lemma for geodesic flows",
  "authors": [
    "Daniel Visscher"
  ],
  "categories": [
    "math.DS"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-12-03T15:03:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "geodesic flow",
      "elementary proof",
      "linear symplectic map",
      "perpendicular part",
      "dense subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.6573V"
    }
  }
}