{
  "id": "1709.02530",
  "version": "v1",
  "published": "2017-09-08T04:23:17.000Z",
  "updated": "2017-09-08T04:23:17.000Z",
  "title": "On compact Hermitian manifolds with flat Gauduchon connections",
  "authors": [
    "Bo Yang",
    "Fangyang Zheng"
  ],
  "comment": "9 pages. This preprint was submitted to Acta Mathematica Sinica, a special issue dedicated to Professor Qikeng Lu",
  "categories": [
    "math.DG"
  ],
  "abstract": "Given a Hermitian manifold $(M^n,g)$, the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call $\\nabla^s = (1-\\frac{s}{2})\\nabla^c + \\frac{s}{2}\\nabla^b$ the $s$-Gauduchon connection of $M$, where $\\nabla^c$ and $\\nabla^b$ are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat $s$-Gauduchon connection. This is related to a question asked by Yau \\cite{Yau}. The cases with $s=0$ (a flat Chern connection) or $s=2$ (a flat Bismut connection) are classified respectively by Boothby \\cite{Boothby} in the 1950s or by Q. Wang and the authors recently \\cite{WYZ}. In this article, we observe that if either $s\\geq 4+2\\sqrt{3} \\approx 7.46$ or $s\\leq 4-2\\sqrt{3}\\approx 0.54$ and $s\\neq 0$, then $g$ is K\\\"ahler. We also show that, when $n=2$, $g$ is always K\\\"ahler unless $s=2$. Note that non-K\\\"ahler compact Bismut flat surfaces are exactly those isosceles Hopf surfaces by \\cite{WYZ}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-08T04:23:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compact hermitian manifold",
      "flat gauduchon connections",
      "compact bismut flat surfaces",
      "chern connection",
      "isosceles hopf surfaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}