{
  "id": "0803.3533",
  "version": "v1",
  "published": "2008-03-25T11:37:09.000Z",
  "updated": "2008-03-25T11:37:09.000Z",
  "title": "Riemannian geometry of Hartogs domains",
  "authors": [
    "Antonio J. Di Scala",
    "Andrea Loi",
    "Fabio Zuddas"
  ],
  "comment": "to appear in International Journal of Mathematics",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $D_F = \\{(z_0, z) \\in {\\C}^{n} | |z_0|^2 < b, \\|z\\|^2 < F(|z_0|^2) \\}$ be a strongly pseudoconvex Hartogs domain endowed with the \\K metric $g_F$ associated to the \\K form $\\omega_F = -\\frac{i}{2} \\partial \\bar{\\partial} \\log (F(|z_0|^2) - \\|z\\|^2)$. This paper contains several results on the Riemannian geometry of these domains. In the first one we prove that if $D_F$ admits a non special geodesic (see definition below) through the origin whose trace is a straight line then $D_F$ is holomorphically isometric to an open subset of the complex hyperbolic space. In the second theorem we prove that all the geodesics through the origin of $D_F$ do not self-intersect, we find necessary and sufficient conditions on $F$ for $D_F$ to be geodesically complete and we prove that $D_F$ is locally irreducible as a Riemannian manifold. Finally, we compare the Bergman metric $g_B$ and the metric $g_F$ in a bounded Hartogs domain and we prove that if $g_B$ is a multiple of $g_F$, namely $g_B=\\lambda g_F$, for some $\\lambda\\in \\R^+$, then $D_F$ is holomorphically isometric to an open subset of the complex hyperbolic space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-03-25T11:37:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C55",
      "32Q15",
      "32T15"
    ],
    "keywords": [
      "riemannian geometry",
      "complex hyperbolic space",
      "pseudoconvex hartogs domain",
      "open subset",
      "holomorphically isometric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0803.3533D"
    }
  }
}