{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:6LYJLABROBMSIVQKEN3QYVO6R3","short_pith_number":"pith:6LYJLABR","canonical_record":{"source":{"id":"1510.06782","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-22T22:41:21Z","cross_cats_sorted":[],"title_canon_sha256":"eba247368f5cc242db822fa420df8b8b5a7b5a42134760e1a706a05ff3411c4b","abstract_canon_sha256":"13ce40eecd544a37cdec553c157d02781f53eeea358577ed0481e18252db7f7f"},"schema_version":"1.0"},"canonical_sha256":"f2f0958031705924560a23770c55de8eff6c1bd876507d16e967f0a92353d35c","source":{"kind":"arxiv","id":"1510.06782","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.06782","created_at":"2026-05-18T01:06:29Z"},{"alias_kind":"arxiv_version","alias_value":"1510.06782v2","created_at":"2026-05-18T01:06:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.06782","created_at":"2026-05-18T01:06:29Z"},{"alias_kind":"pith_short_12","alias_value":"6LYJLABROBMS","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6LYJLABROBMSIVQK","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6LYJLABR","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:6LYJLABROBMSIVQKEN3QYVO6R3","target":"record","payload":{"canonical_record":{"source":{"id":"1510.06782","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-22T22:41:21Z","cross_cats_sorted":[],"title_canon_sha256":"eba247368f5cc242db822fa420df8b8b5a7b5a42134760e1a706a05ff3411c4b","abstract_canon_sha256":"13ce40eecd544a37cdec553c157d02781f53eeea358577ed0481e18252db7f7f"},"schema_version":"1.0"},"canonical_sha256":"f2f0958031705924560a23770c55de8eff6c1bd876507d16e967f0a92353d35c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:06:29.088453Z","signature_b64":"2DOXECGgAj0WUW6K/5NTGgZNWJSE8I2q95lVAU1YjZLZfTPYoopjIallTpycQHkp4g8OHs8/C6s/mGlIklvxCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f2f0958031705924560a23770c55de8eff6c1bd876507d16e967f0a92353d35c","last_reissued_at":"2026-05-18T01:06:29.087811Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:06:29.087811Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.06782","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":"2026-05-18T01:06:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yu8AxiwoHWpI1aL1Z/xO2GocBLxhHufC741fptPpxZ531vadhRSx03K75KhHeIH2fViNjKzv/0n6eMf1alDlDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:45:46.055489Z"},"content_sha256":"57d5f9df7018173a0792fdbbedf18e5c1c3d9274b9a4e0bb78f20ae3180b6504","schema_version":"1.0","event_id":"sha256:57d5f9df7018173a0792fdbbedf18e5c1c3d9274b9a4e0bb78f20ae3180b6504"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:6LYJLABROBMSIVQKEN3QYVO6R3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Pseudo-Riemannian $\\mathrm{G}_{2(2)}$-manifolds with dimension at most $21$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"R. Quiroga-Barranco","submitted_at":"2015-10-22T22:41:21Z","abstract_excerpt":"Let $G_{2(2)}$ be the non-compact connected simple Lie group of type $G_2$ over $\\mathbb{R}$, and let $M$ be a connected analytic complete pseudo-Riemannian manifold that admits an isometric $G_{2(2)}$-action with a dense orbit. For the case $\\dim(M) \\leq 21$, we provide a full description of the manifold $M$, its geometry and its $G_{2(2)}$-action. The latter are always given in terms of a Lie group geometry related to $G_{2(2)}$, and in one case $M$ is essentially the quotient of $\\widetilde{\\mathrm{S0}}_0(3,4)$ by a lattice."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.06782","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":"2026-05-18T01:06:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ebf/x+6ooL/qqcleLtSpeiWxlAGvZkhh1OphSmoGEWjrIaoGGa4FbKYIyZ9WfPWbOp6RYZXGmxYMIE6ralgmDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:45:46.055833Z"},"content_sha256":"3e20a449937bb42b628c3ef6e769ff174d4ccdb05d6e2c6fb710276426fc17c7","schema_version":"1.0","event_id":"sha256:3e20a449937bb42b628c3ef6e769ff174d4ccdb05d6e2c6fb710276426fc17c7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6LYJLABROBMSIVQKEN3QYVO6R3/bundle.json","state_url":"https://pith.science