{"id":"openai-math/Entropy-equality-and-local-symmetry-in-negative-curvature","pdf":"https://github.com/openai/math/blob/main/preprints/Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026","tags":["riemannian-geometry","geodesic-flow","entropy-rigidity"],"score":0,"title":"Entropy equality and local symmetry in negative curvature","domain":"Differential geometry","family":"339","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"For every closed connected smooth Riemannian manifold of dimension at least three with strictly negative sectional curvature, we prove that normalized Liouville measure has maximal entropy for the unit-speed geodesic flow if and only if the metric is locally symmetric. This resolves Katok's entropy rigidity conjecture positively in these dimensions, including all rank-one symmetric types at arbitrary scale.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=The%20conjecture%20was%20settled%20by%20Katok%20for%20surfaces%20and%20mostly%20open%20in%20higher%20dimension","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"The conjecture was settled by Katok for surfaces and mostly open in higher dimension. The main project of my thesis was to prove a local version of the conjecture near complex hyperbolic metrics. After this I was planning on attacking the conjecture more globally. Open AI claimed a proof of the…","added":"2026-10-24","title":"100+ reactions to 100+ solutions"}],"formalized":false,"family_title":"Katok's entropy rigidity conjecture"}