{
  "id": "math-ph/0609075",
  "version": "v3",
  "published": "2006-09-26T19:47:26.000Z",
  "updated": "2008-04-05T23:07:42.000Z",
  "title": "Determination of the metric from the connection",
  "authors": [
    "Richard Atkins"
  ],
  "comment": "21 pages",
  "categories": [
    "math-ph",
    "math.DG",
    "math.MP"
  ],
  "abstract": "As is well known, a metric on a manifold determines a unique symmetric connection for which the metric is parallel: the Levi-Civita connection. In this paper we investigate the inverse problem: to what extent is the metric of a Riemannian manifold determined by its Levi-Civita connection? It is shown that for a generic Levi-Civita connection of a metric $h$ there exists a set of positive semi-definite tensor fields $h_{a}$ such that the parallel metrics are the positive-definite linear combinations of the $h_{a}$. Moreover, the set of all parallel metrics may be constructed by a soley algebraic procedure.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-04-05T23:07:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "determination",
      "parallel metrics",
      "generic levi-civita connection",
      "positive semi-definite tensor fields",
      "unique symmetric connection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.ph...9075A"
    }
  }
}