{
  "abstract": "Ricci curvature measures the average stability of geodesics under transverse perturbations, but how it should be defined in infinite dimensions is often unclear. This paper proposes a definition of Ricci curvature on the space of Hamiltonian diffeomorphisms on the two-dimensional flat torus $\\mathrm{HDiff}(\\mathbb{T}^2)$, the state space for ideal fluids. Our definition is based on Zeitlin's model, which approximates $\\mathrm{HDiff}(\\mathbb{T}^2)$ by finite-dimensional Lie groups $\\mathrm{SU}(N)$. We derive a formula for the Ricci curvature tensor on $\\mathrm{SU}(N)$ and provide numerical evidence for its convergence in the large-$N$ limit to our conjectured finite value. Additionally, we explore potential applications for hydrodynamics through the Lyapunov stability of gravest wave modes and Arnold's tradewind estimates for long-term weather predictability. Our framework extends to a wide range of settings. We demonstrate this by introducing Ricci curvature on the state spaces of fluids on rectangular domains, the Lagrangian averaged Euler equation induced by the $H^1$-Sobolev metric, and the quasi-geostrophic equation incorporating the Coriolis effect.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01259",
  "author_total": 2,
  "authors": [
    {
      "name": "Sadashige Ishida"
    },
    {
      "name": "Alex Suri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/02hny9hyna218.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01259",
  "fulltext_url": null,
  "id": "02hny9hyna218",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01259",
  "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"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7206186213",
  "pdf_url": "https://arxiv.org/pdf/2609.01259",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/02hny9hyna218.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01259",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01259"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Ricci curvature for fluid models on the torus via Zeitlin's quantization",
  "title_text": "Ricci curvature for fluid models on the torus via Zeitlin's quantization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/02hny9hyna218",
  "venue": "arXiv",
  "versions": []
}