{
  "abstract": "We develop a theory of orientability for metric manifolds and singular metric spaces using locally integral currents. For topological $n$-manifolds with locally finite Hausdorff measure, we introduce local degree and measure-theoretic conditions under which topological orientations correspond canonically to boundaryless, locally integer rectifiable $n$-currents of multiplicity one on the rectifiable part. We extend these currents to a class of singular spaces whose manifold part is dense and of full measure, and establish constancy and top-dimensional homology results under local uniqueness of orientation. For orientable non-collapsed $\\RCD(κ,n)$ spaces, we prove a current-theoretic Stokes theorem: the boundary of the orientation current is the induced orientation current of the geometric boundary, with multiplicity one. In particular, its mass measure is precisely the $(n-1)$-dimensional Hausdorff measure on that boundary. For purely $n$-dimensional, locally geodesically complete spaces with local upper curvature bounds, we show that local uniqueness of orientation excludes codimension-one branching and implies a local $1$-Poincaré inequality. The latter result applies beyond homology manifolds and extends the connection between quantitative topology and Poincaré inequalities to this class of singular spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03968",
  "author_total": 4,
  "authors": [
    {
      "name": "Nicola Cavallucci"
    },
    {
      "name": "Denis Marti"
    },
    {
      "name": "Andrea Mondino"
    },
    {
      "name": "Raquel Perales"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9rrdapny8v9nz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03968",
  "fulltext_url": null,
  "id": "9rrdapny8v9nz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03968",
  "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.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7220724666",
  "pdf_url": "https://arxiv.org/pdf/2610.03968",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9rrdapny8v9nz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.03968",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03968"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Orientability of metric manifolds and metric spaces with curvature bounds",
  "title_text": "Orientability of metric manifolds and metric spaces with curvature bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/9rrdapny8v9nz",
  "venue": "arXiv",
  "versions": []
}