{
  "id": "1704.02001",
  "version": "v1",
  "published": "2017-04-06T19:44:46.000Z",
  "updated": "2017-04-06T19:44:46.000Z",
  "title": "Bifurcations of the conjugate locus",
  "authors": [
    "Thomas Waters"
  ],
  "comment": "Accepted in Journal of Geometry and Physics April 2017",
  "categories": [
    "math.DG"
  ],
  "abstract": "The conjugate locus of a point $p$ in a surface $\\mathcal{S}$ will have a certain number of cusps. As the point $p$ is moved in the surface the conjugate locus may spontaneously gain or lose cusps. In this paper we explain this `bifurcation' in terms of the vanishing of higher derivatives of the exponential map; we derive simple equations for these higher derivatives in terms of scalar invariants; we classify the bifurcations of cusps in terms of the local structure of the conjugate locus; and we describe an intuitive picture of the bifurcation as the intersection between certain contours in the tangent plane.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-06T19:44:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "conjugate locus",
      "bifurcation",
      "higher derivatives",
      "local structure",
      "scalar invariants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}