{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:TVYRBC5QWPD27VCEOATLJP7YEP","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"250cd2787152e92cd9eb256b9657a013615bcdd7c6f33c9008bdc3818cb61b88","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-11T03:11:53Z","title_canon_sha256":"552d981ffc6053c50fc7eebb9acdeba76712a25daa1dfb6d55073bb4a7e53f6a"},"schema_version":"1.0","source":{"id":"1506.03535","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.03535","created_at":"2026-05-18T01:15:36Z"},{"alias_kind":"arxiv_version","alias_value":"1506.03535v2","created_at":"2026-05-18T01:15:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.03535","created_at":"2026-05-18T01:15:36Z"},{"alias_kind":"pith_short_12","alias_value":"TVYRBC5QWPD2","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TVYRBC5QWPD27VCE","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TVYRBC5Q","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:d5497ea7a14b0c8d934502396a646c82b76dfdab69dd29bfb8b0db10b6396717","target":"graph","created_at":"2026-05-18T01:15:36Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $M^n\\ (n\\geq3)$ be a complete Riemannian manifold with $\\sec_M\\geq 1$, and let $M_i^{n_i}$ ($i=1,2$) be two comlplete totally geodesic submanifolds in $M$. We prove that if $n_1+n_2=n-2$ and if the distance $|M_1M_2|\\geq\\frac{\\pi}{2}$, then $M_i$ is isometric to $\\Bbb S^{n_i}/\\Bbb Z_h$, ${\\Bbb C\\Bbb P}^{\\frac {n_i}2}$ or ${\\Bbb C\\Bbb P}^{\\frac {n_i}2}/\\Bbb Z_2$ with the canonical metric when $n_i>0$, and thus $M$ is isometric to $\\Bbb S^n/\\Bbb Z_h$, ${\\Bbb C\\Bbb P}^{\\frac n2}$ or ${\\Bbb C\\Bbb P}^{\\frac n2}/\\Bbb Z_2$ except possibly when $n=3$ and $M_1$ (or $M_2$) $\\stackrel{\\rm iso}{\\cong}","authors_text":"Hongwei Sun, Xiaole Su, Yusheng Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-11T03:11:53Z","title":"An Isometrical ${\\Bbb C\\Bbb P}^{n}$-Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.03535","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4a2b2b9b2246c8159df235495178bd18b9cf5c731a9eff2922898db324e3263e","target":"record","created_at":"2026-05-18T01:15:36Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"250cd2787152e92cd9eb256b9657a013615bcdd7c6f33c9008bdc3818cb61b88","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-11T03:11:53Z","title_canon_sha256":"552d981ffc6053c50fc7eebb9acdeba76712a25daa1dfb6d55073bb4a7e53f6a"},"schema_version":"1.0","source":{"id":"1506.03535","kind":"arxiv","version":2}},"canonical_sha256":"9d71108bb0b3c7afd4447026b4bff823fcbf473e378a3c54e5d0d699dbcd3910","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9d71108bb0b3c7afd4447026b4bff823fcbf473e378a3c54e5d0d699dbcd3910","first_computed_at":"2026-05-18T01:15:36.206175Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:15:36.206175Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VYYbyGOz/KwYV4OiFloNT8iIAO4zAxmd67JfYLMAsxBMa8DWJvILTK63ma6+oEciu/rN/fOCv+F8P0V7FkjFBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:15:36.206888Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.03535","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a2b2b9b2246c8159df235495178bd18b9cf5c731a9eff2922898db324e3263e","sha256:d5497ea7a14b0c8d934502396a646c82b76dfdab69dd29bfb8b0db10b6396717"],"state_sha256":"51e219a604fcf7cfb8252a25caee55b339d28dab6195a34979102bdc71d1418d"}