{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:MTKWTTQHXONB5MO4DY2DVIDZTB","short_pith_number":"pith:MTKWTTQH","canonical_record":{"source":{"id":"1403.0305","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-03-03T03:45:22Z","cross_cats_sorted":[],"title_canon_sha256":"32ab643adf3844faccc3a33a46b7ec9f1471ba9a6cf3e94505206a2f06e74fbe","abstract_canon_sha256":"375659fb2ad948f95afe2683a4c7e68d9a8458b5aa021733401eb683d14c0678"},"schema_version":"1.0"},"canonical_sha256":"64d569ce07bb9a1eb1dc1e343aa0799852271cebad3047d8ac29eea68a2bd091","source":{"kind":"arxiv","id":"1403.0305","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.0305","created_at":"2026-05-18T00:29:21Z"},{"alias_kind":"arxiv_version","alias_value":"1403.0305v3","created_at":"2026-05-18T00:29:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.0305","created_at":"2026-05-18T00:29:21Z"},{"alias_kind":"pith_short_12","alias_value":"MTKWTTQHXONB","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MTKWTTQHXONB5MO4","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MTKWTTQH","created_at":"2026-05-18T12:28:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:MTKWTTQHXONB5MO4DY2DVIDZTB","target":"record","payload":{"canonical_record":{"source":{"id":"1403.0305","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-03-03T03:45:22Z","cross_cats_sorted":[],"title_canon_sha256":"32ab643adf3844faccc3a33a46b7ec9f1471ba9a6cf3e94505206a2f06e74fbe","abstract_canon_sha256":"375659fb2ad948f95afe2683a4c7e68d9a8458b5aa021733401eb683d14c0678"},"schema_version":"1.0"},"canonical_sha256":"64d569ce07bb9a1eb1dc1e343aa0799852271cebad3047d8ac29eea68a2bd091","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:21.523763Z","signature_b64":"zq24DnFj+Q3sMlSZQuSWDBIdM/c4Ec9KdjGOVmv9FDNQ1Q/1usRf/xcEpGrRpdXYJVaKARJCkGlEdiDbVTY9DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64d569ce07bb9a1eb1dc1e343aa0799852271cebad3047d8ac29eea68a2bd091","last_reissued_at":"2026-05-18T00:29:21.523063Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:21.523063Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1403.0305","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":"2026-05-18T00:29:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VpNxVYVHNQNNw5INqzdf5AmOqfKUHuhXz6rlu8nUshlVHQUz53SJOFREMbEYSyTA48ie7x2ZLQLXXor+t6aGAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T23:25:11.414669Z"},"content_sha256":"f3a5a1b0408413adbd405580228cc880ff09bd8bb5c889c692f9b14d348861c8","schema_version":"1.0","event_id":"sha256:f3a5a1b0408413adbd405580228cc880ff09bd8bb5c889c692f9b14d348861c8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:MTKWTTQHXONB5MO4DY2DVIDZTB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lagrangian immersions in the product of Lorentzian two manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Nikos Georgiou","submitted_at":"2014-03-03T03:45:22Z","abstract_excerpt":"For Lorentzian 2-manifolds $(\\Sigma_1,g_1)$ and $(\\Sigma_2,g_2)$ we consider the two product para-K\\\"ahler structures $(G^{\\epsilon},J,\\Omega^{\\epsilon})$ defined on the product four manifold $\\Sigma_1\\times\\Sigma_2$, with $\\epsilon=\\pm 1$. We show that the metric $G^{\\epsilon}$ is locally conformally flat (resp. Einstein) if and only if the Gauss curvatures $\\kappa_1,\\kappa_2$ of $g_1,g_2$, respectively, are both constants satisfying $\\kappa_1=-\\epsilon\\kappa_2$ (resp. $\\kappa_1=\\epsilon\\kappa_2$). We give the conditions on the Gauss curvatures for which every Lagrangian surface with parallel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0305","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":"2026-05-18T00:29:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zCul56RlCZ8DNM0GhiE9LRiVMOu8iCeBpBEm+KETF+LCME+JE1Un8whf7TqBfRxP7Moo2Ha+g2BgZ+TCH6e1Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T23:25:11.415400Z"},"content_sha256":"018cc590979d0aa678f467a36c4448d3f5072b36f2be803ddc9a25ba392618e8","schema_version":"1.0","event_id":"sha256:018cc590979d0aa678f467a36c4448d3f5072b36f2be803ddc9a25ba392618e8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MTKWTTQHXONB5MO4DY2DVIDZTB/bundle.json","state_url":"https://pith