{
  "id": "math/0104157",
  "version": "v1",
  "published": "2001-04-14T17:58:30.000Z",
  "updated": "2001-04-14T17:58:30.000Z",
  "title": "An Inverse Problem from Sub-Riemannian Geometry",
  "authors": [
    "Thomas A. Ivey"
  ],
  "comment": "13 pages",
  "categories": [
    "math.DG",
    "math.OC"
  ],
  "abstract": "The geodesics for a sub-Riemannian metric on a three-dimensional contact manifold $M$ form a 1-parameter family of curves along each contact direction. However, a collection of such contact curves on $M$, locally equivalent to the solutions of a fourth-order ODE, are the geodesics of a sub-Riemannian metric only if a sequence of invariants vanish. The first of these, which was earlier identified by Fels, determines if the differential equation is variational. The next two determine if there is a well-defined metric on $M$ and if the given paths are its geodesics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-04-14T17:58:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C17",
      "49N45",
      "34A26",
      "53A55"
    ],
    "keywords": [
      "sub-riemannian geometry",
      "inverse problem",
      "sub-riemannian metric",
      "three-dimensional contact manifold",
      "contact direction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......4157I"
    }
  }
}