{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:L3PRXN7GZJCVJQYB7N56LLEZWB","short_pith_number":"pith:L3PRXN7G","schema_version":"1.0","canonical_sha256":"5edf1bb7e6ca4554c301fb7be5ac99b0604df3c550f13a2e656b8ea9ce216221","source":{"kind":"arxiv","id":"1701.06515","version":3},"attestation_state":"computed","paper":{"title":"A characterization of codimension one collapse under bounded curvature and diameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Saskia Roos","submitted_at":"2017-01-23T17:23:05Z","abstract_excerpt":"Let $\\mathcal{M}(n,D)$ be the space of closed $n$-dimensional Riemannian manifolds $(M,g)$ with $diam(M) \\leq D$ and $| \\sec^M | \\leq 1$. In this paper we consider sequences $(M_i,g_i)$ in $\\mathcal{M}(n,D)$ converging in the Gromov-Hausdorff topology to a compact metric space $Y$. We show on the one hand that the limit space of this sequence has at most codimension $1$ if there is a positive number $r$ such that the quotient $\\frac{vol(B^{M_i}_r(x))}{inj^{M_i}(x)}$ can be uniformly bounded from below by a positive constant $C(n,r,Y)$ for all points $x \\in M_i$. On the other hand, we show that"},"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":"1701.06515","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-01-23T17:23:05Z","cross_cats_sorted":[],"title_canon_sha256":"4a5d6e128306084decdd2fec2b09c3126fa4767738f6eeb1e60a7b5225ddf725","abstract_canon_sha256":"6a6b08eebe95de79bbdcbbdb23f3e2a163ae57ccc317912c473e37248c3e2580"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:07.417984Z","signature_b64":"/z5ks5s0gGOPE+Y0EeqlqcSHNr12JE3nIXGJZXPevh+1++OoOUOzxItHV/8xz7xOqRwSJIID+kvRepp/NCrCDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5edf1bb7e6ca4554c301fb7be5ac99b0604df3c550f13a2e656b8ea9ce216221","last_reissued_at":"2026-05-18T00:40:07.417423Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:07.417423Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A characterization of codimension one collapse under bounded curvature and diameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Saskia Roos","submitted_at":"2017-01-23T17:23:05Z","abstract_excerpt":"Let $\\mathcal{M}(n,D)$ be the space of closed $n$-dimensional Riemannian manifolds $(M,g)$ with $diam(M) \\leq D$ and $| \\sec^M | \\leq 1$. In this paper we consider sequences $(M_i,g_i)$ in $\\mathcal{M}(n,D)$ converging in the Gromov-Hausdorff topology to a compact metric space $Y$. We show on the one hand that the limit space of this sequence has at most codimension $1$ if there is a positive number $r$ such that the quotient $\\frac{vol(B^{M_i}_r(x))}{inj^{M_i}(x)}$ can be uniformly bounded from below by a positive constant $C(n,r,Y)$ for all points $x \\in M_i$. On the other hand, we show that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.06515","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":"1701.06515","created_at":"2026-05-18T00:40:07.417502+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.06515v3","created_at":"2026-05-18T00:40:07.417502+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.06515","created_at":"2026-05-18T00:40:07.417502+00:00"},{"alias_kind":"pith_short_12","alias_value":"L3PRXN7GZJCV","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_16","alias_value":"L3PRXN7GZJCVJQYB","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_8","alias_value":"L3PRXN7G","created_at":"2026-05-18T12:31:28.150371+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/L3PRXN7GZJCVJQYB7N56LLEZWB","json":"https://pith.science/pith/L3PRXN7GZJCVJQYB7N56LLEZWB.json","graph_json":"https://pith.science/api/pith-number/L3PRXN7GZJCVJQYB7N56LLEZWB/graph.json","events_json":"https://pith.science/api/pith-number/L3PRXN7GZJCVJQYB7N56LLEZWB/events.json","paper":"https://pith.science/paper/L3PRXN7G"},"agent_actions":{"view_html":"https://pith.science/pith/L3PRXN7GZJCVJQYB7N56LLEZWB","download_json":"https://pith.science/pith/L3PRXN7GZJCVJQYB7N56LLEZWB.json","view_paper":"https://pith.science/paper/L3PRXN7G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.06515&json=true","fetch_graph":"https://pith.science/api/pith-number/L3PRXN7GZJCVJQYB7N56LLEZWB/graph.json","fetch_events":"https://pith.science/api/pith-number/L3PRXN7GZJCVJQYB7N56LLEZWB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L3PRXN7GZJCVJQYB7N56LLEZWB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L3PRXN7GZJCVJQYB7N56LLEZWB/action/storage_attestation","attest_author":"https://pith.science/pith/L3PRXN7GZJCVJQYB7N56LLEZWB/action/author_attestation","sign_citation":"https://pith.science/pith/L3PRXN7GZJCVJQYB7N56LLEZWB/action/citation_signature","submit_replication":"https://pith.science/pith/L3PRXN7GZJCVJQYB7N56LLEZWB/action/replication_record"}},"created_at":"2026-05-18T00:40:07.417502+00:00","updated_at":"2026-05-18T00:40:07.417502+00:00"}