{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:ZPKRWJI7U5JLJT2C2GB44FTGLE","short_pith_number":"pith:ZPKRWJI7","canonical_record":{"source":{"id":"1506.04292","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-13T16:35:23Z","cross_cats_sorted":[],"title_canon_sha256":"a8f2e31b8d77b56e19acc8ef61a7faf544dcafb582a1fdaea81693e2c95c9caf","abstract_canon_sha256":"b0d52870f2345e379028e0c7d5b20228d0ab42e1027a3328faa69166767a7b3a"},"schema_version":"1.0"},"canonical_sha256":"cbd51b251fa752b4cf42d183ce16665915bd1b5aa8cb736adc3e3b07aa603a23","source":{"kind":"arxiv","id":"1506.04292","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.04292","created_at":"2026-05-17T23:56:56Z"},{"alias_kind":"arxiv_version","alias_value":"1506.04292v1","created_at":"2026-05-17T23:56:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.04292","created_at":"2026-05-17T23:56:56Z"},{"alias_kind":"pith_short_12","alias_value":"ZPKRWJI7U5JL","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"ZPKRWJI7U5JLJT2C","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"ZPKRWJI7","created_at":"2026-05-18T12:29:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:ZPKRWJI7U5JLJT2C2GB44FTGLE","target":"record","payload":{"canonical_record":{"source":{"id":"1506.04292","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-13T16:35:23Z","cross_cats_sorted":[],"title_canon_sha256":"a8f2e31b8d77b56e19acc8ef61a7faf544dcafb582a1fdaea81693e2c95c9caf","abstract_canon_sha256":"b0d52870f2345e379028e0c7d5b20228d0ab42e1027a3328faa69166767a7b3a"},"schema_version":"1.0"},"canonical_sha256":"cbd51b251fa752b4cf42d183ce16665915bd1b5aa8cb736adc3e3b07aa603a23","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:56.332958Z","signature_b64":"0YiMlOT8HtTM9TeibHtD49DIBtlx1LATrOaJdj9/WX00ogexx3BtDF2tYbt9WkCOL8MZO7aKus65HNYQFUwmCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cbd51b251fa752b4cf42d183ce16665915bd1b5aa8cb736adc3e3b07aa603a23","last_reissued_at":"2026-05-17T23:56:56.332374Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:56.332374Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.04292","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:56:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vnj63QtBpEOXtfwqg1zPR1KBHputKC4FaBFn7kXExkQfFutvIe3Qi35dHMSj7TGkPeUoiRSI36Gu2hvl6aZLBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T16:36:19.638794Z"},"content_sha256":"65dbd5f5e8defb64ef0b480b7fecad9e67804dd6ebbf056641f1e37cede108f6","schema_version":"1.0","event_id":"sha256:65dbd5f5e8defb64ef0b480b7fecad9e67804dd6ebbf056641f1e37cede108f6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:ZPKRWJI7U5JLJT2C2GB44FTGLE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Killing 2-forms in dimension 4","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Andrei Moroianu, Paul Gauduchon","submitted_at":"2015-06-13T16:35:23Z","abstract_excerpt":"A Killing $p$-form on a Riemannian manifold is a $p$-form whose covariant derivative is totally anti-symmetric. In this paper we give the complete (local) description of 4-dimensional Riemannian manifolds (M,g) carrying non-parallel Killing 2-forms $\\varphi$. If $M$ is connected and oriented, we show that there exists a dense open subset of $M$ on which one of the three exclusive situations holds: either $\\phi$ is everywhere degenerate and $g$ is conformal to a product metric, or $g$ is conformal to an ambik\\\"ahler metric obtained via the Calabi construction from a polarized Riemannian surface"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.04292","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:56:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Axwa8ONIdytI4g0ECBJYJqz9/acd/2JnKGkt6fAocH2bBjFv/OqEQug4XEnzbh9oVFJB0skS5huFkvGZya88DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T16:36:19.639420Z"},"content_sha256":"73c1603b503a0a4848749e96d830cfb2abe6ca82d5403e41d1c3840a980f1c42","schema_version":"1.0","event_id":"sha256:73c1603b503a0a4848749e96d830cfb2abe6ca82d5403e41d1c3840a980f1c42"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZPKRWJI7U5JLJT2C2GB44FTGLE/bundle.json","state_url":"https://pith.science/pith/ZPKRWJI7U5JLJT2C2GB44