{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:AEOPK3LTC6CW35HTWLSPG5KD7U","short_pith_number":"pith:AEOPK3LT","schema_version":"1.0","canonical_sha256":"011cf56d7317856df4f3b2e4f37543fd123a896198c667b5e5c3ceb2403639e1","source":{"kind":"arxiv","id":"1703.01592","version":3},"attestation_state":"computed","paper":{"title":"Tubular neighborhoods in the sub-Riemannian Heisenberg groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.MG","authors_text":"Manuel Ritor\\'e","submitted_at":"2017-03-05T12:39:36Z","abstract_excerpt":"We consider the Carnot-Carath\\'eodory distance $\\delta_E$ to a closed set $E$ in the sub-Riemannian Heisenberg groups $\\mathbb{H}^n$, $n\\ge 1$. The $\\mathbb{H}$-regularity of $\\delta_E$ is proved under mild conditions involving a general notion of singular points. In case $E$ is a Euclidean $C^k$ submanifold, $k\\ge 2$, we prove that $\\delta_E$ is $C^k$ out of the singular set. Explicit expressions for the volume of the tubular neighborhood when the boundary of $E$ is of class $C^2$ are obtained, out of the singular set, in terms of the horizontal principal curvatures of $\\partial E$ and of the"},"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":"1703.01592","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-03-05T12:39:36Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"e38b6710dae640efd87adcd5fe2b10fad8b29f6ba499c4e9dbfb7d5049fc73c8","abstract_canon_sha256":"5b39055c6d054807ccc275df3ac2c9406bb8f6b953cffd1e5ba5c54659027777"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:55.185249Z","signature_b64":"Ue1Udfu3RlziJ0I0DskG8DvLSNd8Eb9igYsBQ1VvTF7jnZ89zAdX0c9BnvjcipMDj2XXyabRqvx9hC811aC8Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"011cf56d7317856df4f3b2e4f37543fd123a896198c667b5e5c3ceb2403639e1","last_reissued_at":"2026-05-18T00:30:55.184602Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:55.184602Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tubular neighborhoods in the sub-Riemannian Heisenberg groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.MG","authors_text":"Manuel Ritor\\'e","submitted_at":"2017-03-05T12:39:36Z","abstract_excerpt":"We consider the Carnot-Carath\\'eodory distance $\\delta_E$ to a closed set $E$ in the sub-Riemannian Heisenberg groups $\\mathbb{H}^n$, $n\\ge 1$. The $\\mathbb{H}$-regularity of $\\delta_E$ is proved under mild conditions involving a general notion of singular points. In case $E$ is a Euclidean $C^k$ submanifold, $k\\ge 2$, we prove that $\\delta_E$ is $C^k$ out of the singular set. Explicit expressions for the volume of the tubular neighborhood when the boundary of $E$ is of class $C^2$ are obtained, out of the singular set, in terms of the horizontal principal curvatures of $\\partial E$ and of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.01592","kind":"arxiv","version":3},"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":"1703.01592","created_at":"2026-05-18T00:30:55.184697+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.01592v3","created_at":"2026-05-18T00:30:55.184697+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.01592","created_at":"2026-05-18T00:30:55.184697+00:00"},{"alias_kind":"pith_short_12","alias_value":"AEOPK3LTC6CW","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_16","alias_value":"AEOPK3LTC6CW35HT","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_8","alias_value":"AEOPK3LT","created_at":"2026-05-18T12:31:05.417338+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/AEOPK3LTC6CW35HTWLSPG5KD7U","json":"https://pith.science/pith/AEOPK3LTC6CW35HTWLSPG5KD7U.json","graph_json":"https://pith.science/api/pith-number/AEOPK3LTC6CW35HTWLSPG5KD7U/graph.json","events_json":"https://pith.science/api/pith-number/AEOPK3LTC6CW35HTWLSPG5KD7U/events.json","paper":"https://pith.science/paper/AEOPK3LT"},"agent_actions":{"view_html":"https://pith.science/pith/AEOPK3LTC6CW35HTWLSPG5KD7U","download_json":"https://pith.science/pith/AEOPK3LTC6CW35HTWLSPG5KD7U.json","view_paper":"https://pith.science/paper/AEOPK3LT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.01592&json=true","fetch_graph":"https://pith.science/api/pith-number/AEOPK3LTC6CW35HTWLSPG5KD7U/graph.json","fetch_events":"https://pith.science/api/pith-number/AEOPK3LTC6CW35HTWLSPG5KD7U/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AEOPK3LTC6CW35HTWLSPG5KD7U/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AEOPK3LTC6CW35HTWLSPG5KD7U/action/storage_attestation","attest_author":"https://pith.science/pith/AEOPK3LTC6CW35HTWLSPG5KD7U/action/author_attestation","sign_citation":"https://pith.science/pith/AEOPK3LTC6CW35HTWLSPG5KD7U/action/citation_signature","submit_replication":"https://pith.science/pith/AEOPK3LTC6CW35HTWLSPG5KD7U/action/replication_record"}},"created_at":"2026-05-18T00:30:55.184697+00:00","updated_at":"2026-05-18T00:30:55.184697+00:00"}