{
  "id": "math/0409528",
  "version": "v1",
  "published": "2004-09-27T16:55:43.000Z",
  "updated": "2004-09-27T16:55:43.000Z",
  "title": "Magnetic Rigidity of Horocycle flows",
  "authors": [
    "Gabriel P. Paternain"
  ],
  "categories": [
    "math.DS",
    "math.DG"
  ],
  "abstract": "Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$ and let $\\Omega$ be a 2-form. We show that the magnetic flow of the pair $(g,\\Omega)$ has zero asymptotic Maslov index and zero Liouville action if and only $g$ has constant Gaussian curvature, $\\Omega$ is a constant multiple of the area form of $g$ and the magnetic flow is a horocycle flow. This characterization of horocycle flows implies that if the magnetic flow of a pair $(g,\\Omega)$ is $C^1$-conjugate to the horocycle flow of a hyperbolic metric $\\bar{g}$ then there exists a constant $a>0$, such that $ag$ and $\\bar{g}$ are isometric and $a^{-1}\\Omega$ is, up to a sign, the area form of $g$. The characterization also implies that if a magnetic flow is Ma\\~n\\'e critical and uniquely ergodic it must be the horocycle flow. As a by-product we also obtain results on existence of closed magnetic geodesics for almost all energy levels in the case weakly exact magnetic fields on arbitrary manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-09-27T16:55:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "magnetic rigidity",
      "magnetic flow",
      "zero asymptotic maslov index",
      "case weakly exact magnetic fields",
      "area form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......9528P"
    }
  }
}