{
  "abstract": "For $m=2,3$, we prove that every smooth immersion $F:\\mathbb{R}\\mathbb{P}^m\\looparrowright\\overline{\\mathbb B}^{,N}(1)$ satisfies $κ(F)^2\\ge 2m/(m+1)$, with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Euclidean unit ball with $κ(F)\\le\\sqrt{3/2}$ is diffeomorphic to $S^3$, $\\mathbb{R}\\mathbb{P}^3$, or $S^2\\times S^1$. All three possibilities occur, while $κ(F)<\\sqrt{3/2}$ forces $X\\cong S^3$. These results answer a question of Petrunin and prove a conjecture of Chodosh--Li concerning the normal curvature of three-manifolds. The key intrinsic input is the strict scalar--systolic inequality \\[ (\\min_Y R_g)\\text{sys}(g)^2<6π^2 \\] for every spherical three-space form $Y$ with $|π_1(Y)|>2$. Its proof uses systolic monotonicity along Ricci flow with surgery. This strict inequality complements the sharp scalar--systolic inequality for $\\mathbb{R}\\mathbb{P}^3$ of Bray--Brendle--Eichmair--Neves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.18002",
  "author_total": 2,
  "authors": [
    {
      "name": "Tsz-Kiu Aaron Chow"
    },
    {
      "name": "Jingbo Wan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ckvvptj20x2he.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.18002",
  "fulltext_url": null,
  "id": "ckvvptj20x2he",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.18002",
  "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.GT"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7203758617",
  "pdf_url": "https://arxiv.org/pdf/2608.18002",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ckvvptj20x2he.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2608.18002",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.18002"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Small Normal Curvature and Three-Manifold Topology",
  "title_text": "Small Normal Curvature and Three-Manifold Topology",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/ckvvptj20x2he",
  "venue": "arXiv",
  "versions": []
}