/pith/6LYJLABROBMSIVQKEN3QYVO6R3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6LYJLABROBMSIVQKEN3QYVO6R3/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":"2026-06-23T16:45:46Z","links":{"resolver":"https://pith.science/pith/6LYJLABROBMSIVQKEN3QYVO6R3","bundle":"https://pith.science/pith/6LYJLABROBMSIVQKEN3QYVO6R3/bundle.json","state":"https://pith.science/pith/6LYJLABROBMSIVQKEN3QYVO6R3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6LYJLABROBMSIVQKEN3QYVO6R3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6LYJLABROBMSIVQKEN3QYVO6R3","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":"13ce40eecd544a37cdec553c157d02781f53eeea358577ed0481e18252db7f7f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-22T22:41:21Z","title_canon_sha256":"eba247368f5cc242db822fa420df8b8b5a7b5a42134760e1a706a05ff3411c4b"},"schema_version":"1.0","source":{"id":"1510.06782","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.06782","created_at":"2026-05-18T01:06:29Z"},{"alias_kind":"arxiv_version","alias_value":"1510.06782v2","created_at":"2026-05-18T01:06:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.06782","created_at":"2026-05-18T01:06:29Z"},{"alias_kind":"pith_short_12","alias_value":"6LYJLABROBMS","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6LYJLABROBMSIVQK","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6LYJLABR","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:3e20a449937bb42b628c3ef6e769ff174d4ccdb05d6e2c6fb710276426fc17c7","target":"graph","created_at":"2026-05-18T01:06:29Z","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":"Let $G_{2(2)}$ be the non-compact connected simple Lie group of type $G_2$ over $\\mathbb{R}$, and let $M$ be a connected analytic complete pseudo-Riemannian manifold that admits an isometric $G_{2(2)}$-action with a dense orbit. For the case $\\dim(M) \\leq 21$, we provide a full description of the manifold $M$, its geometry and its $G_{2(2)}$-action. The latter are always given in terms of a Lie group geometry related to $G_{2(2)}$, and in one case $M$ is essentially the quotient of $\\widetilde{\\mathrm{S0}}_0(3,4)$ by a lattice.","authors_text":"R. Quiroga-Barranco","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-22T22:41:21Z","title":"Pseudo-Riemannian $\\mathrm{G}_{2(2)}$-manifolds with dimension at most $21$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.06782","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":"sha256:57d5f9df7018173a0792fdbbedf18e5c1c3d9274b9a4e0bb78f20ae3180b6504","target":"record","created_at":"2026-05-18T01:06:29Z","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":"13ce40eecd544a37cdec553c157d02781f53eeea358577ed0481e18252db7f7f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-22T22:41:21Z","title_canon_sha256":"eba247368f5cc242db822fa420df8b8b5a7b5a42134760e1a706a05ff3411c4b"},"schema_version":"1.0","source":{"id":"1510.06782","kind":"arxiv","version":2}},"canonical_sha256":"f2f0958031705924560a23770c55de8eff6c1bd876507d16e967f0a92353d35c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f2f0958031705924560a23770c55de8eff6c1bd876507d16e967f0a92353d35c","first_computed_at":"2026-05-18T01:06:29.087811Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:06:29.087811Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2DOXECGgAj0WUW6K/5NTGgZNWJSE8I2q95lVAU1YjZLZfTPYoopjIallTpycQHkp4g8OHs8/C6s/mGlIklvxCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:06:29.088453Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.06782","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:57d5f9df7018173a0792fdbbedf18e5c1c3d9274b9a4e0bb78f20ae3180b6504","sha256:3e20a449937bb42b628c3ef6e769ff174d4ccdb05d6e2c6fb710276426fc17c7"],"state_sha256":"e96d71cd79d573fe383ee945a9e3e8d7e4f921bf8ce4af1ade50a572c29300f5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KaIaIv1jrpyincbk4SYyXx/RWxZMB2AtiyJBhMfjx1cUzDwlu3y2cLj9m1rVry6GemRr4ZKloXlOiMFFJ6f/Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T16:45:46.057756Z","bundle_sha256":"92b8d9be1bec30ad1632265eb666987ad0e14c208da3ada5da4ad4521f680670"}}