{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:HNLRGISHDGS4PDIPEHB7DVLPBH","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":"ae3b752da388f345d0da47701a497ffe281c45cfd8c343ccc0ce7bda6e3e755a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-11-24T21:40:39Z","title_canon_sha256":"4cf6d04671ad230dc36d237ad74c47db2bd1a0a7c01c27f5f5c15d8888214c74"},"schema_version":"1.0","source":{"id":"1311.6174","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.6174","created_at":"2026-05-18T03:06:14Z"},{"alias_kind":"arxiv_version","alias_value":"1311.6174v1","created_at":"2026-05-18T03:06:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.6174","created_at":"2026-05-18T03:06:14Z"},{"alias_kind":"pith_short_12","alias_value":"HNLRGISHDGS4","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HNLRGISHDGS4PDIP","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HNLRGISH","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:8a367f713804ab88d17a2350372aa669f91d8436a46f97609e6464f71acb6922","target":"graph","created_at":"2026-05-18T03:06:14Z","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":"We call a connected Lie group endowed with a left-invariant Lorentzian flat metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. Moreover, we show that a Lorentzian flat Lie group $(\\mathrm{G},\\mu)$ admits a timelike left-invariant Killing vector field if and only if $\\mathrm{G}$ admits a left-invariant Riemannian metric which has the same Levi-Civita connection of $\\mu$. Finally, we give an useful characte","authors_text":"Hicham Lebzioui","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-11-24T21:40:39Z","title":"Lorentzian Flat Lie Groups Admitting a Timelike Left-Invariant Killing Vector Field"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6174","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:eed5c556a646243b7d6cc5c06ed26db5bbaccda6f48415949a05d418d12381dc","target":"record","created_at":"2026-05-18T03:06:14Z","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":"ae3b752da388f345d0da47701a497ffe281c45cfd8c343ccc0ce7bda6e3e755a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-11-24T21:40:39Z","title_canon_sha256":"4cf6d04671ad230dc36d237ad74c47db2bd1a0a7c01c27f5f5c15d8888214c74"},"schema_version":"1.0","source":{"id":"1311.6174","kind":"arxiv","version":1}},"canonical_sha256":"3b5713224719a5c78d0f21c3f1d56f09c17caf8ae8b4c969381dcfabff3be5e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3b5713224719a5c78d0f21c3f1d56f09c17caf8ae8b4c969381dcfabff3be5e5","first_computed_at":"2026-05-18T03:06:14.711940Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:06:14.711940Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ss6ZmxSB07C16RMoWsmLrb3zTU4hlDNDYB6XpBo2FasIICVFhAkT+jLme6yVL5KsIiwIL83q0Fj6nm0lWUxhDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:06:14.712760Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.6174","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eed5c556a646243b7d6cc5c06ed26db5bbaccda6f48415949a05d418d12381dc","sha256:8a367f713804ab88d17a2350372aa669f91d8436a46f97609e6464f71acb6922"],"state_sha256":"7abd03ea96f2098b1af0ce7629795604cb3d9ab820d408728d5ca830d395ec0f"}