{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:7N7REW7BAQMXKHZN54KEDKORYZ","short_pith_number":"pith:7N7REW7B","canonical_record":{"source":{"id":"1209.3817","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-09-18T00:09:32Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"39db661e7984131b156deabeb4746208848e28e2ac957098041b7c24b8f25d7e","abstract_canon_sha256":"de5c66e3195968288f5673bc62d26d68d0661a4069e3f6fab8a792185da4208e"},"schema_version":"1.0"},"canonical_sha256":"fb7f125be10419751f2def1441a9d1c64b1e20937e5ae39e97a44194e59707cd","source":{"kind":"arxiv","id":"1209.3817","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3817","created_at":"2026-05-17T23:57:51Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3817v5","created_at":"2026-05-17T23:57:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3817","created_at":"2026-05-17T23:57:51Z"},{"alias_kind":"pith_short_12","alias_value":"7N7REW7BAQMX","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7N7REW7BAQMXKHZN","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7N7REW7B","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:7N7REW7BAQMXKHZN54KEDKORYZ","target":"record","payload":{"canonical_record":{"source":{"id":"1209.3817","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-09-18T00:09:32Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"39db661e7984131b156deabeb4746208848e28e2ac957098041b7c24b8f25d7e","abstract_canon_sha256":"de5c66e3195968288f5673bc62d26d68d0661a4069e3f6fab8a792185da4208e"},"schema_version":"1.0"},"canonical_sha256":"fb7f125be10419751f2def1441a9d1c64b1e20937e5ae39e97a44194e59707cd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:51.228778Z","signature_b64":"EId/8W14U2lVyLK9+eGvDyyV69yf/azbRRRe7FMPZzGZ3NF9cyHT7yvdHFxAodMD/zhAexK8qIBcUWItaqcjAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb7f125be10419751f2def1441a9d1c64b1e20937e5ae39e97a44194e59707cd","last_reissued_at":"2026-05-17T23:57:51.228012Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:51.228012Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.3817","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:57:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XzcZYaobBXjnKwfQaGDnXCCcYy49OHs+l2AszaI3EJkUz9DrqJnVanDQYVDudqF8YZY40AzMiVem7a3kssx7Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:03:05.610091Z"},"content_sha256":"ce24428ffab2a977231c7f4a4a1d4202502cd8817b591c97ab1769730cb0d497","schema_version":"1.0","event_id":"sha256:ce24428ffab2a977231c7f4a4a1d4202502cd8817b591c97ab1769730cb0d497"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:7N7REW7BAQMXKHZN54KEDKORYZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Haefliger cohomology of complete Riemannian foliations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.DG","authors_text":"Hiraku Nozawa","submitted_at":"2012-09-18T00:09:32Z","abstract_excerpt":"Haefliger cohomology characterizes taut foliated manifolds by Haefliger's theorem. We show that Haefliger cohomology characterizes strongly tense foliated manifolds, namely, foliated manifolds which admit a Riemannian metric such that the mean curvature form of the leaves is closed and basic. We show that Haefliger cohomology is dual to invariant cohomology for complete Riemannian foliations. As an application of these results, we prove that any complete Riemannian foliation is strongly tense, which is a generalization of Dom\\'{i}nguez's tenseness theorem for Riemannian foliations on closed ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3817","kind":"arxiv","version":5},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:57:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OecajtInWbsZDtGoHhfsydmZpDhrBxSj48c/jD2LeX1K/i4i5A8/MH42pXG62GoMfiJNMV+W1hCCoG1lMEemCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:03:05.610921Z"},"content_sha256":"cf75719618572f2c8f7d53d67b0c7a5cbac750d2ccd816dd1f9c470677eb34ce","schema_version":"1.0","event_id":"sha256:cf75719618572f2c8f7d53d67b0c7a5cbac750d2ccd816dd1f9c470677eb34ce"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7N7REW7BAQMXKHZN54KEDKORYZ/bundle.json","state_url":"https://