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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 $d\\zw_1 =\\zl\\zex\\zw_1$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1501.03932","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-01-16T10:13:05Z","cross_cats_sorted":[],"title_canon_sha256":"2c5680d9bb10a64ef4b6970f05c8a46afa50505306b5045d762a961ab36de904","abstract_canon_sha256":"fc740ffbdd19121575759994a50cc4f261c6c8afb5e17f64a6763a52d21638a8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:16.728120Z","signature_b64":"cR9MUyMyozl391btMyvMlsx5OTjdo2qQUlDHgCh3cs7+U2G84BGXwYdhmFgQ4fgwopFmVRjBOGdEy63WfU+ACg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"84313ec3a2afe3c89b4698f0644fb7fccf9a001b0ccf525ecf62576eb9864f24","last_reissued_at":"2026-05-18T02:29:16.727736Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:16.727736Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1501.03932","created_at":"2026-05-18T02:29:16.727793+00:00"},{"alias_kind":"arxiv_version","alias_value":"1501.03932v1","created_at":"2026-05-18T02:29:16.727793+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.03932","created_at":"2026-05-18T02:29:16.727793+00:00"},{"alias_kind":"pith_short_12","alias_value":"QQYT5Q5CV7R4","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_16","alias_value":"QQYT5Q5CV7R4RG2G","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_8","alias_value":"QQYT5Q5C","created_at":"2026-05-18T12:29:37.295048+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QQYT5Q5CV7R4RG2GTDYGIT5X7T","json":"https://pith.science/pith/QQYT5Q5CV7R4RG2GTDYGIT5X7T.json","graph_json":"https://pith.science/api/pith-number/QQYT5Q5CV7R4RG2GTDYGIT5X7T/graph.json","events_json":"https://pith.science/api/pith-number/QQYT5Q5CV7R4RG2GTDYGIT5X7T/events.json","paper":"https://pith.science/paper/QQYT5Q5C"},"agent_actions":{"view_html":"https://pith.science/pith/QQYT5Q5CV7R4RG2GTDYGIT5X7T","download_json":"https://pith.science/pith/QQYT5Q5CV7R4RG2GTDYGIT5X7T.json","view_paper":"https://pith.science/paper/QQYT5Q5C","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1501.03932&json=true","fetch_graph":"https://pith.science/api/pith-number/QQYT5Q5CV7R4RG2GTDYGIT5X7T/graph.json","fetch_events":"https://pith.science/api/pith-number/QQYT5Q5CV7R4RG2GTDYGIT5X7T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QQYT5Q5CV7R4RG2GTDYGIT5X7T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QQYT5Q5CV7R4RG2GTDYGIT5X7T/action/storage_attestation","attest_author":"https://pith.science/pith/QQYT5Q5CV7R4RG2GTDYGIT5X7T/action/author_attestation","sign_citation":"https://pith.science/pith/QQYT5Q5CV7R4RG2GTDYGIT5X7T/action/citation_signature","submit_replication":"https://pith.science/pith/QQYT5Q5CV7R4RG2GTDYGIT5X7T/action/replication_record"}},"created_at":"2026-05-18T02:29:16.727793+00:00","updated_at":"2026-05-18T02:29:16.727793+00:00"}