{
  "id": "2402.13407",
  "version": "v3",
  "published": "2024-02-20T22:25:07.000Z",
  "updated": "2024-10-15T14:39:13.000Z",
  "title": "Einstein metrics on homogeneous spaces $H\\times H/ΔK$",
  "authors": [
    "Jorge Lauret",
    "Cynthia Will"
  ],
  "comment": "33 pages, 11 tables, 1 figure. Final version accepted in Communications in Contemporary Mathematics. New statements for Theorems 1.3 and 7.4 and some remarks added",
  "categories": [
    "math.DG"
  ],
  "abstract": "Given any compact homogeneous space $H/K$ with $H$ simple, we consider the new space $M=H\\times H/\\Delta K$, where $\\Delta K$ denotes diagonal embedding, and study the existence, classification and stability of $H\\times H$-invariant Einstein metrics on $M$, as a first step into the largely unexplored case of homogeneous spaces of compact non-simple Lie groups. We find unstable Einstein metrics on $M$ for most spaces $H/K$ such that their standard metric is Einstein (e.g., isotropy irreducible) and the Killing form of $\\mathfrak{k}$ is a multiple of the Killing form of $\\mathfrak{h}$ (e.g., $K$ simple), a class which contains $17$ families and $50$ individual examples. A complete classification is obtained in the case when $H/K$ is an irreducible symmetric space with $K$ simple. We also study the behavior of the scalar curvature function on the space of all normal metrics on $M=H\\times H/\\Delta K$ (none of which is Einstein), obtaining that the standard metric is a global minimum.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2024-10-15T14:39:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "standard metric",
      "compact non-simple lie groups",
      "scalar curvature function",
      "invariant einstein metrics",
      "killing form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}