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We show that a neighborhood of any point in some open dense subset of $M$ is locally isometric to one of the following five types; {\\rm (i)} $ \\mathbb{S}^2(\\frac{R}{6}) \\times \\mathbb{S}^2(\\frac{R}{3})$ with $R>0$, {\\rm (ii)} $ \\m"},"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":"1604.03241","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-04-12T04:12:55Z","cross_cats_sorted":[],"title_canon_sha256":"fb9a94198fa493687982b41b1e32c7cad6510ab32e1a8ac79bae7fd60fdde326","abstract_canon_sha256":"82240c84540c05284193968fc2eee36070310528899a845107cebdee09ceed05"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:15.974040Z","signature_b64":"2awH+lIiXAKs8SqcrxFnClSeG1lnxXgByZaPGGrwpZXPICbYzx4zmUD7JDCd2ilPW4GJyDuoXu30Ar0tXYvqDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2e155472e1cac543457c5dcc26d034f86d6fd083b641dcd58d13718096d7c9dc","last_reissued_at":"2026-05-18T01:17:15.973360Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:15.973360Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Four dimensional static and related critical spaces with harmonic curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jinwoo Shin, Jongsu Kim","submitted_at":"2016-04-12T04:12:55Z","abstract_excerpt":"In this article we study any 4-dimensional Riemannian manifold $(M,g)$ with harmonic curvature which admits a smooth nonzero solution $f$ to the following equation \\begin{eqnarray} \\label{0002bx} \\nabla df = f(Rc -\\frac{R}{n-1} g) + x Rc+ y(R) g. \\end{eqnarray} where $Rc$ is the Ricci tensor of $g$, $x$ is a constant and $y(R)$ a function of the scalar curvature $R$. We show that a neighborhood of any point in some open dense subset of $M$ is locally isometric to one of the following five types; {\\rm (i)} $ \\mathbb{S}^2(\\frac{R}{6}) \\times \\mathbb{S}^2(\\frac{R}{3})$ with $R>0$, {\\rm (ii)} $ \\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.03241","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":"1604.03241","created_at":"2026-05-18T01:17:15.973458+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.03241v1","created_at":"2026-05-18T01:17:15.973458+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.03241","created_at":"2026-05-18T01:17:15.973458+00:00"},{"alias_kind":"pith_short_12","alias_value":"FYKVI4XBZLCU","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"FYKVI4XBZLCUGRL4","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"FYKVI4XB","created_at":"2026-05-18T12:30:15.759754+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/FYKVI4XBZLCUGRL4LXGCNUBU7B","json":"https://pith.science/pith/FYKVI4XBZLCUGRL4LXGCNUBU7B.json","graph_json":"https://pith.science/api/pith-number/FYKVI4XBZLCUGRL4LXGCNUBU7B/graph.json","events_json":"https://pith.science/api/pith-number/FYKVI4XBZLCUGRL4LXGCNUBU7B/events.json","paper":"https://pith.science/paper/FYKVI4XB"},"agent_actions":{"view_html":"https://pith.science/pith/FYKVI4XBZLCUGRL4LXGCNUBU7B","download_json":"https://pith.science/pith/FYKVI4XBZLCUGRL4LXGCNUBU7B.json","view_paper":"https://pith.science/paper/FYKVI4XB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.03241&json=true","fetch_graph":"https://pith.science/api/pith-number/FYKVI4XBZLCUGRL4LXGCNUBU7B/graph.json","fetch_events":"https://pith.science/api/pith-number/FYKVI4XBZLCUGRL4LXGCNUBU7B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FYKVI4XBZLCUGRL4LXGCNUBU7B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FYKVI4XBZLCUGRL4LXGCNUBU7B/action/storage_attestation","attest_author":"https://pith.science/pith/FYKVI4XBZLCUGRL4LXGCNUBU7B/action/author_attestation","sign_citation":"https://pith.science/pith/FYKVI4XBZLCUGRL4LXGCNUBU7B/action/citation_signature","submit_replication":"https://pith.science/pith/FYKVI4XBZLCUGRL4LXGCNUBU7B/action/replication_record"}},"created_at":"2026-05-18T01:17:15.973458+00:00","updated_at":"2026-05-18T01:17:15.973458+00:00"}