{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:AWTV4BOPWMNSYD4R4SXGHBM5CY","short_pith_number":"pith:AWTV4BOP","schema_version":"1.0","canonical_sha256":"05a75e05cfb31b2c0f91e4ae63859d161e07af025eefb72385335f5112a8046f","source":{"kind":"arxiv","id":"1701.06045","version":2},"attestation_state":"computed","paper":{"title":"Umbilical spacelike submanifolds of arbitrary co-dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["gr-qc"],"primary_cat":"math.DG","authors_text":"Jos\\'e M. M. Senovilla, Nastassja Cipriani","submitted_at":"2017-01-21T15:57:04Z","abstract_excerpt":"Given a semi-Riemannian manifold, we give necessary and sufficient conditions for a Riemannian submanifold of arbitrary co-dimension to be umbilical along normal directions. We do that by using the so-called \\emph{total shear tensor}, i.e., the trace-free part of the second fundamental form. We define the \\emph{shear space} and the \\emph{umbilical space} as the spaces generated by the total shear tensor and by the umbilical vector fields, respectively. We show that the sum of their dimensions must equal the co-dimension."},"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":"1701.06045","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-01-21T15:57:04Z","cross_cats_sorted":["gr-qc"],"title_canon_sha256":"cbcf8b539d8b0d114dc80cfb3b32a0cfbe0592db8f8c84e09c33423f2af46048","abstract_canon_sha256":"ff59976e5ebfa715817ed29942ebe62fb21948346b6fc30898e83c4ba811eb28"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:42.084988Z","signature_b64":"Jqn04Rc4LFrRY86hBL60PWJa+crbyqTVx96DLwFsbopCA+8ViXjCogfMWLu4qPvWTjWybzicn+VClRSnWYHvAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"05a75e05cfb31b2c0f91e4ae63859d161e07af025eefb72385335f5112a8046f","last_reissued_at":"2026-05-18T00:50:42.084239Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:42.084239Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Umbilical spacelike submanifolds of arbitrary co-dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["gr-qc"],"primary_cat":"math.DG","authors_text":"Jos\\'e M. M. Senovilla, Nastassja Cipriani","submitted_at":"2017-01-21T15:57:04Z","abstract_excerpt":"Given a semi-Riemannian manifold, we give necessary and sufficient conditions for a Riemannian submanifold of arbitrary co-dimension to be umbilical along normal directions. We do that by using the so-called \\emph{total shear tensor}, i.e., the trace-free part of the second fundamental form. We define the \\emph{shear space} and the \\emph{umbilical space} as the spaces generated by the total shear tensor and by the umbilical vector fields, respectively. We show that the sum of their dimensions must equal the co-dimension."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.06045","kind":"arxiv","version":2},"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":"1701.06045","created_at":"2026-05-18T00:50:42.084365+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.06045v2","created_at":"2026-05-18T00:50:42.084365+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.06045","created_at":"2026-05-18T00:50:42.084365+00:00"},{"alias_kind":"pith_short_12","alias_value":"AWTV4BOPWMNS","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"AWTV4BOPWMNSYD4R","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"AWTV4BOP","created_at":"2026-05-18T12:31:08.081275+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/AWTV4BOPWMNSYD4R4SXGHBM5CY","json":"https://pith.science/pith/AWTV4BOPWMNSYD4R4SXGHBM5CY.json","graph_json":"https://pith.science/api/pith-number/AWTV4BOPWMNSYD4R4SXGHBM5CY/graph.json","events_json":"https://pith.science/api/pith-number/AWTV4BOPWMNSYD4R4SXGHBM5CY/events.json","paper":"https://pith.science/paper/AWTV4BOP"},"agent_actions":{"view_html":"https://pith.science/pith/AWTV4BOPWMNSYD4R4SXGHBM5CY","download_json":"https://pith.science/pith/AWTV4BOPWMNSYD4R4SXGHBM5CY.json","view_paper":"https://pith.science/paper/AWTV4BOP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.06045&json=true","fetch_graph":"https://pith.science/api/pith-number/AWTV4BOPWMNSYD4R4SXGHBM5CY/graph.json","fetch_events":"https://pith.science/api/pith-number/AWTV4BOPWMNSYD4R4SXGHBM5CY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AWTV4BOPWMNSYD4R4SXGHBM5CY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AWTV4BOPWMNSYD4R4SXGHBM5CY/action/storage_attestation","attest_author":"https://pith.science/pith/AWTV4BOPWMNSYD4R4SXGHBM5CY/action/author_attestation","sign_citation":"https://pith.science/pith/AWTV4BOPWMNSYD4R4SXGHBM5CY/action/citation_signature","submit_replication":"https://pith.science/pith/AWTV4BOPWMNSYD4R4SXGHBM5CY/action/replication_record"}},"created_at":"2026-05-18T00:50:42.084365+00:00","updated_at":"2026-05-18T00:50:42.084365+00:00"}