{"paper":{"title":"On the universality of the incompressible Euler equation on compact manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AP","authors_text":"Terence Tao","submitted_at":"2017-07-25T03:58:24Z","abstract_excerpt":"The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form \\begin{align*} \\partial_t u + \\nabla_u u &= - \\mathrm{grad}_g p \\mathrm{div}_g u &= 0. \\end{align*} We show that any quadratic ODE $\\partial_t y = B(y,y)$, where $B : {\\bf R}^n \\times {\\bf R}^n \\to {\\bf R}^n$ is a symmetric bilinear map, can be linearly embedded into the incompressible Euler equations for some manifold $M$ if and only if $B$ obeys the cancellation condition $\\langle B(y,y), y \\rangle = 0$ for some positive definite inner product $\\langle,\\rangle$ on $ {\\bf R}^n$. This allows one to constr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.07807","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}