{
  "id": "2402.12209",
  "version": "v1",
  "published": "2024-02-19T15:12:12.000Z",
  "updated": "2024-02-19T15:12:12.000Z",
  "title": "Some Riemannian properties of $\\mathbf{SU_n}$ endowed with a bi-invariant metric",
  "authors": [
    "Donato Pertici",
    "Alberto Dolcetti"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "We study some properties of $SU_n$ endowed with the Frobenius metric $\\phi$, which is, up to a positive constant multiple, the unique bi-invariant Riemannian metric on $SU_n$. In particular we express the distance between $P, Q \\in SU_n$ in terms of eigenvalues of $P^*Q$; we compute the diameter of $(SU_n, \\phi)$ and we determine its diametral pairs; we prove that the set of all minimizing geodesic segments with endpoints $P$, $Q$ can be parametrized by means of a compact connected submanifold of $\\mathfrak{su}_n$, diffeomorphic to a suitable complex Grassmannian depending on $P$ and $Q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-19T15:12:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C35",
      "15B30",
      "22E15"
    ],
    "keywords": [
      "bi-invariant metric",
      "riemannian properties",
      "unique bi-invariant riemannian metric",
      "frobenius metric",
      "positive constant multiple"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}