{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:V5SEF7UDYX4JAHV5WBHBJSQ2QZ","short_pith_number":"pith:V5SEF7UD","schema_version":"1.0","canonical_sha256":"af6442fe83c5f8901ebdb04e14ca1a8644019aee670be06a319c031ebcf88431","source":{"kind":"arxiv","id":"2001.01398","version":1},"attestation_state":"computed","paper":{"title":"Integral geometric Hopf conjectures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Oliver Knill","submitted_at":"2020-01-06T05:00:25Z","abstract_excerpt":"The Hopf sign conjecture states that a compact Riemannian 2d-manifold M of positive curvature has Euler characteristic X(M)>0 and that in the case of negative curvature X(M) (-1)^d >0. The Hopf product conjecture asks whether a positive curvature metric can exist on product manifolds like S^2 x S^2. By formulating curvature integral geometrically, these questions can be explored for finite simple graphs, where it leads to linear programming problems. In this more expository document we aim to explore also a bit of the history of the Hopf conjecture and mention some strategies of attacks which "},"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":"2001.01398","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2020-01-06T05:00:25Z","cross_cats_sorted":[],"title_canon_sha256":"5df242f1a80e2256a682fd4174acb47016b5e7796fa46fba0b93cb24cf88adcc","abstract_canon_sha256":"8623504410bcdbf058f005d2dfabba051d0a0485be4ddfb4b16e5f3084a78eda"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:29:46.408648Z","signature_b64":"mTGHRf4sj8jYhDHGsKHacN5cAk7s2r9ceZCVeyf7bvmkZl6ZFEpubSZRHLb7ClcelGUHFZ+ezlVsAEqRJS+2CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af6442fe83c5f8901ebdb04e14ca1a8644019aee670be06a319c031ebcf88431","last_reissued_at":"2026-07-05T00:29:46.408247Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:29:46.408247Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Integral geometric Hopf conjectures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Oliver Knill","submitted_at":"2020-01-06T05:00:25Z","abstract_excerpt":"The Hopf sign conjecture states that a compact Riemannian 2d-manifold M of positive curvature has Euler characteristic X(M)>0 and that in the case of negative curvature X(M) (-1)^d >0. The Hopf product conjecture asks whether a positive curvature metric can exist on product manifolds like S^2 x S^2. By formulating curvature integral geometrically, these questions can be explored for finite simple graphs, where it leads to linear programming problems. In this more expository document we aim to explore also a bit of the history of the Hopf conjecture and mention some strategies of attacks which "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2001.01398","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2001.01398/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2001.01398","created_at":"2026-07-05T00:29:46.408310+00:00"},{"alias_kind":"arxiv_version","alias_value":"2001.01398v1","created_at":"2026-07-05T00:29:46.408310+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2001.01398","created_at":"2026-07-05T00:29:46.408310+00:00"},{"alias_kind":"pith_short_12","alias_value":"V5SEF7UDYX4J","created_at":"2026-07-05T00:29:46.408310+00:00"},{"alias_kind":"pith_short_16","alias_value":"V5SEF7UDYX4JAHV5","created_at":"2026-07-05T00:29:46.408310+00:00"},{"alias_kind":"pith_short_8","alias_value":"V5SEF7UD","created_at":"2026-07-05T00:29:46.408310+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/V5SEF7UDYX4JAHV5WBHBJSQ2QZ","json":"https://pith.science/pith/V5SEF7UDYX4JAHV5WBHBJSQ2QZ.json","graph_json":"https://pith.science/api/pith-number/V5SEF7UDYX4JAHV5WBHBJSQ2QZ/graph.json","events_json":"https://pith.science/api/pith-number/V5SEF7UDYX4JAHV5WBHBJSQ2QZ/events.json","paper":"https://pith.science/paper/V5SEF7UD"},"agent_actions":{"view_html":"https://pith.science/pith/V5SEF7UDYX4JAHV5WBHBJSQ2QZ","download_json":"https://pith.science/pith/V5SEF7UDYX4JAHV5WBHBJSQ2QZ.json","view_paper":"https://pith.science/paper/V5SEF7UD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2001.01398&json=true","fetch_graph":"https://pith.science/api/pith-number/V5SEF7UDYX4JAHV5WBHBJSQ2QZ/graph.json","fetch_events":"https://pith.science/api/pith-number/V5SEF7UDYX4JAHV5WBHBJSQ2QZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V5SEF7UDYX4JAHV5WBHBJSQ2QZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V5SEF7UDYX4JAHV5WBHBJSQ2QZ/action/storage_attestation","attest_author":"https://pith.science/pith/V5SEF7UDYX4JAHV5WBHBJSQ2QZ/action/author_attestation","sign_citation":"https://pith.science/pith/V5SEF7UDYX4JAHV5WBHBJSQ2QZ/action/citation_signature","submit_replication":"https://pith.science/pith/V5SEF7UDYX4JAHV5WBHBJSQ2QZ/action/replication_record"}},"created_at":"2026-07-05T00:29:46.408310+00:00","updated_at":"2026-07-05T00:29:46.408310+00:00"}