{
  "id": "math/9905096",
  "version": "v3",
  "published": "1999-05-17T13:52:25.000Z",
  "updated": "2000-04-24T17:28:21.000Z",
  "title": "Stability of the Focal and Geometric Index in semi-Riemannian Geometry via the Maslov Index",
  "authors": [
    "Francesco Mercuri",
    "Paolo Piccione",
    "Daniel V. Tausk"
  ],
  "comment": "58 pages, LaTeX2e, amsart class some references updated and minor changes in the abstract and in the Introduction in the replacement of Aug. 13th, 1999 Revised version of April 24th, 2000: added Remark 6.2.2, minor modifications in the Introduction",
  "categories": [
    "math.DG",
    "math.SG"
  ],
  "abstract": "We investigate the problem of the stability of the number of conjugate or focal points (counted with multiplicity) along a semi-Riemannian geodesic $\\gamma$. For a Riemannian or a non spacelike Lorentzian geodesic, such number is equal to the intersection number (Maslov index) of a continuous curve with a subvariety of codimension one of the Lagrangian Grassmannian of a symplectic space. Such intersection number is proven to be stable in a large variety of circumstances. In the general semi-Riemannian case, under suitable hypotheses this number is equal to an algebraic count of the multiplicities of the conjugate points, and it is related to the spectral properties of a non self-adjoint differential operator. This last relation gives a weak extension of the classical Morse Index Theorem in Riemannian and Lorentzian geometry. In this paper we reprove some results that were incorrectly stated by Helfer in a previous reference; in particular, a counterexample to one of Helfer's results, which is essential for the theory, is given. In the last part of the paper we discuss a general technique for the construction of examples and counterexamples in the index theory for semi-Riemannian metrics, in which some new phenomena appear.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2000-04-24T17:28:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34B24",
      "34C10",
      "34L05",
      "53C22",
      "53C50",
      "53C80"
    ],
    "keywords": [
      "maslov index",
      "semi-riemannian geometry",
      "geometric index",
      "intersection number",
      "non self-adjoint differential operator"
    ],
    "note": {
      "typesetting": "AMS-TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math......5096M"
    }
  }
}