{
  "id": "math/0605611",
  "version": "v1",
  "published": "2006-05-23T11:12:59.000Z",
  "updated": "2006-05-23T11:12:59.000Z",
  "title": "Integrability Conditions For Almost Hermitian And Almost Kaehler 4-Manifolds",
  "authors": [
    "Klaus-Dieter Kirchberg"
  ],
  "comment": "19 pages, Latex",
  "categories": [
    "math.DG"
  ],
  "abstract": "If $W_+$ denotes the self dual part of the Weyl tensor of any K\\\"ahler 4-manifold and $S$ its scalar curvature, then the relation $|W_+|^2 = S^2/6$ is well-known. For any almost K\\\"ahler 4-manifold with $S \\ge 0$, this condition forces the K\\\"ahler property. A compact almost K\\\"ahler 4-manifold is already K\\\"ahler if it satisfies the conditions $| W_+ |^2 = S^2/6$ and $\\delta W_+=0$ and also if it is Einstein and $| W_+|$ is constant. Some further results of this type are proved. An almost Hermitian 4-manifold $(M,g,J)$ with $\\mathrm{supp} (W_+)=M$ is already K\\\"ahler if it satisfies the condition $| W_+ |^2 = 3 (S_{\\star} - S/3)^2 /8$ together with $|\\nabla W_+ | = | \\nabla |W_+||$ or with $\\delta W_+ + \\nabla \\log | W_+ | \\lrcorner W_+ =0$, respectively. The almost complex structure $J$ enters here explicitely via the star scalar curvature $S_{\\star}$ only.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-05-23T11:12:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53B20",
      "53C25"
    ],
    "keywords": [
      "integrability conditions",
      "star scalar curvature",
      "self dual part",
      "weyl tensor",
      "condition forces"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......5611K"
    }
  }
}