{
  "abstract": "We study the classical problem of sectional curvature pinching in the class of 2-step nilmanifolds, which are necessarily of mixed curvature. We show that the pinching constant of any 2-step nilmanifold lies in the compact interval $[-3, -\\frac{3}{2}]$. The upper bound $-\\frac{3}{2}$ is achieved by the complex Heisenberg group $\\mathrm{Heis}_3(\\mathbb{C})$ with a Ricci soliton metric. The upper bound exhibits rigidity: if a simply connected 2-step nilmanifold $N$ has the pinching constant $-\\frac{3}{2}$, then $N$ admits a Ricci soliton complex Heisenberg group as a totally geodesic subgroup. On the other hand, the lower bound satisfies non-rigidity: any 2-step nilpotent Lie group admits a metric with pinching constant $-3$. This is derived by showing that there is an open neighborhood $U$ of $\\mathrm{Heis}_3(\\mathbb{R})\\times \\mathbb{R}^{n-3}$ in the space of $n$-dimensional 2-step nilmanifolds such that the pinching constant is $-3$ on $U$, and any 2-step nilpotent Lie group $N$ has a metric $g$ such that $(N,g)$ lies in $U$. In fact, if $N$ is not isomorphic to $\\mathrm{Heis}_3(\\mathbb{R})\\times \\mathbb{R}^{n-3}$, then there is a curve $g_t$ of metrics on $N$ with $(N,g_t)\\in U$, showing that there are uncountably many left-invariant metrics on $N$ such that the pinching constant is $-3$. An algebraic characterization of a 2-step nilpotent Lie group that admits a metric with the pinching constant $-\\frac{3}{2}$ is also given, and the pinching constants of various examples are computed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13052",
  "author_total": 1,
  "authors": [
    {
      "name": "Tomoya Tatsuno"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0b0m071zzzgkb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13052",
  "fulltext_url": null,
  "id": "0b0m071zzzgkb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13052",
  "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": "W7212782537",
  "pdf_url": "https://arxiv.org/pdf/2609.13052",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0b0m071zzzgkb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.13052",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13052"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Sectional Curvature Pinching of Two-Step Nilmanifolds",
  "title_text": "Sectional Curvature Pinching of Two-Step Nilmanifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/0b0m071zzzgkb",
  "venue": "arXiv",
  "versions": []
}