{
  "id": "1901.04829",
  "version": "v1",
  "published": "2019-01-15T14:08:17.000Z",
  "updated": "2019-01-15T14:08:17.000Z",
  "title": "A note on the structure of prescribed gradient--like domains of non--integrable vector fields",
  "authors": [
    "Razvan M. Tudoran"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Given a geometric structure on $\\mathbb{R}^{n}$ with $n$ even (e.g. Euclidean, symplectic, Minkowski, pseudo-Euclidean), we analyze the set of points inside the domain of definition of an arbitrary given $\\mathcal{C}^1$ vector field, where the value of the vector field equals the value of the left/right gradient--like vector field of some fixed $\\mathcal{C}^2$ potential function, although a non-integrability condition holds at each such a point. Particular examples of gradient--like vector fields include the class of gradient vector fields with respect to Euclidean or pseudo-Euclidean inner products, and the class of Hamiltonian vector fields associated to symplectic structures on $\\mathbb{R}^{n}$ (with $n$ even). The main result of this article provides a geometric version of the main result of [1].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-15T14:08:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-integrable vector fields",
      "prescribed gradient-like domains",
      "main result",
      "hamiltonian vector fields",
      "left/right gradient-like vector field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}