{
  "id": "0908.4545",
  "version": "v1",
  "published": "2009-08-31T14:31:00.000Z",
  "updated": "2009-08-31T14:31:00.000Z",
  "title": "Generalized Obata theorem and its applications on foliations",
  "authors": [
    "Seoung Dal Jung",
    "Keum Ran Lee",
    "Ken Richardson"
  ],
  "comment": "16 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "We prove the generalized Obata theorem on foliations. Let M be a complete Riemannian manifold with a foliation F of codimension $q>1$ and a bundle-like metric. Then $(M, F)$ is transversally isometric to the q-sphere of radius 1/c in (q+1)-dimensional Euclidean space endowed with the action of a discrete subgroup of the orthogonal group O(q), if and only if there exists a non-constant basic function f such that $\\nabla_X df = -c^2 f X^\\flat for all basic normal vector fields X, where c is a positive constant and \\nabla is the connection on the normal bundle. By the generalized Obata theorem, we classify such manifolds which admit transversal non-isometric conformal fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-08-31T14:31:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C12",
      "53C27",
      "57R30"
    ],
    "keywords": [
      "generalized obata theorem",
      "admit transversal non-isometric conformal fields",
      "applications",
      "basic normal vector fields",
      "complete riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0908.4545D"
    }
  }
}