{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:6MKP6WQ3O4LBVUJZTZCRMKFC7B","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":"11d1a633e6a1e3f463ded4ab68f2a62776fcacae686c7cd484ed413a1577e159","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-07-17T08:11:14Z","title_canon_sha256":"cc79b67d5afad405211ff7e754abacb67157118a8b69d383db10d81c65ad2e14"},"schema_version":"1.0","source":{"id":"1707.05036","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.05036","created_at":"2026-05-18T00:08:36Z"},{"alias_kind":"arxiv_version","alias_value":"1707.05036v2","created_at":"2026-05-18T00:08:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.05036","created_at":"2026-05-18T00:08:36Z"},{"alias_kind":"pith_short_12","alias_value":"6MKP6WQ3O4LB","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_16","alias_value":"6MKP6WQ3O4LBVUJZ","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_8","alias_value":"6MKP6WQ3","created_at":"2026-05-18T12:31:03Z"}],"graph_snapshots":[{"event_id":"sha256:50da9a749e3bc7d4bc634b9c856fbd0497e9929d2f34d41084c318295f16be0d","target":"graph","created_at":"2026-05-18T00:08:36Z","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":"For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality involving $L^{\\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant.","authors_text":"Bingqing Ma, Guangyue Huang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-07-17T08:11:14Z","title":"Rigidity of complete Riemannian manifolds with vanishing Bach tensor"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.05036","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":"sha256:6d52f7508d79824604c0bef278c3bddf9312f1b49707dd2abc3caea4302caf51","target":"record","created_at":"2026-05-18T00:08:36Z","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":"11d1a633e6a1e3f463ded4ab68f2a62776fcacae686c7cd484ed413a1577e159","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-07-17T08:11:14Z","title_canon_sha256":"cc79b67d5afad405211ff7e754abacb67157118a8b69d383db10d81c65ad2e14"},"schema_version":"1.0","source":{"id":"1707.05036","kind":"arxiv","version":2}},"canonical_sha256":"f314ff5a1b77161ad1399e451628a2f876cf12db6c9bf77cd75301ccc9b8c0d2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f314ff5a1b77161ad1399e451628a2f876cf12db6c9bf77cd75301ccc9b8c0d2","first_computed_at":"2026-05-18T00:08:36.495109Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:08:36.495109Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IyW1XCiXC1MenuVsbmxwiOpx+LogS37uXe/rq9wFKz+R6cppt+QazwhaAtCcg21JIEcGX/yUO+UDGDxTyvPKBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:08:36.495631Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.05036","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6d52f7508d79824604c0bef278c3bddf9312f1b49707dd2abc3caea4302caf51","sha256:50da9a749e3bc7d4bc634b9c856fbd0497e9929d2f34d41084c318295f16be0d"],"state_sha256":"e876c2cadb98dccff53184e8effd0148ca4fe40d645d4887f9e38adbcfb94ada"}