{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:TT26SBRVXX4W6SW5RL7SHZM762","short_pith_number":"pith:TT26SBRV","schema_version":"1.0","canonical_sha256":"9cf5e90635bdf96f4add8aff23e59ff6b5831cd57bf6b280c6e0669165b8ac25","source":{"kind":"arxiv","id":"1612.06500","version":2},"attestation_state":"computed","paper":{"title":"Minimal submanifolds in certain types of kaehler product manifold","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Bang Xiao, Xingda Liu","submitted_at":"2016-12-20T04:02:02Z","abstract_excerpt":"Let $M$ be a real $l$-dimensional minimal submanifold with flat normal connection in a kaehler product manifold $\\overline{M}^m\\times \\overline{M}^n$ where $\\overline{M}^m$ and $\\overline{M}^n$ are complex $m$-dimensional and complex $n$-dimensional kaehler manifolds with constant holomorphic sectional curvature $c_1$ and $c_2$ respectively. We give a formula for the Laplacian of the second fundamental form of $M$. Specially we discuss the F-anti invariant case. We also give some applications of this formula."},"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":"1612.06500","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.DG","submitted_at":"2016-12-20T04:02:02Z","cross_cats_sorted":[],"title_canon_sha256":"ab353028ecd802a77a947d9649bf0ac5c81b7fa5bc5b4eac3be610f9cd4ba25d","abstract_canon_sha256":"fb048357a410cce27a084a7c141f1a7a708c421acd65f3431d9806872b600588"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:20.413776Z","signature_b64":"t6xzXBrqfqJDr+9HZmr8EOr/Ry/OA31pt/nmi1JVh9dTIb1h8xQoqDx/dgRfSoA6yJugkqXm0J5/ezr7SVI0Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9cf5e90635bdf96f4add8aff23e59ff6b5831cd57bf6b280c6e0669165b8ac25","last_reissued_at":"2026-05-18T00:53:20.413223Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:20.413223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimal submanifolds in certain types of kaehler product manifold","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Bang Xiao, Xingda Liu","submitted_at":"2016-12-20T04:02:02Z","abstract_excerpt":"Let $M$ be a real $l$-dimensional minimal submanifold with flat normal connection in a kaehler product manifold $\\overline{M}^m\\times \\overline{M}^n$ where $\\overline{M}^m$ and $\\overline{M}^n$ are complex $m$-dimensional and complex $n$-dimensional kaehler manifolds with constant holomorphic sectional curvature $c_1$ and $c_2$ respectively. We give a formula for the Laplacian of the second fundamental form of $M$. Specially we discuss the F-anti invariant case. We also give some applications of this formula."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.06500","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":"1612.06500","created_at":"2026-05-18T00:53:20.413302+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.06500v2","created_at":"2026-05-18T00:53:20.413302+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.06500","created_at":"2026-05-18T00:53:20.413302+00:00"},{"alias_kind":"pith_short_12","alias_value":"TT26SBRVXX4W","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"TT26SBRVXX4W6SW5","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"TT26SBRV","created_at":"2026-05-18T12:30:46.583412+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/TT26SBRVXX4W6SW5RL7SHZM762","json":"https://pith.science/pith/TT26SBRVXX4W6SW5RL7SHZM762.json","graph_json":"https://pith.science/api/pith-number/TT26SBRVXX4W6SW5RL7SHZM762/graph.json","events_json":"https://pith.science/api/pith-number/TT26SBRVXX4W6SW5RL7SHZM762/events.json","paper":"https://pith.science/paper/TT26SBRV"},"agent_actions":{"view_html":"https://pith.science/pith/TT26SBRVXX4W6SW5RL7SHZM762","download_json":"https://pith.science/pith/TT26SBRVXX4W6SW5RL7SHZM762.json","view_paper":"https://pith.science/paper/TT26SBRV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.06500&json=true","fetch_graph":"https://pith.science/api/pith-number/TT26SBRVXX4W6SW5RL7SHZM762/graph.json","fetch_events":"https://pith.science/api/pith-number/TT26SBRVXX4W6SW5RL7SHZM762/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TT26SBRVXX4W6SW5RL7SHZM762/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TT26SBRVXX4W6SW5RL7SHZM762/action/storage_attestation","attest_author":"https://pith.science/pith/TT26SBRVXX4W6SW5RL7SHZM762/action/author_attestation","sign_citation":"https://pith.science/pith/TT26SBRVXX4W6SW5RL7SHZM762/action/citation_signature","submit_replication":"https://pith.science/pith/TT26SBRVXX4W6SW5RL7SHZM762/action/replication_record"}},"created_at":"2026-05-18T00:53:20.413302+00:00","updated_at":"2026-05-18T00:53:20.413302+00:00"}