{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:AONJFI6I32FYXULV2RCWNHRFDI","short_pith_number":"pith:AONJFI6I","schema_version":"1.0","canonical_sha256":"039a92a3c8de8b8bd175d445669e251a2f33c8ef3c95630bcc0c39f1da3d3891","source":{"kind":"arxiv","id":"0901.1406","version":1},"attestation_state":"computed","paper":{"title":"Sub-Riemannian geometry of parallelizable spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Irina Markina, Mauricio Godoy Molina","submitted_at":"2009-01-11T01:49:49Z","abstract_excerpt":"The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere $S^3$ originating from different constructions. Namely, we describe the sub-Riemannian geometry of $S^3$ arising through its right Lie group action over itself, the one inherited from the natural complex structure of the open unit ball in $\\comp^2$ and the geometry that appears when considering the Hopf map as a principal bundle. The main result of this comparison is that in fact those three structures coincide.\n  In the second place, we present two bracket generating distributi"},"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":"0901.1406","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-01-11T01:49:49Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"5435da0fdb71388f59bef18c1a1d4850a17f836540b8a31c93a81a86e1b4b87b","abstract_canon_sha256":"97a053cb58bd9a835bab492e5d9c61ffb45b62f9fdf52a9b87dd14797275c2e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:33.356951Z","signature_b64":"jlFgc5EGUam+jU8d0cD7qEdi9qQgaJuutaOD93HFiIPFaL0U/mplKvyMaZPrkIIYWb+DRd/1u0PwBME40AhjDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"039a92a3c8de8b8bd175d445669e251a2f33c8ef3c95630bcc0c39f1da3d3891","last_reissued_at":"2026-05-18T01:35:33.356444Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:33.356444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sub-Riemannian geometry of parallelizable spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Irina Markina, Mauricio Godoy Molina","submitted_at":"2009-01-11T01:49:49Z","abstract_excerpt":"The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere $S^3$ originating from different constructions. Namely, we describe the sub-Riemannian geometry of $S^3$ arising through its right Lie group action over itself, the one inherited from the natural complex structure of the open unit ball in $\\comp^2$ and the geometry that appears when considering the Hopf map as a principal bundle. The main result of this comparison is that in fact those three structures coincide.\n  In the second place, we present two bracket generating distributi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.1406","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":"0901.1406","created_at":"2026-05-18T01:35:33.356540+00:00"},{"alias_kind":"arxiv_version","alias_value":"0901.1406v1","created_at":"2026-05-18T01:35:33.356540+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.1406","created_at":"2026-05-18T01:35:33.356540+00:00"},{"alias_kind":"pith_short_12","alias_value":"AONJFI6I32FY","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_16","alias_value":"AONJFI6I32FYXULV","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_8","alias_value":"AONJFI6I","created_at":"2026-05-18T12:25:58.837520+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/AONJFI6I32FYXULV2RCWNHRFDI","json":"https://pith.science/pith/AONJFI6I32FYXULV2RCWNHRFDI.json","graph_json":"https://pith.science/api/pith-number/AONJFI6I32FYXULV2RCWNHRFDI/graph.json","events_json":"https://pith.science/api/pith-number/AONJFI6I32FYXULV2RCWNHRFDI/events.json","paper":"https://pith.science/paper/AONJFI6I"},"agent_actions":{"view_html":"https://pith.science/pith/AONJFI6I32FYXULV2RCWNHRFDI","download_json":"https://pith.science/pith/AONJFI6I32FYXULV2RCWNHRFDI.json","view_paper":"https://pith.science/paper/AONJFI6I","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0901.1406&json=true","fetch_graph":"https://pith.science/api/pith-number/AONJFI6I32FYXULV2RCWNHRFDI/graph.json","fetch_events":"https://pith.science/api/pith-number/AONJFI6I32FYXULV2RCWNHRFDI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AONJFI6I32FYXULV2RCWNHRFDI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AONJFI6I32FYXULV2RCWNHRFDI/action/storage_attestation","attest_author":"https://pith.science/pith/AONJFI6I32FYXULV2RCWNHRFDI/action/author_attestation","sign_citation":"https://pith.science/pith/AONJFI6I32FYXULV2RCWNHRFDI/action/citation_signature","submit_replication":"https://pith.science/pith/AONJFI6I32FYXULV2RCWNHRFDI/action/replication_record"}},"created_at":"2026-05-18T01:35:33.356540+00:00","updated_at":"2026-05-18T01:35:33.356540+00:00"}