{
  "id": "1912.11315",
  "version": "v1",
  "published": "2019-12-24T12:20:58.000Z",
  "updated": "2019-12-24T12:20:58.000Z",
  "title": "Constant index expectation curvature for graphs or Riemannian manifolds",
  "authors": [
    "Oliver Knill"
  ],
  "comment": "17 pages, 2 figures",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "An integral geometric curvature is defined as the index expectation K(x) = E[i(x)] if a probability measure m is given on vector fields on a Riemannian manifold or on a finite simple graph. Such curvatures are local, satisfy Gauss-Bonnet and are independent of any embedding in an ambient space. While realizing constant Gauss-Bonnet-Chern curvature is not possible in general already for 4-manifolds, we prove that for compact connected manifolds, constant curvature K_m can always be realized with m supported on Morse gradient fields. We give examples of finite simple graphs which do not allow for any constant m-curvature and prove that for one-dimensional connected graphs, there is a convex set of constant curvature configurations with dimension of the first Betti number of the graph. In particular, there is always a unique constant curvature solution for trees.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-24T12:20:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C10",
      "57M15",
      "03H05",
      "53Axx"
    ],
    "keywords": [
      "constant index expectation curvature",
      "riemannian manifold",
      "finite simple graph",
      "unique constant curvature solution",
      "first betti number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}