pith.science/pith/7N7REW7BAQMXKHZN54KEDKORYZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7N7REW7BAQMXKHZN54KEDKORYZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-11T06:03:05Z","links":{"resolver":"https://pith.science/pith/7N7REW7BAQMXKHZN54KEDKORYZ","bundle":"https://pith.science/pith/7N7REW7BAQMXKHZN54KEDKORYZ/bundle.json","state":"https://pith.science/pith/7N7REW7BAQMXKHZN54KEDKORYZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7N7REW7BAQMXKHZN54KEDKORYZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:7N7REW7BAQMXKHZN54KEDKORYZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"de5c66e3195968288f5673bc62d26d68d0661a4069e3f6fab8a792185da4208e","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-09-18T00:09:32Z","title_canon_sha256":"39db661e7984131b156deabeb4746208848e28e2ac957098041b7c24b8f25d7e"},"schema_version":"1.0","source":{"id":"1209.3817","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3817","created_at":"2026-05-17T23:57:51Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3817v5","created_at":"2026-05-17T23:57:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3817","created_at":"2026-05-17T23:57:51Z"},{"alias_kind":"pith_short_12","alias_value":"7N7REW7BAQMX","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7N7REW7BAQMXKHZN","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7N7REW7B","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:cf75719618572f2c8f7d53d67b0c7a5cbac750d2ccd816dd1f9c470677eb34ce","target":"graph","created_at":"2026-05-17T23:57:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Haefliger cohomology characterizes taut foliated manifolds by Haefliger's theorem. We show that Haefliger cohomology characterizes strongly tense foliated manifolds, namely, foliated manifolds which admit a Riemannian metric such that the mean curvature form of the leaves is closed and basic. We show that Haefliger cohomology is dual to invariant cohomology for complete Riemannian foliations. As an application of these results, we prove that any complete Riemannian foliation is strongly tense, which is a generalization of Dom\\'{i}nguez's tenseness theorem for Riemannian foliations on closed ma","authors_text":"Hiraku Nozawa","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-09-18T00:09:32Z","title":"Haefliger cohomology of complete Riemannian foliations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3817","kind":"arxiv","version":5},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ce24428ffab2a977231c7f4a4a1d4202502cd8817b591c97ab1769730cb0d497","target":"record","created_at":"2026-05-17T23:57:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"de5c66e3195968288f5673bc62d26d68d0661a4069e3f6fab8a792185da4208e","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-09-18T00:09:32Z","title_canon_sha256":"39db661e7984131b156deabeb4746208848e28e2ac957098041b7c24b8f25d7e"},"schema_version":"1.0","source":{"id":"1209.3817","kind":"arxiv","version":5}},"canonical_sha256":"fb7f125be10419751f2def1441a9d1c64b1e20937e5ae39e97a44194e59707cd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb7f125be10419751f2def1441a9d1c64b1e20937e5ae39e97a44194e59707cd","first_computed_at":"2026-05-17T23:57:51.228012Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:57:51.228012Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EId/8W14U2lVyLK9+eGvDyyV69yf/azbRRRe7FMPZzGZ3NF9cyHT7yvdHFxAodMD/zhAexK8qIBcUWItaqcjAA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:57:51.228778Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.3817","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ce24428ffab2a977231c7f4a4a1d4202502cd8817b591c97ab1769730cb0d497","sha256:cf75719618572f2c8f7d53d67b0c7a5cbac750d2ccd816dd1f9c470677eb34ce"],"state_sha256":"e9cf3fd1b22df3ca4a802b4255fe49f2ed144f7e994bd8089a6aee9f957034b5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IJh0bbsUB+7goFNOxM+VVouu9jF7479HV9MCOm7AY33bowDwprk23FN38Dd7ggi4rKB+H0hY6KyNFos9Ok0XDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T06:03:05.615072Z","bundle_sha256":"73429078a583f664b29ad9619e4d72e619db124836d3dd886e1023336adfe576"}}