{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:2E76X7LPG54RRAUI2RI4N6RUWU","short_pith_number":"pith:2E76X7LP","canonical_record":{"source":{"id":"1106.4720","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-06-23T13:54:01Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"549875465961864479be6ed562aef358802bc51a798aa1913db81d7add5ea788","abstract_canon_sha256":"abf1083c9f7de05e18b2fb562c0ec9ba72a758113aad022ef55327c7e5b10084"},"schema_version":"1.0"},"canonical_sha256":"d13febfd6f3779188288d451c6fa34b50a87633ca3565c5d8b8db6bedb0ebf0a","source":{"kind":"arxiv","id":"1106.4720","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.4720","created_at":"2026-05-18T04:19:22Z"},{"alias_kind":"arxiv_version","alias_value":"1106.4720v2","created_at":"2026-05-18T04:19:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.4720","created_at":"2026-05-18T04:19:22Z"},{"alias_kind":"pith_short_12","alias_value":"2E76X7LPG54R","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"2E76X7LPG54RRAUI","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"2E76X7LP","created_at":"2026-05-18T12:26:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:2E76X7LPG54RRAUI2RI4N6RUWU","target":"record","payload":{"canonical_record":{"source":{"id":"1106.4720","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-06-23T13:54:01Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"549875465961864479be6ed562aef358802bc51a798aa1913db81d7add5ea788","abstract_canon_sha256":"abf1083c9f7de05e18b2fb562c0ec9ba72a758113aad022ef55327c7e5b10084"},"schema_version":"1.0"},"canonical_sha256":"d13febfd6f3779188288d451c6fa34b50a87633ca3565c5d8b8db6bedb0ebf0a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:22.860051Z","signature_b64":"FedQ2D9lALvkr1gmg5Vm9n/XQ8314goG3/2RHCjPdDIUG2mgDtpVtLnTBENA3ZKrK2RrgeuJ2y1shYuRnKgWDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d13febfd6f3779188288d451c6fa34b50a87633ca3565c5d8b8db6bedb0ebf0a","last_reissued_at":"2026-05-18T04:19:22.859672Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:22.859672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1106.4720","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:19:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YcB6cG8s/ZzGxT7Z5th45ejja4YBX6+J9hQtIoRI8jKZ937MgXgg5OnIboTCPT/LWB2H0en2l0tlia/eGmadDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T15:41:46.447845Z"},"content_sha256":"c50214c751711e441ba207d50fb55d8fe62a3c5dce73ca458aea8a87f0d120e8","schema_version":"1.0","event_id":"sha256:c50214c751711e441ba207d50fb55d8fe62a3c5dce73ca458aea8a87f0d120e8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:2E76X7LPG54RRAUI2RI4N6RUWU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Homogeneous spaces, dynamics, cosmology: Geometric flows and rational dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.DG","authors_text":"Abdelghani Zeghib (UMPA-ENSL)","submitted_at":"2011-06-23T13:54:01Z","abstract_excerpt":"The Ricci flow is a parabolic evolution equation in the space of Riemannian metrics of a smooth manifold. To some extent, Einstein equations give rise to a similar hyperbolic evolution. The present text is an introductory exposition to Bianchi-Ricci and Bianchi-Einstein flows, that is, the restricted finitely dimensional dynamical systems, obtained by considering homogeneous metrics."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.4720","kind":"arxiv","version":2},"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":"2026-05-18T04:19:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9Mw+jZknHfuekXbin3/7UKzcZyW+MyDFsVxx4t7YS1b7fdoYwgmjyCo1mAEm7LJDqfteqo8sKryCoTsBkNy/Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T15:41:46.448180Z"},"content_sha256":"2fe752d998ab9cb46087b6d06456e328c9db1e85775980d144bbed825becc580","schema_version":"1.0","event_id":"sha256:2fe752d998ab9cb46087b6d06456e328c9db1e85775980d144bbed825becc580"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2E76X7LPG54RRAUI2RI4N6RUWU/bundle.json","state_url":"https