{
  "id": "1912.00577",
  "version": "v1",
  "published": "2019-12-02T04:10:08.000Z",
  "updated": "2019-12-02T04:10:08.000Z",
  "title": "More on Poincare-Hopf and Gauss-Bonnet",
  "authors": [
    "Oliver Knill"
  ],
  "comment": "12 pages, 11 figures",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "We illustrate connections between differential geometry on finite simple graphs G=(V,E) and Riemannian manifolds (M,g). The link is that curvature can be defined integral geometrically as an expectation in a probability space of Poincare-Hopf indices of coloring or Morse functions. Regge calculus with an isometric Nash embedding links then the Gauss-Bonnet-Chern integrand of a Riemannian manifold with the graph curvature. There is also a direct nonstandard approach: if V is a finite set containing all standard points of M and E contains pairs which are infinitesimally close in the sense of internal set theory, one gets a finite simple graph (V,E) which gets a curvature which as a measure corresponds to the standard curvature. The probabilistic approach is an umbrella framework which covers discrete spaces, piecewise linear spaces, manifolds or varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-02T04:10:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C10",
      "57M15",
      "03H05",
      "62-07",
      "62-04"
    ],
    "keywords": [
      "finite simple graph",
      "riemannian manifold",
      "gauss-bonnet",
      "internal set theory",
      "isometric nash embedding links"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}