{"paper":{"title":"Flatness of generic Poisson pairs in odd dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Francisco-Javier Turiel","submitted_at":"2015-01-16T10:13:05Z","abstract_excerpt":"Given a $(m-2)$-form $\\zw$ and a volume form $\\zW$ on a $m$-manifold one defines a bi-vector $\\zL$ by setting $\\zL(\\za,\\zb)={\\frac {\\za\\zex\\zb\\zex\\zw} {\\zW}}$ for any $1$-forms $\\za,\\zb$. In this way, locally, a Poisson pair, or bi-Hamiltonian structure, $(\\zL,\\zL_1 )$ is always represented by a couple of $(m-2)$-forms $\\zw,\\zw_1$ and a volume form $\\zW$. Here one shows that, for $m\\zmai 5$ and odd and $(\\zL,\\zL_1 )$ generic, $(\\zL,\\zL_1 )$ is flat if and only if there exists a $1$-form $\\zl$ such that $d\\zw=\\zl\\zex\\zw$ and $d\\zw_1 =\\zl\\zex\\zw_1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.03932","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":""},"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"}