{"paper":{"title":"A Riemannian viewpoint on the Amari-Cencov $\\alpha$-connections and Proudman-Johnson equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Alice Le Brigant, Cy Maor, Martin Bauer","submitted_at":"2025-08-01T07:03:36Z","abstract_excerpt":"We give a new geometric interpretation of the Amari-Cencov $\\alpha$-connections $\\nabla^{(\\alpha)}$ from information geometry: On the space of densities $\\operatorname{Dens}_+(M)$, 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 $\\operatorname{Diff}(M)$, even though the connections are invariant. This gives a new way of interpreting the geodesics of the $\\nabl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.00371","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.00371/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}