{
  "abstract": "We classify, up to equivalence, all left-invariant complex structures on the three-dimensional Euclidean rigid motion group $SE(3).$ Our approach is based on the correspondence between complex structures on a real Lie algebra $\\mathfrak{e}(3)$ of the motion group and complex Lie subalgebras of the complexification of $\\mathfrak{e}(3).$ The resulting structures form a one-parameter family $J_ϕ.$ We further establish a connection with the classical Calabi-Eckmann complex structures on $S^3\\times S^3.$ More precisely, we construct a one-parameter deformation of Lie brackets connecting $\\mathfrak{so}(3)\\oplus \\mathfrak{so}(3)$ to $\\mathfrak{e}(3),$ for which each endomorphism $J_ϕ$ remains integrable. Consequently, every left-invariant complex structure on $SE(3)$ arises as a contraction limit of Calabi-Eckmann structures. The same family of endomorphisms also defines complex structures on $\\mathfrak{so}(3,1).$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20232",
  "author_total": 2,
  "authors": [
    {
      "name": "Neda Bokan"
    },
    {
      "name": "Srdjan Vukmirovic"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7pcnj2a5ffy2m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20232",
  "fulltext_url": null,
  "id": "7pcnj2a5ffy2m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20232",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7222324550",
  "pdf_url": "https://arxiv.org/pdf/2610.20232",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7pcnj2a5ffy2m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.20232",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20232"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Invariant complex structures on the Euclidean motion group as deformations of Calabi-Eckmann structures",
  "title_text": "Invariant complex structures on the Euclidean motion group as deformations of Calabi-Eckmann structures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/7pcnj2a5ffy2m",
  "venue": "arXiv",
  "versions": []
}