{
  "abstract": "We study the topology of isometric classes of tensorial $H$-structures for closed and connected subgroups $H\\leqslant\\mathrm{SO}(n)$. The map assigning an $H$-structure its induced metric admits a global trivialization; consequently, the space of $H$-structures is homotopy equivalent to any fixed isometric class. On flat tori, these classes are mapping spaces into $\\mathrm{SO}(n)/H$. We give examples with infinitely many connected components for $\\mathrm{G}_2$, $\\mathrm{Spin}(7)$, and almost Hermitian structures, determine the number of components for $\\mathrm{SU}(m)$-structures on flat tori, and study the corresponding moduli spaces. For closed manifolds of dimension $n>2$, the intrinsic torsion energy on the full space of $H$-structures has infimum zero on every nonempty path component, and its critical points are precisely the torsion-free structures. In fixed isometric classes, we contrast the concentration phenomena underlying finite-time singularities with cohomological classes admitting positive energy lower bounds and harmonic representatives. Finally, we establish a lifting principle for metric-dependent flows, derive an evolution identity for isometric flows, and extend harmonic-flow estimates to general $H$-structures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.12031",
  "author_total": 2,
  "authors": [
    {
      "name": "Daniel Fadel"
    },
    {
      "name": "Eric Loubeau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b1rt204kg2vy1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.12031",
  "fulltext_url": null,
  "id": "b1rt204kg2vy1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.12031",
  "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": "W7164410512",
  "pdf_url": "https://arxiv.org/pdf/2606.12031",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b1rt204kg2vy1.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2606.12031",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.12031"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Topology of isometric classes and flows of geometric structures",
  "title_text": "Topology of isometric classes and flows of geometric structures",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:51Z",
  "url": "https://research.draconitic.com/paper/b1rt204kg2vy1",
  "venue": "arXiv",
  "versions": []
}