FTGLE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZPKRWJI7U5JLJT2C2GB44FTGLE/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":"2026-05-26T16:36:19Z","links":{"resolver":"https://pith.science/pith/ZPKRWJI7U5JLJT2C2GB44FTGLE","bundle":"https://pith.science/pith/ZPKRWJI7U5JLJT2C2GB44FTGLE/bundle.json","state":"https://pith.science/pith/ZPKRWJI7U5JLJT2C2GB44FTGLE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZPKRWJI7U5JLJT2C2GB44FTGLE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:ZPKRWJI7U5JLJT2C2GB44FTGLE","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":"b0d52870f2345e379028e0c7d5b20228d0ab42e1027a3328faa69166767a7b3a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-13T16:35:23Z","title_canon_sha256":"a8f2e31b8d77b56e19acc8ef61a7faf544dcafb582a1fdaea81693e2c95c9caf"},"schema_version":"1.0","source":{"id":"1506.04292","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.04292","created_at":"2026-05-17T23:56:56Z"},{"alias_kind":"arxiv_version","alias_value":"1506.04292v1","created_at":"2026-05-17T23:56:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.04292","created_at":"2026-05-17T23:56:56Z"},{"alias_kind":"pith_short_12","alias_value":"ZPKRWJI7U5JL","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"ZPKRWJI7U5JLJT2C","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"ZPKRWJI7","created_at":"2026-05-18T12:29:52Z"}],"graph_snapshots":[{"event_id":"sha256:73c1603b503a0a4848749e96d830cfb2abe6ca82d5403e41d1c3840a980f1c42","target":"graph","created_at":"2026-05-17T23:56:56Z","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":"A Killing $p$-form on a Riemannian manifold is a $p$-form whose covariant derivative is totally anti-symmetric. In this paper we give the complete (local) description of 4-dimensional Riemannian manifolds (M,g) carrying non-parallel Killing 2-forms $\\varphi$. If $M$ is connected and oriented, we show that there exists a dense open subset of $M$ on which one of the three exclusive situations holds: either $\\phi$ is everywhere degenerate and $g$ is conformal to a product metric, or $g$ is conformal to an ambik\\\"ahler metric obtained via the Calabi construction from a polarized Riemannian surface","authors_text":"Andrei Moroianu, Paul Gauduchon","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-13T16:35:23Z","title":"Killing 2-forms in dimension 4"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.04292","kind":"arxiv","version":1},"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":"sha256:65dbd5f5e8defb64ef0b480b7fecad9e67804dd6ebbf056641f1e37cede108f6","target":"record","created_at":"2026-05-17T23:56:56Z","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":"b0d52870f2345e379028e0c7d5b20228d0ab42e1027a3328faa69166767a7b3a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-13T16:35:23Z","title_canon_sha256":"a8f2e31b8d77b56e19acc8ef61a7faf544dcafb582a1fdaea81693e2c95c9caf"},"schema_version":"1.0","source":{"id":"1506.04292","kind":"arxiv","version":1}},"canonical_sha256":"cbd51b251fa752b4cf42d183ce16665915bd1b5aa8cb736adc3e3b07aa603a23","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cbd51b251fa752b4cf42d183ce16665915bd1b5aa8cb736adc3e3b07aa603a23","first_computed_at":"2026-05-17T23:56:56.332374Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:56:56.332374Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0YiMlOT8HtTM9TeibHtD49DIBtlx1LATrOaJdj9/WX00ogexx3BtDF2tYbt9WkCOL8MZO7aKus65HNYQFUwmCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:56:56.332958Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.04292","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:65dbd5f5e8defb64ef0b480b7fecad9e67804dd6ebbf056641f1e37cede108f6","sha256:73c1603b503a0a4848749e96d830cfb2abe6ca82d5403e41d1c3840a980f1c42"],"state_sha256":"4ab669eb08aa5260c0c80d1a0742ea65ab732c7df20ae6a3b677f12eda47be65"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zbUPzeM9SVa5nJAlxxZC7dLBwdoctmsmYes1jjs7dM+6/Ub2Z1MWQiFXtecbtRy1mAyHRVV2W6DIEiAB4PuKDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T16:36:19.642256Z","bundle_sha256":"7ca8d07a746e2fee9414a19291cb144cf8f1ca501b68402d2326786f2ef94112"}}