://pith.science/pith/2E76X7LPG54RRAUI2RI4N6RUWU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2E76X7LPG54RRAUI2RI4N6RUWU/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":"2026-06-21T15:41:46Z","links":{"resolver":"https://pith.science/pith/2E76X7LPG54RRAUI2RI4N6RUWU","bundle":"https://pith.science/pith/2E76X7LPG54RRAUI2RI4N6RUWU/bundle.json","state":"https://pith.science/pith/2E76X7LPG54RRAUI2RI4N6RUWU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2E76X7LPG54RRAUI2RI4N6RUWU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:2E76X7LPG54RRAUI2RI4N6RUWU","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":"abf1083c9f7de05e18b2fb562c0ec9ba72a758113aad022ef55327c7e5b10084","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-06-23T13:54:01Z","title_canon_sha256":"549875465961864479be6ed562aef358802bc51a798aa1913db81d7add5ea788"},"schema_version":"1.0","source":{"id":"1106.4720","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.4720","created_at":"2026-05-18T04:19:22Z"},{"alias_kind":"arxiv_version","alias_value":"1106.4720v2","created_at":"2026-05-18T04:19:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.4720","created_at":"2026-05-18T04:19:22Z"},{"alias_kind":"pith_short_12","alias_value":"2E76X7LPG54R","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"2E76X7LPG54RRAUI","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"2E76X7LP","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:2fe752d998ab9cb46087b6d06456e328c9db1e85775980d144bbed825becc580","target":"graph","created_at":"2026-05-18T04:19:22Z","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":"The Ricci flow is a parabolic evolution equation in the space of Riemannian metrics of a smooth manifold. To some extent, Einstein equations give rise to a similar hyperbolic evolution. The present text is an introductory exposition to Bianchi-Ricci and Bianchi-Einstein flows, that is, the restricted finitely dimensional dynamical systems, obtained by considering homogeneous metrics.","authors_text":"Abdelghani Zeghib (UMPA-ENSL)","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-06-23T13:54:01Z","title":"Homogeneous spaces, dynamics, cosmology: Geometric flows and rational dynamics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.4720","kind":"arxiv","version":2},"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:c50214c751711e441ba207d50fb55d8fe62a3c5dce73ca458aea8a87f0d120e8","target":"record","created_at":"2026-05-18T04:19:22Z","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":"abf1083c9f7de05e18b2fb562c0ec9ba72a758113aad022ef55327c7e5b10084","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-06-23T13:54:01Z","title_canon_sha256":"549875465961864479be6ed562aef358802bc51a798aa1913db81d7add5ea788"},"schema_version":"1.0","source":{"id":"1106.4720","kind":"arxiv","version":2}},"canonical_sha256":"d13febfd6f3779188288d451c6fa34b50a87633ca3565c5d8b8db6bedb0ebf0a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d13febfd6f3779188288d451c6fa34b50a87633ca3565c5d8b8db6bedb0ebf0a","first_computed_at":"2026-05-18T04:19:22.859672Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:19:22.859672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FedQ2D9lALvkr1gmg5Vm9n/XQ8314goG3/2RHCjPdDIUG2mgDtpVtLnTBENA3ZKrK2RrgeuJ2y1shYuRnKgWDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:19:22.860051Z","signed_message":"canonical_sha256_bytes"},"source_id":"1106.4720","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c50214c751711e441ba207d50fb55d8fe62a3c5dce73ca458aea8a87f0d120e8","sha256:2fe752d998ab9cb46087b6d06456e328c9db1e85775980d144bbed825becc580"],"state_sha256":"e83f3c532dd65978a92524edb97b4b475bf55d6ca02c0abfd1f2dd135721c802"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ibBQrduWFXxIDP2IvFTPW56lCU/FtjSIm1lFGMGEHD5pu2ga1kpnPxCCFg282uk983WCHpnVNgA6PFEjgnOfCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T15:41:46.450120Z","bundle_sha256":"743eb0b3813eb8484d41d3b0e5b6b12c33ebcad1c1e118adcbbc73fa22041a3c"}}