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Consider the two canonical K\\\"ahler structures $(G^{\\epsilon},J,\\Omega^{\\epsilon})$ on the product 4-manifold $\\Sigma_1\\times\\Sigma_2$ given by $ G^{\\epsilon}=g_1\\oplus \\epsilon g_2$, $\\epsilon=\\pm 1$ and $J$ is the canonical product complex structure. Thus for $\\epsilon=1$ the K\\\"ahler metric $G^+$ is Riemannian while for $\\epsilon=-1$, $G^-$ is of neutral signature. We show that the metric $G^{\\epsilon}$ is locally conformally flat iff the Gauss curvatures $\\kappa(g_1)$ and $\\kappa(g_2)$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1305.1561","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-05-07T15:34:22Z","cross_cats_sorted":[],"title_canon_sha256":"c6109080cfc9edd39adab4d3958570762e9a1b007e66e07bd711d209ebd065d3","abstract_canon_sha256":"ae662440562030d3adf5a9b517492059fae38ddf35bb388802ba1df93692416c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:16.049577Z","signature_b64":"VT3ISc1BzK+Q1e9dzaFkiBC2H23jcSDh0D8NCAiQ/ZXfgyk6vMBRZe6mW7c/dwne5O5bQZcWsdvZjtV/b9ljAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2485a34ceee6d41fdce8dbdb04ae17b9fe2730ec01dd7e5f76970fc8099806b3","last_reissued_at":"2026-05-18T00:29:16.048983Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:16.048983Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On minimal Lagrangian surfaces in the product of Riemannian two manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Nikos Georgiou","submitted_at":"2013-05-07T15:34:22Z","abstract_excerpt":"Let $(\\Sigma_1,g_1)$ and $(\\Sigma_2,g_2)$ be connected, complete and orientable Riemannian two manifolds. Consider the two canonical K\\\"ahler structures $(G^{\\epsilon},J,\\Omega^{\\epsilon})$ on the product 4-manifold $\\Sigma_1\\times\\Sigma_2$ given by $ G^{\\epsilon}=g_1\\oplus \\epsilon g_2$, $\\epsilon=\\pm 1$ and $J$ is the canonical product complex structure. Thus for $\\epsilon=1$ the K\\\"ahler metric $G^+$ is Riemannian while for $\\epsilon=-1$, $G^-$ is of neutral signature. We show that the metric $G^{\\epsilon}$ is locally conformally flat iff the Gauss curvatures $\\kappa(g_1)$ and $\\kappa(g_2)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.1561","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1305.1561","created_at":"2026-05-18T00:29:16.049073+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.1561v2","created_at":"2026-05-18T00:29:16.049073+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.1561","created_at":"2026-05-18T00:29:16.049073+00:00"},{"alias_kind":"pith_short_12","alias_value":"ESC2GTHO43KB","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_16","alias_value":"ESC2GTHO43KB7XHI","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_8","alias_value":"ESC2GTHO","created_at":"2026-05-18T12:27:43.054852+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ESC2GTHO43KB7XHI3PNQJLQXXH","json":"https://pith.science/pith/ESC2GTHO43KB7XHI3PNQJLQXXH.json","graph_json":"https://pith.science/api/pith-number/ESC2GTHO43KB7XHI3PNQJLQXXH/graph.json","events_json":"https://pith.science/api/pith-number/ESC2GTHO43KB7XHI3PNQJLQXXH/events.json","paper":"https://pith.science/paper/ESC2GTHO"},"agent_actions":{"view_html":"https://pith.science/pith/ESC2GTHO43KB7XHI3PNQJLQXXH","download_json":"https://pith.science/pith/ESC2GTHO43KB7XHI3PNQJLQXXH.json","view_paper":"https://pith.science/paper/ESC2GTHO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.1561&json=true","fetch_graph":"https://pith.science/api/pith-number/ESC2GTHO43KB7XHI3PNQJLQXXH/graph.json","fetch_events":"https://pith.science/api/pith-number/ESC2GTHO43KB7XHI3PNQJLQXXH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ESC2GTHO43KB7XHI3PNQJLQXXH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ESC2GTHO43KB7XHI3PNQJLQXXH/action/storage_attestation","attest_author":"https://pith.science/pith/ESC2GTHO43KB7XHI3PNQJLQXXH/action/author_attestation","sign_citation":"https://pith.science/pith/ESC2GTHO43KB7XHI3PNQJLQXXH/action/citation_signature","submit_replication":"https://pith.science/pith/ESC2GTHO43KB7XHI3PNQJLQXXH/action/replication_record"}},"created_at":"2026-05-18T00:29:16.049073+00:00","updated_at":"2026-05-18T00:29:16.049073+00:00"}