{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:J47XJGKSTHUYR2X45VPCJCIES5","short_pith_number":"pith:J47XJGKS","canonical_record":{"source":{"id":"1109.4878","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-09-22T16:34:41Z","cross_cats_sorted":[],"title_canon_sha256":"36db7d240521cf669a18454404780c8d25cc90a0c6a3d5a7df7119a5bd34d347","abstract_canon_sha256":"2066f957b87eb610056964cc0fce9c564a5fbc6523ffa9015dba333deef327a9"},"schema_version":"1.0"},"canonical_sha256":"4f3f74995299e988eafced5e2489049745fe3cd61e53e4dc123df4140e65ff8d","source":{"kind":"arxiv","id":"1109.4878","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.4878","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"arxiv_version","alias_value":"1109.4878v3","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.4878","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"pith_short_12","alias_value":"J47XJGKSTHUY","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"J47XJGKSTHUYR2X4","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"J47XJGKS","created_at":"2026-05-18T12:26:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:J47XJGKSTHUYR2X45VPCJCIES5","target":"record","payload":{"canonical_record":{"source":{"id":"1109.4878","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-09-22T16:34:41Z","cross_cats_sorted":[],"title_canon_sha256":"36db7d240521cf669a18454404780c8d25cc90a0c6a3d5a7df7119a5bd34d347","abstract_canon_sha256":"2066f957b87eb610056964cc0fce9c564a5fbc6523ffa9015dba333deef327a9"},"schema_version":"1.0"},"canonical_sha256":"4f3f74995299e988eafced5e2489049745fe3cd61e53e4dc123df4140e65ff8d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:00.444497Z","signature_b64":"uCDlgnpq5dPjeGS2yKsNzvx/LsWyRCebO1UuA2nJUp54uJtoHa+pohwTecVnS3mY2TtiWXMFy0Mk8SyTPjv1CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4f3f74995299e988eafced5e2489049745fe3cd61e53e4dc123df4140e65ff8d","last_reissued_at":"2026-05-18T02:58:00.443803Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:00.443803Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.4878","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:58:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"siLvWcTGyCxdSb5Nj9WetwerNPyONKGBuSHgngnViWJ/QsghVu2YAnukbXIAG+dx/9O9Qt4c0zrto1AQNk4RAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T14:43:14.682191Z"},"content_sha256":"a3a268c4ab5203597c22367210078c40b422104487b150aad45697779fde767a","schema_version":"1.0","event_id":"sha256:a3a268c4ab5203597c22367210078c40b422104487b150aad45697779fde767a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:J47XJGKSTHUYR2X45VPCJCIES5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Isometry groups of Alexandrov spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Fernando Galaz-Garcia, Luis Guijarro","submitted_at":"2011-09-22T16:34:41Z","abstract_excerpt":"For an Alexandrov space (with curvature bounded below), we determine the maximal dimension of its isometry group and show that the space is isometric to a Riemannian manifold, provided the dimension of its isometry group is maximal. We also determine a gap in the possible dimensions of the isometry groups and show that if the Alexandrov space is symmetric, then it is isometric to a Riemannian manifold."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4878","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:58:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fEKZaNuSgLaTQsr+Kgt4/bV+9gbgre0+QRq3UNqV5D+ptKtFl8JU6UxogwUHsEUJ2gFNs4oYCq53F+by9FzABA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T14:43:14.682548Z"},"content_sha256":"2dd43bac746cb31424aea22086178a7836266d8cf72e780b59bd25a2aa0e9f28","schema_version":"1.0","event_id":"sha256:2dd43bac746cb31424aea22086178a7836266d8cf72e780b59bd25a2aa0e9f28"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J47XJGKSTHUYR2X45VPCJCIES5/bundle.json","state_url":"https://pith.science/pith/J47XJGKSTHUYR2X45VPCJCIES5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J47XJGKSTHUYR2X45VPCJCIES5/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":"2026-06-07T14:43:14Z","links":{"resolver":"https://pith.science/pith/J47XJGKSTHUYR2X45VPCJCIES5","bundle":"https://pith.science/pith/J47XJGKSTHUYR2X45VPCJCIES5/bundle.json","state":"https://pith.science/pith/J47XJGKSTHUYR2X45VPCJCIES5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J47XJGKSTHUYR2X45VPCJCIES5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:J47XJGKSTHUYR2X45VPCJCIES5","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":"2066f957b87eb610056964cc0fce9c564a5fbc6523ffa9015dba333deef327a9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-09-22T16:34:41Z","title_canon_sha256":"36db7d240521cf669a18454404780c8d25cc90a0c6a3d5a7df7119a5bd34d347"},"schema_version":"1.0","source":{"id":"1109.4878","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.4878","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"arxiv_version","alias_value":"1109.4878v3","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.4878","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"pith_short_12","alias_value":"J47XJGKSTHUY","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"J47XJGKSTHUYR2X4","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"J47XJGKS","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:2dd43bac746cb31424aea22086178a7836266d8cf72e780b59bd25a2aa0e9f28","target":"graph","created_at":"2026-05-18T02:58:00Z","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":"For an Alexandrov space (with curvature bounded below), we determine the maximal dimension of its isometry group and show that the space is isometric to a Riemannian manifold, provided the dimension of its isometry group is maximal. We also determine a gap in the possible dimensions of the isometry groups and show that if the Alexandrov space is symmetric, then it is isometric to a Riemannian manifold.","authors_text":"Fernando Galaz-Garcia, Luis Guijarro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-09-22T16:34:41Z","title":"Isometry groups of Alexandrov spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4878","kind":"arxiv","version":3},"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:a3a268c4ab5203597c22367210078c40b422104487b150aad45697779fde767a","target":"record","created_at":"2026-05-18T02:58:00Z","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":"2066f957b87eb610056964cc0fce9c564a5fbc6523ffa9015dba333deef327a9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-09-22T16:34:41Z","title_canon_sha256":"36db7d240521cf669a18454404780c8d25cc90a0c6a3d5a7df7119a5bd34d347"},"schema_version":"1.0","source":{"id":"1109.4878","kind":"arxiv","version":3}},"canonical_sha256":"4f3f74995299e988eafced5e2489049745fe3cd61e53e4dc123df4140e65ff8d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4f3f74995299e988eafced5e2489049745fe3cd61e53e4dc123df4140e65ff8d","first_computed_at":"2026-05-18T02:58:00.443803Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:00.443803Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uCDlgnpq5dPjeGS2yKsNzvx/LsWyRCebO1UuA2nJUp54uJtoHa+pohwTecVnS3mY2TtiWXMFy0Mk8SyTPjv1CA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:00.444497Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.4878","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a3a268c4ab5203597c22367210078c40b422104487b150aad45697779fde767a","sha256:2dd43bac746cb31424aea22086178a7836266d8cf72e780b59bd25a2aa0e9f28"],"state_sha256":"88c65cf62225dfead50cd5c86e65759299ff0851999f5ad57131762e0ae83fa8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8EqDycqdzCf3cb2yn5zwDUyRi2DIQyMaNgHWUKBht0/Kdndz4yGHoCwVwekajISlV6HIoMYM4UF+1meq2pjhDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T14:43:14.684762Z","bundle_sha256":"860192f8be2f05bce35b488062f74c38d7cb842564ede5999785f89e7d193623"}}