{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:72VKYGNQR4CDRXADOR6JLJC2D2","short_pith_number":"pith:72VKYGNQ","canonical_record":{"source":{"id":"1003.2759","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-03-14T05:43:28Z","cross_cats_sorted":[],"title_canon_sha256":"a8aadcd6043afa3746f3d14a52057b19dea6d32bd3abfed9e44d51c135b0db22","abstract_canon_sha256":"e6aea97f17601422b22a034b11b133b3fe06ad9f9f3046da920257c64d1c7295"},"schema_version":"1.0"},"canonical_sha256":"feaaac19b08f0438dc03747c95a45a1e8c5e9dd2c34f9ee672abb26baac0a538","source":{"kind":"arxiv","id":"1003.2759","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.2759","created_at":"2026-05-18T04:18:34Z"},{"alias_kind":"arxiv_version","alias_value":"1003.2759v2","created_at":"2026-05-18T04:18:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.2759","created_at":"2026-05-18T04:18:34Z"},{"alias_kind":"pith_short_12","alias_value":"72VKYGNQR4CD","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"72VKYGNQR4CDRXAD","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"72VKYGNQ","created_at":"2026-05-18T12:26:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:72VKYGNQR4CDRXADOR6JLJC2D2","target":"record","payload":{"canonical_record":{"source":{"id":"1003.2759","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-03-14T05:43:28Z","cross_cats_sorted":[],"title_canon_sha256":"a8aadcd6043afa3746f3d14a52057b19dea6d32bd3abfed9e44d51c135b0db22","abstract_canon_sha256":"e6aea97f17601422b22a034b11b133b3fe06ad9f9f3046da920257c64d1c7295"},"schema_version":"1.0"},"canonical_sha256":"feaaac19b08f0438dc03747c95a45a1e8c5e9dd2c34f9ee672abb26baac0a538","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:34.807236Z","signature_b64":"GUViUJot38LAQdKV2/SU8BfxqyloyXbrzLJMPATA85lhNs0budWoAKmd1XsiVIKXYRUFNEil7xU0QX4f5LlYBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"feaaac19b08f0438dc03747c95a45a1e8c5e9dd2c34f9ee672abb26baac0a538","last_reissued_at":"2026-05-18T04:18:34.806736Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:34.806736Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1003.2759","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":"2026-05-18T04:18:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xaH5Yzdo20je9oGkeivXxeI++LvfvdIqmJM9CjcZfql0JALONjEL2VZS4YGvBMtkUQJK3i8o+GO/LMOlDhFUAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T08:56:29.460364Z"},"content_sha256":"84214dc0df86c799c105c51d426cbe969e9356bf9b7902643f57a7ae3a159f7d","schema_version":"1.0","event_id":"sha256:84214dc0df86c799c105c51d426cbe969e9356bf9b7902643f57a7ae3a159f7d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:72VKYGNQR4CDRXADOR6JLJC2D2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Locally homogeneous geometric manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"William M. Goldman","submitted_at":"2010-03-14T05:43:28Z","abstract_excerpt":"Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology S and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of G/H into S. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidea"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.2759","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":"2026-05-18T04:18:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pTWw7xfTyTJ3RC91yq9mkfCunYdnYUq9P0H53yuBKyIOuHaaV4So0YuYq/kQUKKQWzC7HdBlMCtkmkl2xNNqAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T08:56:29.460720Z"},"content_sha256":"79c5ecd61052055106e08b750459718b0b0d20c5619ee8b3a0c8302744b4ce01","schema_version":"1.0","event_id":"sha256:79c5ecd61052055106e08b750459718b0b0d20c5619ee8b3a0c8302744b4ce01"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/72VKYGNQR4CDRXADOR6JLJC2D2/bundle.json","state_url":"https://pith.science/pith/72VKYGNQR4CDRXADOR6JLJC2D2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/72VKYGNQR4CDRXADOR6JLJC2D2/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":"2026-05-30T08:56:29Z","links":{"resolver":"https://pith.science/pith/72VKYGNQR4CDRXADOR6JLJC2D2","bundle":"https://pith.science/pith/72VKYGNQR4CDRXADOR6JLJC2D2/bundle.json","state":"https://pith.science/pith/72VKYGNQR4CDRXADOR6JLJC2D2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/72VKYGNQR4CDRXADOR6JLJC2D2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:72VKYGNQR4CDRXADOR6JLJC2D2","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":"e6aea97f17601422b22a034b11b133b3fe06ad9f9f3046da920257c64d1c7295","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-03-14T05:43:28Z","title_canon_sha256":"a8aadcd6043afa3746f3d14a52057b19dea6d32bd3abfed9e44d51c135b0db22"},"schema_version":"1.0","source":{"id":"1003.2759","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.2759","created_at":"2026-05-18T04:18:34Z"},{"alias_kind":"arxiv_version","alias_value":"1003.2759v2","created_at":"2026-05-18T04:18:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.2759","created_at":"2026-05-18T04:18:34Z"},{"alias_kind":"pith_short_12","alias_value":"72VKYGNQR4CD","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"72VKYGNQR4CDRXAD","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"72VKYGNQ","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:79c5ecd61052055106e08b750459718b0b0d20c5619ee8b3a0c8302744b4ce01","target":"graph","created_at":"2026-05-18T04:18:34Z","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":"Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology S and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of G/H into S. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidea","authors_text":"William M. Goldman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-03-14T05:43:28Z","title":"Locally homogeneous geometric manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.2759","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":"sha256:84214dc0df86c799c105c51d426cbe969e9356bf9b7902643f57a7ae3a159f7d","target":"record","created_at":"2026-05-18T04:18:34Z","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":"e6aea97f17601422b22a034b11b133b3fe06ad9f9f3046da920257c64d1c7295","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-03-14T05:43:28Z","title_canon_sha256":"a8aadcd6043afa3746f3d14a52057b19dea6d32bd3abfed9e44d51c135b0db22"},"schema_version":"1.0","source":{"id":"1003.2759","kind":"arxiv","version":2}},"canonical_sha256":"feaaac19b08f0438dc03747c95a45a1e8c5e9dd2c34f9ee672abb26baac0a538","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"feaaac19b08f0438dc03747c95a45a1e8c5e9dd2c34f9ee672abb26baac0a538","first_computed_at":"2026-05-18T04:18:34.806736Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:18:34.806736Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GUViUJot38LAQdKV2/SU8BfxqyloyXbrzLJMPATA85lhNs0budWoAKmd1XsiVIKXYRUFNEil7xU0QX4f5LlYBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:18:34.807236Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.2759","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:84214dc0df86c799c105c51d426cbe969e9356bf9b7902643f57a7ae3a159f7d","sha256:79c5ecd61052055106e08b750459718b0b0d20c5619ee8b3a0c8302744b4ce01"],"state_sha256":"23524cbc723524be037a724ca15dc9944ab96385aa85495e9b1bf1523a02f42b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AGOR+KzjBH0gvr9ruQHYygiGZKVGEpe3ZVoTA179uL7p8DekEeGf/5l5i5kr6HzXZ/19ApFTN22lA89WWi2LAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T08:56:29.462835Z","bundle_sha256":"83a31fb3a13e8b3f0092cee8fa58cefc52490646757f71a12d72e2a20a2a257d"}}