{
  "abstract": "The oscillator groups are four-dimensional solvable Lie groups, extensions of the Heisenberg Lie group. We present them as circle bundles of the standard contact sub-Riemannian structure on $\\mathbb{R}^{3}$. We define sub-Riemannian and sub-Lorentzian structures on them, describe their geodesics explicitly and determine which of them are periodic. We also study a generalization: Let $G$ be a semisimple Lie group and $K$ a compact subgroup such that $\\left( G,K\\right) $ is a symmetric pair of the compact or the noncompact type. Let $\\mathfrak{g}$ and $% \\mathfrak{k}$ be the Lie algebras of $G$ and $K$, respectively. We consider on $\\mathfrak{g}$ the usual nilpotent Lie group structure with center $\\mathfrak{k}$ and call it $N\\left( \\mathfrak{g},\\mathfrak{k}\\right)$. A suitable semidirect product with $\\operatorname{Ad}\\left( K\\right) $ yields a solvable Lie group $\\operatorname{Osc}\\left( G,K\\right) $, which generalizes the oscillator groups and is also quadratic (that is, it possesses bi-invariant metrics; in particular, a canonical one). We define nonholonomic pseudo-Riemannian structures on it and find their geodesics explicitly. In doing so, we obtain a result that may not be merely auxiliary: formulas for the monoparametric subgroups of $\\operatorname{Osc}\\left( G,K\\right) $ and for the sub-Riemannian geodesics of $N\\left( \\mathfrak{g},\\mathfrak{k}\\right) $ with the standard left-invariant distribution $\\mathcal{D}$. Moreover, when $G/K$ is compact with rank one, we realize $\\operatorname{Osc}\\left( G,K\\right) $ as a Stiefel bundle of $\\left( N\\left( \\mathfrak{g},\\mathfrak{k}\\right) ,\\mathcal{D}\\right) $, up to finite coverings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12027",
  "author_total": 2,
  "authors": [
    {
      "name": "Mauricio Godoy Molina"
    },
    {
      "name": "Marcos Salvai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3956draf36ec7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12027",
  "fulltext_url": null,
  "id": "3956draf36ec7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12027",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "3103|Advanced Differential Geometry Research"
    ]
  ],
  "openalex": "W7221173907",
  "pdf_url": "https://arxiv.org/pdf/2610.12027",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3956draf36ec7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12027",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12027"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Sub-Riemannian and sub-Lorentzian geodesics of the oscillator groups",
  "title_text": "Sub-Riemannian and sub-Lorentzian geodesics of the oscillator groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/3956draf36ec7",
  "venue": "arXiv",
  "versions": []
}