.science/pith/MTKWTTQHXONB5MO4DY2DVIDZTB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MTKWTTQHXONB5MO4DY2DVIDZTB/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":"2026-06-04T23:25:11Z","links":{"resolver":"https://pith.science/pith/MTKWTTQHXONB5MO4DY2DVIDZTB","bundle":"https://pith.science/pith/MTKWTTQHXONB5MO4DY2DVIDZTB/bundle.json","state":"https://pith.science/pith/MTKWTTQHXONB5MO4DY2DVIDZTB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MTKWTTQHXONB5MO4DY2DVIDZTB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:MTKWTTQHXONB5MO4DY2DVIDZTB","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":"375659fb2ad948f95afe2683a4c7e68d9a8458b5aa021733401eb683d14c0678","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-03-03T03:45:22Z","title_canon_sha256":"32ab643adf3844faccc3a33a46b7ec9f1471ba9a6cf3e94505206a2f06e74fbe"},"schema_version":"1.0","source":{"id":"1403.0305","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.0305","created_at":"2026-05-18T00:29:21Z"},{"alias_kind":"arxiv_version","alias_value":"1403.0305v3","created_at":"2026-05-18T00:29:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.0305","created_at":"2026-05-18T00:29:21Z"},{"alias_kind":"pith_short_12","alias_value":"MTKWTTQHXONB","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MTKWTTQHXONB5MO4","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MTKWTTQH","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:018cc590979d0aa678f467a36c4448d3f5072b36f2be803ddc9a25ba392618e8","target":"graph","created_at":"2026-05-18T00:29:21Z","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 Lorentzian 2-manifolds $(\\Sigma_1,g_1)$ and $(\\Sigma_2,g_2)$ we consider the two product para-K\\\"ahler structures $(G^{\\epsilon},J,\\Omega^{\\epsilon})$ defined on the product four manifold $\\Sigma_1\\times\\Sigma_2$, with $\\epsilon=\\pm 1$. We show that the metric $G^{\\epsilon}$ is locally conformally flat (resp. Einstein) if and only if the Gauss curvatures $\\kappa_1,\\kappa_2$ of $g_1,g_2$, respectively, are both constants satisfying $\\kappa_1=-\\epsilon\\kappa_2$ (resp. $\\kappa_1=\\epsilon\\kappa_2$). We give the conditions on the Gauss curvatures for which every Lagrangian surface with parallel","authors_text":"Nikos Georgiou","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-03-03T03:45:22Z","title":"Lagrangian immersions in the product of Lorentzian two manifold"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0305","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:f3a5a1b0408413adbd405580228cc880ff09bd8bb5c889c692f9b14d348861c8","target":"record","created_at":"2026-05-18T00:29:21Z","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":"375659fb2ad948f95afe2683a4c7e68d9a8458b5aa021733401eb683d14c0678","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-03-03T03:45:22Z","title_canon_sha256":"32ab643adf3844faccc3a33a46b7ec9f1471ba9a6cf3e94505206a2f06e74fbe"},"schema_version":"1.0","source":{"id":"1403.0305","kind":"arxiv","version":3}},"canonical_sha256":"64d569ce07bb9a1eb1dc1e343aa0799852271cebad3047d8ac29eea68a2bd091","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"64d569ce07bb9a1eb1dc1e343aa0799852271cebad3047d8ac29eea68a2bd091","first_computed_at":"2026-05-18T00:29:21.523063Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:29:21.523063Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zq24DnFj+Q3sMlSZQuSWDBIdM/c4Ec9KdjGOVmv9FDNQ1Q/1usRf/xcEpGrRpdXYJVaKARJCkGlEdiDbVTY9DA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:29:21.523763Z","signed_message":"canonical_sha256_bytes"},"source_id":"1403.0305","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f3a5a1b0408413adbd405580228cc880ff09bd8bb5c889c692f9b14d348861c8","sha256:018cc590979d0aa678f467a36c4448d3f5072b36f2be803ddc9a25ba392618e8"],"state_sha256":"23c801d2e69538695752ecfe4fd046159130674fd1263b6a51509f1f99817205"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JvqTWjdl2yqRrCOXLzx4FFNLfm2xOYNRTn7z7p45HEyTWf/h/1x8+rLvo2WDXsN42O7F7SDI+OXZE6UI7gtABA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T23:25:11.418642Z","bundle_sha256":"feec0574e991126af6cf74cca30d079b3c9c312c2cd9c04f272a190d0df8a068"}}