{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:WDFF6YLNS3J2BT5Z7JJNRVNZQS","short_pith_number":"pith:WDFF6YLN","canonical_record":{"source":{"id":"1903.00209","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-03-01T08:52:31Z","cross_cats_sorted":[],"title_canon_sha256":"642507d351ba68a20dab7b84a5171799c07739d07b84e6bbc5dfa42a079a1585","abstract_canon_sha256":"977d50b8b55373b74186a61ceb96b3a24e6347ce6cceaf261d123c7605d3e3c2"},"schema_version":"1.0"},"canonical_sha256":"b0ca5f616d96d3a0cfb9fa52d8d5b984944d92efc030c5d6b9ebd6998a70ef84","source":{"kind":"arxiv","id":"1903.00209","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.00209","created_at":"2026-05-17T23:52:20Z"},{"alias_kind":"arxiv_version","alias_value":"1903.00209v1","created_at":"2026-05-17T23:52:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.00209","created_at":"2026-05-17T23:52:20Z"},{"alias_kind":"pith_short_12","alias_value":"WDFF6YLNS3J2","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"WDFF6YLNS3J2BT5Z","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"WDFF6YLN","created_at":"2026-05-18T12:33:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:WDFF6YLNS3J2BT5Z7JJNRVNZQS","target":"record","payload":{"canonical_record":{"source":{"id":"1903.00209","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-03-01T08:52:31Z","cross_cats_sorted":[],"title_canon_sha256":"642507d351ba68a20dab7b84a5171799c07739d07b84e6bbc5dfa42a079a1585","abstract_canon_sha256":"977d50b8b55373b74186a61ceb96b3a24e6347ce6cceaf261d123c7605d3e3c2"},"schema_version":"1.0"},"canonical_sha256":"b0ca5f616d96d3a0cfb9fa52d8d5b984944d92efc030c5d6b9ebd6998a70ef84","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:20.528127Z","signature_b64":"bSA4IFJnp8GFJ6cBXpz4BmYfYSZVjfLWq7xpycpUzHVWLjtZNuk6+zF5r223saQepZuL8ryn2EmKZAkDCtACDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0ca5f616d96d3a0cfb9fa52d8d5b984944d92efc030c5d6b9ebd6998a70ef84","last_reissued_at":"2026-05-17T23:52:20.527620Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:20.527620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1903.00209","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:52:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sD1FN6tqxgFEawZWebPCSxYXr4SohdJ3L4tVesnJMp7/XPEEK84PyMnpGShUZkB1a029EJ/RcFJ30KzbtiEiCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T16:31:24.880993Z"},"content_sha256":"b4771bef88ee387739f90332e69f94700d5353d8bf13dbb8616dac2840d92b34","schema_version":"1.0","event_id":"sha256:b4771bef88ee387739f90332e69f94700d5353d8bf13dbb8616dac2840d92b34"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:WDFF6YLNS3J2BT5Z7JJNRVNZQS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The topological rigidity theorem for submanifolds in space forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hongwei Xu, Juanru Gu","submitted_at":"2019-03-01T08:52:31Z","abstract_excerpt":"Let $M$ be an $n(\\geq 4)$-dimensional compact submanifold\n  in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\\geq 0$, where $H$ is the mean curvature of $M$. We verify that if the scalar curvature of $M$ satisfies\n  $R>n(n-2)(c+H^2)$, and if $Ric_M\\geq (n-2-\\frac{2\\sigma_n}{2n-\\sigma_n})(c+H^2)$,\n  then $M$ is homeomorphic to a sphere. Here $\\sigma_n=sgn(n-4)((-1)^n+3)$, and $sgn(\\cdot)$ is the standard sign function. This improves our previous sphere theorem \\cite{XG2}. It should be emphasized that our pinching conditions above are optimal. We also obtain some new top"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.00209","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:52:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"euPD766qHx/Gzu9HxaiEcKvVcbm1L42Ls1Ftj9OtJSeLsiJTgW3N6JMOBBBUKWabjt7+wuTwOkNJ8DcWMtIQCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T16:31:24.881678Z"},"content_sha256":"21172150ffa577fb9450c6a90b71dd75107cc663672eec8a7b83e547b38bf903","schema_version":"1.0","event_id":"sha256:21172150ffa577fb9450c6a90b71dd75107cc663672eec8a7b83e547b38bf903"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WDFF6YLNS3J2BT5Z7JJNRVNZQS/bundle.json","state_url":"https://pith.science/pith/WDFF6YLNS3J2BT5Z7JJNRVNZQS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WDFF6YLNS3J2BT5Z7JJNRVNZQS/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-29T16:31:24Z","links":{"resolver":"https://pith.science/pith/WDFF6YLNS3J2BT5Z7JJNRVNZQS","bundle":"https://pith.science/pith/WDFF6YLNS3J2BT5Z7JJNRVNZQS/bundle.json","state":"https://pith.science/pith/WDFF6YLNS3J2BT5Z7JJNRVNZQS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WDFF6YLNS3J2BT5Z7JJNRVNZQS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:WDFF6YLNS3J2BT5Z7JJNRVNZQS","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":"977d50b8b55373b74186a61ceb96b3a24e6347ce6cceaf261d123c7605d3e3c2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-03-01T08:52:31Z","title_canon_sha256":"642507d351ba68a20dab7b84a5171799c07739d07b84e6bbc5dfa42a079a1585"},"schema_version":"1.0","source":{"id":"1903.00209","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.00209","created_at":"2026-05-17T23:52:20Z"},{"alias_kind":"arxiv_version","alias_value":"1903.00209v1","created_at":"2026-05-17T23:52:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.00209","created_at":"2026-05-17T23:52:20Z"},{"alias_kind":"pith_short_12","alias_value":"WDFF6YLNS3J2","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"WDFF6YLNS3J2BT5Z","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"WDFF6YLN","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:21172150ffa577fb9450c6a90b71dd75107cc663672eec8a7b83e547b38bf903","target":"graph","created_at":"2026-05-17T23:52:20Z","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 $M$ be an $n(\\geq 4)$-dimensional compact submanifold\n  in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\\geq 0$, where $H$ is the mean curvature of $M$. We verify that if the scalar curvature of $M$ satisfies\n  $R>n(n-2)(c+H^2)$, and if $Ric_M\\geq (n-2-\\frac{2\\sigma_n}{2n-\\sigma_n})(c+H^2)$,\n  then $M$ is homeomorphic to a sphere. Here $\\sigma_n=sgn(n-4)((-1)^n+3)$, and $sgn(\\cdot)$ is the standard sign function. This improves our previous sphere theorem \\cite{XG2}. It should be emphasized that our pinching conditions above are optimal. We also obtain some new top","authors_text":"Hongwei Xu, Juanru Gu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-03-01T08:52:31Z","title":"The topological rigidity theorem for submanifolds in space forms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.00209","kind":"arxiv","version":1},"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:b4771bef88ee387739f90332e69f94700d5353d8bf13dbb8616dac2840d92b34","target":"record","created_at":"2026-05-17T23:52:20Z","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":"977d50b8b55373b74186a61ceb96b3a24e6347ce6cceaf261d123c7605d3e3c2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-03-01T08:52:31Z","title_canon_sha256":"642507d351ba68a20dab7b84a5171799c07739d07b84e6bbc5dfa42a079a1585"},"schema_version":"1.0","source":{"id":"1903.00209","kind":"arxiv","version":1}},"canonical_sha256":"b0ca5f616d96d3a0cfb9fa52d8d5b984944d92efc030c5d6b9ebd6998a70ef84","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b0ca5f616d96d3a0cfb9fa52d8d5b984944d92efc030c5d6b9ebd6998a70ef84","first_computed_at":"2026-05-17T23:52:20.527620Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:52:20.527620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bSA4IFJnp8GFJ6cBXpz4BmYfYSZVjfLWq7xpycpUzHVWLjtZNuk6+zF5r223saQepZuL8ryn2EmKZAkDCtACDA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:52:20.528127Z","signed_message":"canonical_sha256_bytes"},"source_id":"1903.00209","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b4771bef88ee387739f90332e69f94700d5353d8bf13dbb8616dac2840d92b34","sha256:21172150ffa577fb9450c6a90b71dd75107cc663672eec8a7b83e547b38bf903"],"state_sha256":"92d876a946bd771250f5a935c6e9305f509877648c4ad924a37b6f200ab39324"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C/1TKcdu/HGIJs/m7KzCqF14kDx1RpP2Ds0tf4uEWkCgJ5jqdAHn0dPIn49XxXMi+k5+51QKbkz1T5oAB8mcAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T16:31:24.885552Z","bundle_sha256":"4079dbecf2da0c2c0d0115ba03b618d81c503ac805ed4056e17142b3924f5eee"}}