{
  "id": "2008.07028",
  "version": "v1",
  "published": "2020-08-16T23:32:00.000Z",
  "updated": "2020-08-16T23:32:00.000Z",
  "title": "Rigidity results for complete manifolds with nonnegative scalar curvature",
  "authors": [
    "Jintian Zhu"
  ],
  "comment": "18 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper, we are going to show some rigidity results for complete open Riemannian manifolds with nonnegative scalar curvature. Without using the famous Cheeger-Gromoll splitting theorem we give a new proof to a rigidity result for complete manifolds with nonnegative scalar curvature admitting a proper smooth map to $T^{n-1}\\times \\mathbf R$ with nonzero degree. Here we introduce a trick to obtain the compactness of limit hypersurface from locally graphical convergence. Based on the same idea we also establish an optimal $2$-systole inequality for several classes of complete Riemannian manifolds with positive scalar curvature and further prove a rigidity result for the equality case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-16T23:32:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C24",
      "53C21"
    ],
    "keywords": [
      "nonnegative scalar curvature",
      "rigidity result",
      "complete manifolds",
      "complete open riemannian manifolds",
      "proper smooth map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}