{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:GXWO4BXWBXSHLZQ7FPL3D6ESEL","short_pith_number":"pith:GXWO4BXW","schema_version":"1.0","canonical_sha256":"35ecee06f60de475e61f2bd7b1f89222f9d08a29524e168f838235695b6f79c5","source":{"kind":"arxiv","id":"1307.2778","version":3},"attestation_state":"computed","paper":{"title":"Reconstruction and quantization of Riemannian structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["gr-qc"],"primary_cat":"math.QA","authors_text":"Shahn Majid","submitted_at":"2013-07-10T12:51:38Z","abstract_excerpt":"We study how the Riemannian structure on a manifold can be usefully reconstructed from its codifferential $\\delta$, including a formula $\\nabla_\\omega\\eta={1\\over 2}( \\delta(\\omega\\eta)-(\\delta\\omega)\\eta+\\omega(\\delta\\eta) +L_\\omega(\\eta)+i_\\eta d \\omega)$ for the Levi-Civita covariant derivative in terms of 1-forms, where $L, i$ are respectively the Lie derivative and interior product along the corresponding vector fields. The covariant derivative extends naturally along forms of any degree and to possibly degenerate $(\\ ,\\ )$. In the nondegenerate case, $\\delta$ makes the exterior algebra i"},"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":"1307.2778","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2013-07-10T12:51:38Z","cross_cats_sorted":["gr-qc"],"title_canon_sha256":"59572d8bc809482dcd12d81b5773a0d39a016bf71f612cb8a6fee0b1556ef3c5","abstract_canon_sha256":"8a6f6d501778aa09b78b81ee5ee6faef7911f2a9662267a14fa7d061b6b23d73"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:03:31.966792Z","signature_b64":"BNpy0tCS30XQ3NNPhVULHb97DpRWUO4+7TZ8cklHbScCSmSViq1tj6a7rW4vTmckIqhiYYCx4IrGVf2zOAJtAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"35ecee06f60de475e61f2bd7b1f89222f9d08a29524e168f838235695b6f79c5","last_reissued_at":"2026-05-18T03:03:31.964904Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:03:31.964904Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reconstruction and quantization of Riemannian structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["gr-qc"],"primary_cat":"math.QA","authors_text":"Shahn Majid","submitted_at":"2013-07-10T12:51:38Z","abstract_excerpt":"We study how the Riemannian structure on a manifold can be usefully reconstructed from its codifferential $\\delta$, including a formula $\\nabla_\\omega\\eta={1\\over 2}( \\delta(\\omega\\eta)-(\\delta\\omega)\\eta+\\omega(\\delta\\eta) +L_\\omega(\\eta)+i_\\eta d \\omega)$ for the Levi-Civita covariant derivative in terms of 1-forms, where $L, i$ are respectively the Lie derivative and interior product along the corresponding vector fields. The covariant derivative extends naturally along forms of any degree and to possibly degenerate $(\\ ,\\ )$. In the nondegenerate case, $\\delta$ makes the exterior algebra i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.2778","kind":"arxiv","version":3},"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":"1307.2778","created_at":"2026-05-18T03:03:31.965986+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.2778v3","created_at":"2026-05-18T03:03:31.965986+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.2778","created_at":"2026-05-18T03:03:31.965986+00:00"},{"alias_kind":"pith_short_12","alias_value":"GXWO4BXWBXSH","created_at":"2026-05-18T12:27:46.883200+00:00"},{"alias_kind":"pith_short_16","alias_value":"GXWO4BXWBXSHLZQ7","created_at":"2026-05-18T12:27:46.883200+00:00"},{"alias_kind":"pith_short_8","alias_value":"GXWO4BXW","created_at":"2026-05-18T12:27:46.883200+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/GXWO4BXWBXSHLZQ7FPL3D6ESEL","json":"https://pith.science/pith/GXWO4BXWBXSHLZQ7FPL3D6ESEL.json","graph_json":"https://pith.science/api/pith-number/GXWO4BXWBXSHLZQ7FPL3D6ESEL/graph.json","events_json":"https://pith.science/api/pith-number/GXWO4BXWBXSHLZQ7FPL3D6ESEL/events.json","paper":"https://pith.science/paper/GXWO4BXW"},"agent_actions":{"view_html":"https://pith.science/pith/GXWO4BXWBXSHLZQ7FPL3D6ESEL","download_json":"https://pith.science/pith/GXWO4BXWBXSHLZQ7FPL3D6ESEL.json","view_paper":"https://pith.science/paper/GXWO4BXW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.2778&json=true","fetch_graph":"https://pith.science/api/pith-number/GXWO4BXWBXSHLZQ7FPL3D6ESEL/graph.json","fetch_events":"https://pith.science/api/pith-number/GXWO4BXWBXSHLZQ7FPL3D6ESEL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GXWO4BXWBXSHLZQ7FPL3D6ESEL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GXWO4BXWBXSHLZQ7FPL3D6ESEL/action/storage_attestation","attest_author":"https://pith.science/pith/GXWO4BXWBXSHLZQ7FPL3D6ESEL/action/author_attestation","sign_citation":"https://pith.science/pith/GXWO4BXWBXSHLZQ7FPL3D6ESEL/action/citation_signature","submit_replication":"https://pith.science/pith/GXWO4BXWBXSHLZQ7FPL3D6ESEL/action/replication_record"}},"created_at":"2026-05-18T03:03:31.965986+00:00","updated_at":"2026-05-18T03:03:31.965986+00:00"}