{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:VJLZGNWDPTEMWISFXITSS7C2MZ","short_pith_number":"pith:VJLZGNWD","canonical_record":{"source":{"id":"1112.6302","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.DG","submitted_at":"2011-12-29T13:15:39Z","cross_cats_sorted":[],"title_canon_sha256":"6444e50b368973685df7b0e27fb9447461fdca0929880c571407906c0a1eed1b","abstract_canon_sha256":"f4e19bdaa99f109082ec537834d7aab674460729230cede407999fc2eee3bfd1"},"schema_version":"1.0"},"canonical_sha256":"aa579336c37cc8cb2245ba27297c5a6656143be3fef1d4e99233f53ea071abf4","source":{"kind":"arxiv","id":"1112.6302","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.6302","created_at":"2026-05-18T04:05:36Z"},{"alias_kind":"arxiv_version","alias_value":"1112.6302v1","created_at":"2026-05-18T04:05:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.6302","created_at":"2026-05-18T04:05:36Z"},{"alias_kind":"pith_short_12","alias_value":"VJLZGNWDPTEM","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VJLZGNWDPTEMWISF","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VJLZGNWD","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:VJLZGNWDPTEMWISFXITSS7C2MZ","target":"record","payload":{"canonical_record":{"source":{"id":"1112.6302","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.DG","submitted_at":"2011-12-29T13:15:39Z","cross_cats_sorted":[],"title_canon_sha256":"6444e50b368973685df7b0e27fb9447461fdca0929880c571407906c0a1eed1b","abstract_canon_sha256":"f4e19bdaa99f109082ec537834d7aab674460729230cede407999fc2eee3bfd1"},"schema_version":"1.0"},"canonical_sha256":"aa579336c37cc8cb2245ba27297c5a6656143be3fef1d4e99233f53ea071abf4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:36.283988Z","signature_b64":"KG+hUrXDU8ZhI790EpN0wmhv3LKm7w5FaH5bbzKy62sfZ0D68PhDLD93cTWH3xs6ac6Hg3+yfFL6FW1St6+fDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa579336c37cc8cb2245ba27297c5a6656143be3fef1d4e99233f53ea071abf4","last_reissued_at":"2026-05-18T04:05:36.283518Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:36.283518Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.6302","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":"2026-05-18T04:05:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RcDYOVKT5HIYIkueAYpf6Mgdck+xd63FV5I4QpXt7YdiUfoaME8w50RtARectBYoS6kx7i80pKNTKZtoorgICA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T19:26:54.510434Z"},"content_sha256":"1029dd0160f3c8a21e6fe1bb4cc2f3c78da35d19b3ff7f59374fc70765b1cff0","schema_version":"1.0","event_id":"sha256:1029dd0160f3c8a21e6fe1bb4cc2f3c78da35d19b3ff7f59374fc70765b1cff0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:VJLZGNWDPTEMWISFXITSS7C2MZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Splitting theorems on complete manifolds with Bakry-\\'Emery curvature","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jia-Yong Wu","submitted_at":"2011-12-29T13:15:39Z","abstract_excerpt":"In this paper we study some splitting properties on complete noncompact manifolds with smooth measures when $\\infty$-dimensional Bakry-\\'Emery Ricci curvature is bounded from below by some negative constant and spectrum of the weighted Laplacian has a positive lower bound. These results extend the cases of Ricci curvature and $m$-dimensional Bakry-\\'Emery Ricci curvature."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.6302","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":"2026-05-18T04:05:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"goenmEfHcYiPseKY5haGKOvX61iASAm5NxjzVaKYAvjc6hCaEoXy0hj9C6yM98a4jNFhuYmCSLywCOfjM6SfCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T19:26:54.510780Z"},"content_sha256":"eabdbfd0ad3c3135035263fc341ef973e8742e30309cba49718622102d8cce02","schema_version":"1.0","event_id":"sha256:eabdbfd0ad3c3135035263fc341ef973e8742e30309cba49718622102d8cce02"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VJLZGNWDPTEMWISFXITSS7C2MZ/bundle.json","state_url":"https://pith.science/pith/VJLZGNWDPTEMWISFXITSS7C2MZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VJLZGNWDPTEMWISFXITSS7C2MZ/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":"2026-06-29T19:26:54Z","links":{"resolver":"https://pith.science/pith/VJLZGNWDPTEMWISFXITSS7C2MZ","bundle":"https://pith.science/pith/VJLZGNWDPTEMWISFXITSS7C2MZ/bundle.json","state":"https://pith.science/pith/VJLZGNWDPTEMWISFXITSS7C2MZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VJLZGNWDPTEMWISFXITSS7C2MZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:VJLZGNWDPTEMWISFXITSS7C2MZ","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":"f4e19bdaa99f109082ec537834d7aab674460729230cede407999fc2eee3bfd1","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.DG","submitted_at":"2011-12-29T13:15:39Z","title_canon_sha256":"6444e50b368973685df7b0e27fb9447461fdca0929880c571407906c0a1eed1b"},"schema_version":"1.0","source":{"id":"1112.6302","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.6302","created_at":"2026-05-18T04:05:36Z"},{"alias_kind":"arxiv_version","alias_value":"1112.6302v1","created_at":"2026-05-18T04:05:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.6302","created_at":"2026-05-18T04:05:36Z"},{"alias_kind":"pith_short_12","alias_value":"VJLZGNWDPTEM","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VJLZGNWDPTEMWISF","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VJLZGNWD","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:eabdbfd0ad3c3135035263fc341ef973e8742e30309cba49718622102d8cce02","target":"graph","created_at":"2026-05-18T04:05:36Z","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":"In this paper we study some splitting properties on complete noncompact manifolds with smooth measures when $\\infty$-dimensional Bakry-\\'Emery Ricci curvature is bounded from below by some negative constant and spectrum of the weighted Laplacian has a positive lower bound. These results extend the cases of Ricci curvature and $m$-dimensional Bakry-\\'Emery Ricci curvature.","authors_text":"Jia-Yong Wu","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.DG","submitted_at":"2011-12-29T13:15:39Z","title":"Splitting theorems on complete manifolds with Bakry-\\'Emery curvature"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.6302","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":"sha256:1029dd0160f3c8a21e6fe1bb4cc2f3c78da35d19b3ff7f59374fc70765b1cff0","target":"record","created_at":"2026-05-18T04:05:36Z","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":"f4e19bdaa99f109082ec537834d7aab674460729230cede407999fc2eee3bfd1","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.DG","submitted_at":"2011-12-29T13:15:39Z","title_canon_sha256":"6444e50b368973685df7b0e27fb9447461fdca0929880c571407906c0a1eed1b"},"schema_version":"1.0","source":{"id":"1112.6302","kind":"arxiv","version":1}},"canonical_sha256":"aa579336c37cc8cb2245ba27297c5a6656143be3fef1d4e99233f53ea071abf4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aa579336c37cc8cb2245ba27297c5a6656143be3fef1d4e99233f53ea071abf4","first_computed_at":"2026-05-18T04:05:36.283518Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:05:36.283518Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KG+hUrXDU8ZhI790EpN0wmhv3LKm7w5FaH5bbzKy62sfZ0D68PhDLD93cTWH3xs6ac6Hg3+yfFL6FW1St6+fDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:05:36.283988Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.6302","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1029dd0160f3c8a21e6fe1bb4cc2f3c78da35d19b3ff7f59374fc70765b1cff0","sha256:eabdbfd0ad3c3135035263fc341ef973e8742e30309cba49718622102d8cce02"],"state_sha256":"41e9e4000e56a13c20c22062c0f7db6ab0a420e1b322fec76e1e1afd1ec2bf12"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nHSmVaEBkHF6M68o3kW0+4b0saxRP2bgJRJgzOjVWdmk15ZXfmRl3KcMEZ2lbliolCT5+w1ysOnhwgRnG8anBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T19:26:54.512672Z","bundle_sha256":"0c34701fc7b408cd1cd34c0538a4efa6fe7788cb89ec2f73f57d110a357c8e0c"}}