{
  "id": "1707.09618",
  "version": "v1",
  "published": "2017-07-30T13:22:10.000Z",
  "updated": "2017-07-30T13:22:10.000Z",
  "title": "Critical values of homology classes of loops and positive curvature",
  "authors": [
    "Hans-Bert Rademacher"
  ],
  "comment": "14 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "We study compact and simply-connected Riemannian manifolds with positive sectional curvature $K\\ge 1.$ For a non-trivial homology class of lowest dimension in the space of loops based at a point $p$ or in the free loop space one can define a critical length ${\\sf crl}_p\\left(M,g\\right)$ resp. ${\\sf crl}\\left(M,g\\right).$ Then ${\\sf crl}_p\\left(M,g\\right)$ equals the length of a geodesic loop and ${\\sf crl}\\left(M,g\\right)$ equals the length of a closed geodesic. This is the idea of the proof of the existence of a closed geodesic of positive length presented by Birkhoff in case of a sphere and by Lusternik and Fet in the general case. It is the main result of the paper that the numbers ${\\sf crl}_p\\left(M,g\\right)$ resp. ${\\sf crl}\\left(M,g\\right)$ attain its maximal value $2\\pi$ only for the round metric on the $n$-sphere. Under the additional assumption $K \\le 4$ this result for ${\\sf crl}\\left(M,g\\right)$ follows from results by Sugimoto in even dimensions and Ballmann, Thorbergsson and Ziller in odd dimensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-30T13:22:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20",
      "53C21",
      "53C22",
      "53C24",
      "58E10"
    ],
    "keywords": [
      "positive curvature",
      "critical values",
      "free loop space",
      "non-trivial homology class",
      "closed geodesic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}