{
  "abstract": "We prove that the exotic smooth $8$-sphere and both oriented homotopy $10$-spheres representing elements of order three admit Riemannian metrics with strictly positive sectional curvature. The construction combines Sperança's special $S^3$-$S^3$ bundle models with the compatible-disk construction of He, Liu and Yau. We show that the relevant bundles admit equivariant polar normal forms, with transition functions that are constant along meridians and conjugation-equivariant. We also show that the representation-dependent part of the He--Liu--Yau construction requires only a uniform bound on the infinitesimal action fields. Sperança's $8$- and $10$-dimensional examples satisfy this bound. The resulting northern and southern metrics have matching boundary metrics and compatible second fundamental forms, so the Reiser--Wraith gluing theorem gives the required positively curved metrics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38126",
  "author_total": 1,
  "authors": [
    {
      "name": "Fernando Galaz-García"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/chsf2pamnjmy2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38126",
  "fulltext_url": null,
  "id": "chsf2pamnjmy2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38126",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "3103|Advanced Differential Geometry Research"
    ]
  ],
  "openalex": "W7214941955",
  "pdf_url": "https://arxiv.org/pdf/2609.38126",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/chsf2pamnjmy2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2609.38126",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38126"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Positive sectional curvature on the exotic $8$-sphere and order-three homotopy $10$-spheres",
  "title_text": "Positive sectional curvature on the exotic $8$-sphere and order-three homotopy $10$-spheres",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/chsf2pamnjmy2",
  "venue": "arXiv",
  "versions": []
}