{
  "id": "math/0508164",
  "version": "v2",
  "published": "2005-08-09T11:59:29.000Z",
  "updated": "2006-03-09T22:48:58.000Z",
  "title": "A Cohomology (p+1) Form Canonically Associated with Certain Codimension-q Foliations on a Riemannian Manifold",
  "authors": [
    "Gabriel Baditoiu",
    "Richard H. Escobales Jr.",
    "Stere Ianus"
  ],
  "comment": "20 pages",
  "journal": "Tokyo Journal of Mathematics, vol 29 (2006), No. 1, 247-270",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $(M^{n},g)$ be a closed, connected, oriented, $C^{\\infty}$, Riemannian, n-manifold with a transversely oriented foliation $\\boldkey F$. We show that if $\\lbrace X,Y \\rbrace$ are basic vector fields, the leaf component of $[X,Y]$, $\\Cal{V}[X,Y]$, has vanishing leaf divergence whenever ${\\kappa}\\wedge \\chi_{\\boldkey F}$ is a closed (possibly zero) de Rham cohomology (p+1)-form. Here ${\\kappa}$ is the mean curvature one-form of the foliation ${\\boldkey F}$ and ${\\chi_{\\boldkey F}}$ is its characteristic form. In the codimension-2 case, ${\\kappa}\\wedge \\chi_{\\boldkey F}$ is closed if and only if ${\\kappa}$ is horizontally closed. In certain restricted cases, we give necessary and sufficient conditions for ${\\kappa}\\wedge{\\chi_{\\boldkey F}}$ to be harmonic. As an application, we give a characterization of when certain closed 3-manifolds are locally Riemannian products. We show that bundle-like foliations with totally umbilical leaves with leaf dimension greater than or equal to two on a constant curvature manifold, with non-integrable transversal distribution, and with Einstein-like transversal geometry are totally geodesic.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-03-09T22:48:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R30"
    ],
    "keywords": [
      "riemannian manifold",
      "codimension-q foliations",
      "constant curvature manifold",
      "mean curvature one-form"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......8164B"
    }
  }
}