{
  "id": "2002.10359",
  "version": "v1",
  "published": "2020-02-24T16:40:38.000Z",
  "updated": "2020-02-24T16:40:38.000Z",
  "title": "Invariant Einstein metrics on SU(N) and complex Stiefel manifolds",
  "authors": [
    "Andreas Arvanitoyeorgos",
    "Yusuke Sakane",
    "Marina Statha"
  ],
  "comment": "50 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "We study existence of invariant Einstein metrics on complex Stiefel manifolds $G/K = \\SU(\\ell+m+n)/\\SU(n) $ and the special unitary groups $G = \\SU(\\ell+m+n)$. We decompose the Lie algebra $\\frak g$ of $G$ and the tangent space $\\frak p$ of $G/K$, by using the generalized flag manifolds $G/H = \\SU(\\ell+m+n)/\\s(\\U(\\ell)\\times\\U(m)\\times\\U(n))$. We parametrize scalar products on the 2-dimensional center of the Lie algebra of $H$, and we consider $G$-invariant and left invariant metrics determined by $\\Ad(\\s(\\U(\\ell)\\times\\U(m)\\times\\U(n))$-invariant scalar products on $\\frak g$ and $\\frak p$ respectively. Then we compute their Ricci tensor for such metrics. We prove existence of $\\Ad(\\s(\\U(1)\\times\\U(2)\\times\\U(2))$-invariant Einstein metrics on $V_3\\bb{C}^{5}=\\SU(5)/\\SU(2)$, $\\Ad(\\s(\\U(2)\\times\\U(2)\\times\\U(2))$-invariant Einstein metrics on $V_4\\bb{C}^{6}=\\SU(6)/\\SU(2)$, and $\\Ad(\\s(\\U(m)\\times\\U(m)\\times\\U(n))$-invariant Einstein metrics on $V_{2m}\\bb{C}^{2m+n}=\\SU(2m+n)/\\SU(n)$. We also prove existence of $\\Ad(\\s(\\U(1)\\times\\U(2)\\times\\U(2))$-invariant Einstein metrics on the compact Lie group $\\SU(5)$, which are not naturally reductive. The Lie group $\\SU(5)$ is the special unitary group of smallest rank known for the moment, admitting non naturally reductive Einstein metrics. Finally, we show that the compact Lie group $\\SU(4+n)$ admits two non naturally reductive $\\Ad(\\s(\\U(2)\\times\\U(2)\\times\\U(n)))$-invariant Einstein metrics for $ 2 \\leq n \\leq 25$, and four non naturally reductive Einstein metrics for $n\\ge 26$. This extends previous results of K.~ Mori about non naturally reductive Einstein metrics on $\\SU(4+n)$ ($n \\geq 2$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-24T16:40:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C25",
      "53C30",
      "13P10",
      "65H10",
      "68W30"
    ],
    "keywords": [
      "invariant einstein metrics",
      "complex stiefel manifolds",
      "reductive einstein metrics",
      "compact lie group",
      "special unitary group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}