{
  "id": "2303.16365",
  "version": "v1",
  "published": "2023-03-29T00:18:10.000Z",
  "updated": "2023-03-29T00:18:10.000Z",
  "title": "On the Homogeneity Conjecture",
  "authors": [
    "Joseph A. Wolf"
  ],
  "comment": "This is a survey with new results and open problems",
  "categories": [
    "math.DG",
    "math.GR"
  ],
  "abstract": "Consider a connected homogeneous Riemannian manifold $(M,ds^2)$ and a Riemannian covering $(M,ds^2) \\to \\Gamma \\backslash (M,ds^2)$. If $\\Gamma \\backslash (M,ds^2)$ is homogeneous then every $\\gamma \\in \\Gamma$ is an isometry of constant displacement. The Homogeneity Conjecture suggests the converse: if every $\\gamma \\in \\Gamma$ is an isometry of constant displacement on $(M,ds^2)$ then $\\Gamma \\backslash (M,ds^2)$ is homogeneous. We survey the cases in which the Homogeneity Conjecture has been verified, including some new results, and suggest some related open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-29T00:18:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E40",
      "22F30",
      "22F50",
      "22D45",
      "53C30",
      "53C35"
    ],
    "keywords": [
      "homogeneity conjecture",
      "constant displacement",
      "connected homogeneous riemannian manifold",
      "related open problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}