{
  "abstract": "We give a new geometric interpretation of the Amari--Cencov $\\alpha$-connections $\\nabla^{(\\alpha)}$ from information geometry: On the space of densities, we show that there exist Riemannian metrics $G^\\alpha$, which we call $\\alpha$-Fisher--Rao metrics, whose Levi-Civita connections are $\\nabla^{(\\alpha)}$. With the exception of $\\alpha=0$ (the Fisher--Rao metric), these metrics are non-invariant to the action of the diffeomorphism group, even though the connections are invariant. This gives a new way of interpreting the geodesics of the $\\nabla^{(\\alpha)}$ as energy-minimizing curves. On the space of probability densities, we show that the same phenomenon holds for $\\alpha\\in \\{-1,0,1\\}$ and that the $\\alpha$-connections are not metric otherwise. We show that $\\nabla^{(\\alpha)}$-geodesics on this space can be interpreted as radial projections of straight lines on appropriate hyper-surfaces, and use this geometric picture to obtain geodesic convexity for any $\\alpha\\in \\R$. In addition, we prove analogous results for appropriate metrics and connections on the space of diffeomorphisms, which, for the case $M=\\R$, imply that the generalized Proudman--Johnson equations on the real line are the Euler--Arnold-type equations of non-right invariant metrics. Finally, in the finite-dimensional case, we show that $\\nabla^{(\\alpha)}$ can be metric or non-metric depending on the statistical model considered.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "name": "Martin Bauer"
    },
    {
      "name": "Alice Le Brigant"
    },
    {
      "name": "Cy Maor"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a8c6g4wngtgy6.bib",
  "cited_by": 0,
  "doi": "10.1088/1361-6544/ae806f",
  "fulltext_url": null,
  "id": "a8c6g4wngtgy6",
  "issns": [
    "0951-7715",
    "1361-6544"
  ],
  "landing_url": "https://hal.science/hal-05791793v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
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    [
      "hal",
      "1.math.math-dg"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
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  "openalex": "W7167287728",
  "pdf_url": "https://hal.science/hal-05791793/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-03",
  "publisher": "IOP Publishing",
  "ris": "https://research.draconitic.com/paper/a8c6g4wngtgy6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:53Z",
      "native_id": "hal-05791793",
      "source": "hal",
      "url": "https://hal.science/hal-05791793v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-dg",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Riemannian viewpoint on the Amari–Cencov $\\alpha$-connections and Proudman–Johnson equations",
  "title_text": "A Riemannian viewpoint on the Amari–Cencov α-connections and Proudman–Johnson equations",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:17:53Z",
  "url": "https://research.draconitic.com/paper/a8c6g4wngtgy6",
  "venue": "Nonlinearity",
  "versions": []
}