{
  "id": "1704.03269",
  "version": "v1",
  "published": "2017-04-11T12:51:01.000Z",
  "updated": "2017-04-11T12:51:01.000Z",
  "title": "Local Estimate on Convexity Radius and decay of injectivity radius in a Riemannian manifold",
  "authors": [
    "Shicheng Xu"
  ],
  "comment": "19 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$, $\\operatorname{conv}(p)\\ge \\min\\{\\frac{1}{2}\\operatorname{inj}(p),\\operatorname{foc}(B_{\\operatorname{inj}(p)}(p))\\}$, where $\\operatorname{inj}(p)$ is the injectivity radius of $p$ and $\\operatorname{foc}(B_r(p))$ is the focal radius of open ball centered at $p$ with radius $r$; 2) for any two points $p,q$ in $M$, $\\operatorname{inj}(q)\\ge \\min\\{\\operatorname{inj}(p), \\operatorname{conj}(q)\\}-d(p,q),$ where $\\operatorname{conj}(q)$ is the conjugate radius of $q$; 3) for any $0<r<\\min\\{\\operatorname{inj}(p),\\frac{1}{2}\\operatorname{conj}(B_{\\operatorname{inj}(p)}(p))\\}$, any (not necessarily minimizing) geodesic in $B_r(p)$ has length $\\le 2r$. We also clarify two different concepts on convexity radius and give examples to illustrate that the one more frequently used in literature is not continuous.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-11T12:51:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C22",
      "53C20"
    ],
    "keywords": [
      "convexity radius",
      "injectivity radius",
      "local estimate",
      "complete riemannian manifold",
      "curvature-free estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}