{
  "id": "2308.15454",
  "version": "v1",
  "published": "2023-08-29T17:27:22.000Z",
  "updated": "2023-08-29T17:27:22.000Z",
  "title": "Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds",
  "authors": [
    "Mohammad Ghomi"
  ],
  "comment": "14 pages",
  "categories": [
    "math.DG",
    "math.AP",
    "math.MG"
  ],
  "abstract": "We show that in Cartan-Hadamard manifolds $M^n$, $n\\geq 3$, closed infinitesimally convex hypersurfaces $\\Gamma$ bound convex flat regions, if curvature of $M^n$ vanishes on tangent planes of $\\Gamma$. This encompasses Chern-Lashof characterization of convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Cartan-Hadamard manifolds. It follows that closed simply connected surfaces in $M^3$ with minimal total absolute curvature bound Euclidean convex bodies, as stated by M. Gromov in 1985. The proofs employ the Gauss-Codazzi equations, a generalization of Schur comparison theorem to CAT($k$) spaces, and other techniques from Alexandrov geometry outlined by A. Petrunin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-29T17:27:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20",
      "58J05",
      "53C44",
      "52A15"
    ],
    "keywords": [
      "cartan-hadamard manifolds",
      "hypersurfaces",
      "absolute curvature bound euclidean convex",
      "curvature bound euclidean convex bodies",
      "total absolute curvature bound euclidean"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}