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Then we obtain:\n  1) q^{4}=id; g(qx, qy)=g(x,y), x, y are arbitrary vector fields on M,\n  2) nabla q =0 if and only if grad A=(grad C)q^{2}; 2.grad B= (grad C)(q+q^{3}),"},"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":"1106.2758","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-06-14T16:31:28Z","cross_cats_sorted":[],"title_canon_sha256":"b81915ebf2b342eae9051689b2afddf3e7057f2dace05e3188c4b21749315fcb","abstract_canon_sha256":"b931cd83983d5afffeafb940a3aceb6d918132b0b46f6861eee58c973fab0c74"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:04.202408Z","signature_b64":"QEGHA0k6vOCJ8fkH/Bk1AJiSMiyHgwgbKG52GUXTaFyRd9498DmBxzhdoildwvma/jJqcWl8ijeDL6KxhYTRBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f3fb5f3dfeb53a4b64b68cdcf1ba681719b2689dede15ca9c3263510a0efbc9f","last_reissued_at":"2026-05-18T04:20:04.201963Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:04.201963Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a Class of Special Riemannian Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Dimitar Razpopov","submitted_at":"2011-06-14T16:31:28Z","abstract_excerpt":"We consider a four dimensional Riemannian manifold M with a metric g and an affinor structure q. We note the local coordinates of g and q are circulant matrices. Their first orders are (A, B, C, B)(A, B, C are smooth functions on M) and (0, 1, 0, 0), respectively.\n  Let nabla be the connection of g. Then we obtain:\n  1) q^{4}=id; g(qx, qy)=g(x,y), x, y are arbitrary vector fields on M,\n  2) nabla q =0 if and only if grad A=(grad C)q^{2}; 2.grad B= (grad C)(q+q^{3}),"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.2758","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":"1106.2758","created_at":"2026-05-18T04:20:04.202023+00:00"},{"alias_kind":"arxiv_version","alias_value":"1106.2758v1","created_at":"2026-05-18T04:20:04.202023+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.2758","created_at":"2026-05-18T04:20:04.202023+00:00"},{"alias_kind":"pith_short_12","alias_value":"6P5V6PP6WU5E","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_16","alias_value":"6P5V6PP6WU5EWZFW","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_8","alias_value":"6P5V6PP6","created_at":"2026-05-18T12:26:22.705136+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/6P5V6PP6WU5EWZFWRTOPDOTIC4","json":"https://pith.science/pith/6P5V6PP6WU5EWZFWRTOPDOTIC4.json","graph_json":"https://pith.science/api/pith-number/6P5V6PP6WU5EWZFWRTOPDOTIC4/graph.json","events_json":"https://pith.science/api/pith-number/6P5V6PP6WU5EWZFWRTOPDOTIC4/events.json","paper":"https://pith.science/paper/6P5V6PP6"},"agent_actions":{"view_html":"https://pith.science/pith/6P5V6PP6WU5EWZFWRTOPDOTIC4","download_json":"https://pith.science/pith/6P5V6PP6WU5EWZFWRTOPDOTIC4.json","view_paper":"https://pith.science/paper/6P5V6PP6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1106.2758&json=true","fetch_graph":"https://pith.science/api/pith-number/6P5V6PP6WU5EWZFWRTOPDOTIC4/graph.json","fetch_events":"https://pith.science/api/pith-number/6P5V6PP6WU5EWZFWRTOPDOTIC4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6P5V6PP6WU5EWZFWRTOPDOTIC4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6P5V6PP6WU5EWZFWRTOPDOTIC4/action/storage_attestation","attest_author":"https://pith.science/pith/6P5V6PP6WU5EWZFWRTOPDOTIC4/action/author_attestation","sign_citation":"https://pith.science/pith/6P5V6PP6WU5EWZFWRTOPDOTIC4/action/citation_signature","submit_replication":"https://pith.science/pith/6P5V6PP6WU5EWZFWRTOPDOTIC4/action/replication_record"}},"created_at":"2026-05-18T04:20:04.202023+00:00","updated_at":"2026-05-18T04:20:04.202023+00:00"}