{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:Z6QRDT2OFJES2E57QZNWTR6Q5B","short_pith_number":"pith:Z6QRDT2O","schema_version":"1.0","canonical_sha256":"cfa111cf4e2a492d13bf865b69c7d0e873046ab70b6af5d1039fa46037a7a5c5","source":{"kind":"arxiv","id":"1106.3674","version":1},"attestation_state":"computed","paper":{"title":"Geometry of PR-warped products in para-Kaehler manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Bang-Yen Chen, Marian Ioan Munteanu","submitted_at":"2011-06-18T18:22:45Z","abstract_excerpt":"In this paper, we initiate the study of $\\p R$-warped products in para-K\\\"ahler manifolds and prove some fundamental results on such submanifolds. In particular, we establish a general optimal inequality for $\\p R$-warped products in para-K\\\"ahler manifolds involving only the warping function and the second fundamental form. Moreover, we completely classify $\\p R$-warped products in the flat para-K\\\"ahler manifold with least codimension which satisfy the equality case of the inequality."},"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":"1106.3674","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-06-18T18:22:45Z","cross_cats_sorted":[],"title_canon_sha256":"9ad866f7432266e8a347fe2fca74be60b4629e0b7b9d497fa0928aba71c90263","abstract_canon_sha256":"5dbdc68423c291880c81879014704660ab0d13220f5b305d89bfd725e55237c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:42.430557Z","signature_b64":"IcO5HPnkB8m0MHBu2BVH7slJTQvRhqt3NAzASrdnH2iZq59sg7eHSKib46pnUcnSow7BTU6+SX6f+9w4G0AMAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cfa111cf4e2a492d13bf865b69c7d0e873046ab70b6af5d1039fa46037a7a5c5","last_reissued_at":"2026-05-18T04:19:42.429829Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:42.429829Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Geometry of PR-warped products in para-Kaehler manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Bang-Yen Chen, Marian Ioan Munteanu","submitted_at":"2011-06-18T18:22:45Z","abstract_excerpt":"In this paper, we initiate the study of $\\p R$-warped products in para-K\\\"ahler manifolds and prove some fundamental results on such submanifolds. In particular, we establish a general optimal inequality for $\\p R$-warped products in para-K\\\"ahler manifolds involving only the warping function and the second fundamental form. Moreover, we completely classify $\\p R$-warped products in the flat para-K\\\"ahler manifold with least codimension which satisfy the equality case of the inequality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.3674","kind":"arxiv","version":1},"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":"1106.3674","created_at":"2026-05-18T04:19:42.429955+00:00"},{"alias_kind":"arxiv_version","alias_value":"1106.3674v1","created_at":"2026-05-18T04:19:42.429955+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.3674","created_at":"2026-05-18T04:19:42.429955+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z6QRDT2OFJES","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z6QRDT2OFJES2E57","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z6QRDT2O","created_at":"2026-05-18T12:26:47.523578+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/Z6QRDT2OFJES2E57QZNWTR6Q5B","json":"https://pith.science/pith/Z6QRDT2OFJES2E57QZNWTR6Q5B.json","graph_json":"https://pith.science/api/pith-number/Z6QRDT2OFJES2E57QZNWTR6Q5B/graph.json","events_json":"https://pith.science/api/pith-number/Z6QRDT2OFJES2E57QZNWTR6Q5B/events.json","paper":"https://pith.science/paper/Z6QRDT2O"},"agent_actions":{"view_html":"https://pith.science/pith/Z6QRDT2OFJES2E57QZNWTR6Q5B","download_json":"https://pith.science/pith/Z6QRDT2OFJES2E57QZNWTR6Q5B.json","view_paper":"https://pith.science/paper/Z6QRDT2O","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1106.3674&json=true","fetch_graph":"https://pith.science/api/pith-number/Z6QRDT2OFJES2E57QZNWTR6Q5B/graph.json","fetch_events":"https://pith.science/api/pith-number/Z6QRDT2OFJES2E57QZNWTR6Q5B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z6QRDT2OFJES2E57QZNWTR6Q5B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z6QRDT2OFJES2E57QZNWTR6Q5B/action/storage_attestation","attest_author":"https://pith.science/pith/Z6QRDT2OFJES2E57QZNWTR6Q5B/action/author_attestation","sign_citation":"https://pith.science/pith/Z6QRDT2OFJES2E57QZNWTR6Q5B/action/citation_signature","submit_replication":"https://pith.science/pith/Z6QRDT2OFJES2E57QZNWTR6Q5B/action/replication_record"}},"created_at":"2026-05-18T04:19:42.429955+00:00","updated_at":"2026-05-18T04:19:42.429955+00:00"}