{
  "abstract": "Let g be a Lie algebra of dimension 2n+1, let beta be a contact form, and let alpha be a nonzero integrable 1-form. We study when the affine path alpha_t = (1-t)alpha + t beta consists of contact forms with the orientation determined by beta \\wedge (d beta)^n. We first establish an algebraic criterion: affine deformability is equivalent to the non-negativity of a scalar obtained from the interaction form Theta(alpha, beta) = alpha \\wedge (d beta)^n + n beta \\wedge d alpha \\wedge (d beta)^{n-1}. For unimodular Lie algebras this scalar is (n+1)alpha(Z_beta), where Z_beta is the Reeb vector. We also derive a formula involving the modular character tau(X) = tr(ad_X) in the non-unimodular case. On nilpotent contact Lie algebras, every integrable 1-form is closed; using the standard structural properties of nilpotent contact Lie algebras, this implies that every left-invariant foliation is an affine origin for every fixed contact form. An explicit three-dimensional example illustrates that the modular term can yield affine deformability even when alpha(Z_beta) = 0. We discuss the implications for left-invariant forms on Lie groups and compact quotients. We do not claim a general obstruction example in the non-unimodular case; establishing such an example requires a separate verified construction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35940",
  "author_total": 2,
  "authors": [
    {
      "name": "Cheikh Khoule"
    },
    {
      "name": "Rafik Khalfi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6w07m5hc208df.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35940",
  "fulltext_url": null,
  "id": "6w07m5hc208df",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35940",
  "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"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "3103|Advanced Differential Geometry Research"
    ]
  ],
  "openalex": "W7224135472",
  "pdf_url": "https://arxiv.org/pdf/2610.35940",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6w07m5hc208df.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.35940",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35940"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Rigidity and Affine Deformations of Foliations on Nilpotent and Solvable Contact Lie Algebras",
  "title_text": "Rigidity and Affine Deformations of Foliations on Nilpotent and Solvable Contact Lie Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/6w07m5hc208df",
  "venue": "arXiv",
  "versions": []
}