{
  "id": "1401.5549",
  "version": "v2",
  "published": "2014-01-22T03:47:01.000Z",
  "updated": "2014-01-23T01:23:51.000Z",
  "title": "On conjugate points and geodesic loops in a complete Riemannian manifold",
  "authors": [
    "Shicheng Xu"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "A well-known Lemma in Riemannian geometry by Klingenberg says that if $x_0$ is a minimum point of the distance function $d(p,\\cdot)$ to $p$ in the cut locus $C_p$ of $p$, then either there is a minimal geodesic from $p$ to $x_0$ along which they are conjugate, or there is a geodesic loop at $p$ that smoothly goes through $x_0$. In this paper, we prove that: for any point $q$ and any local minimum point $x_0$ of $F_q(\\cdot)=d(p,\\cdot)+d(q,\\cdot)$ in $C_p$, either $x_0$ is conjugate to $p$ along each minimal geodesic connecting them, or there is a geodesic from $p$ to $q$ passing through $x_0$. In particular, for any local minimum point $x_0$ of $d(p,\\cdot)$ in $C_p$, either $p$ and $x_0$ are conjugate along every minimal geodesic from $p$ to $x_0$, or there is a geodesic loop at $p$ that smoothly goes through $x_0$. Earlier results based on injective radius estimate would hold under weaker conditions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-01-23T01:23:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C22",
      "53C20"
    ],
    "keywords": [
      "complete riemannian manifold",
      "geodesic loop",
      "conjugate points",
      "minimal geodesic",
      "local minimum point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.5549X"
    }
  }
}