{
  "id": "2011.01144",
  "version": "v1",
  "published": "2020-11-02T17:37:53.000Z",
  "updated": "2020-11-02T17:37:53.000Z",
  "title": "Killing vector fields on Riemannian and Lorentzian 3-manifolds",
  "authors": [
    "Amir Babak Aazami",
    "Robert Ream"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "We give a complete local classification of all Riemannian 3-manifolds $(M,g)$ admitting a nonvanishing Killing vector field $T$. We then extend this classification to timelike Killing vector fields on Lorentzian 3-manifolds, which are automatically nonvanishing. The two key ingredients needed in our classification are the scalar curvature $S$ of $g$ and the function $\\text{Ric}(T,T)$, where $\\text{Ric}$ is the Ricci tensor; in fact their sum appears as the Gaussian curvature of the quotient metric obtained from the action of $T$. Our classification generalizes that of Sasakian structures, which is the special case when $\\text{Ric}(T,T) = 2$. We also give necessary, and separately, sufficient conditions, both expressed in terms of $\\text{Ric}(T,T)$, for $g$ to be locally conformally flat. We then move from the local to the global setting, and prove two results: in the event that $T$ has unit length and the coordinates derived in our classification are globally defined on $\\mathbb{R}^3$, we show that the sum $S + \\text{Ric}(T,T)$ completely determines when the metric will be geodesically complete. In the event that the 3-manifold $M$ is compact, we give a condition stating when it admits a metric of constant positive sectional curvature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-02T17:37:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riemannian",
      "lorentzian",
      "constant positive sectional curvature",
      "complete local classification",
      "timelike killing vector fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}