{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:DN6EX3VA2FXFF2YMKHQCSW7T56","short_pith_number":"pith:DN6EX3VA","canonical_record":{"source":{"id":"1506.03800","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-11T19:51:04Z","cross_cats_sorted":[],"title_canon_sha256":"b550d948438dcbf85dc57c799e93bd62609ba3b5396d07ac29e2a82fc1ec2a49","abstract_canon_sha256":"f92ad9e269340de525fc719710740ce623c425f53373aca18afecdb26a9c06ef"},"schema_version":"1.0"},"canonical_sha256":"1b7c4beea0d16e52eb0c51e0295bf3ef8a233d4ee1be368cc81b483e33720e7a","source":{"kind":"arxiv","id":"1506.03800","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.03800","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"arxiv_version","alias_value":"1506.03800v2","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.03800","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"pith_short_12","alias_value":"DN6EX3VA2FXF","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"DN6EX3VA2FXFF2YM","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"DN6EX3VA","created_at":"2026-05-18T12:29:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:DN6EX3VA2FXFF2YMKHQCSW7T56","target":"record","payload":{"canonical_record":{"source":{"id":"1506.03800","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-11T19:51:04Z","cross_cats_sorted":[],"title_canon_sha256":"b550d948438dcbf85dc57c799e93bd62609ba3b5396d07ac29e2a82fc1ec2a49","abstract_canon_sha256":"f92ad9e269340de525fc719710740ce623c425f53373aca18afecdb26a9c06ef"},"schema_version":"1.0"},"canonical_sha256":"1b7c4beea0d16e52eb0c51e0295bf3ef8a233d4ee1be368cc81b483e33720e7a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:43.833222Z","signature_b64":"d7oEmIIj1fMX3lZG7u+uHYYlOWS5Qgi+cVSFYxI8xi//5LdV6GEs5X6H7bPSOb6GLIDQu6wmgzskTmH5Z1k+Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b7c4beea0d16e52eb0c51e0295bf3ef8a233d4ee1be368cc81b483e33720e7a","last_reissued_at":"2026-05-18T01:11:43.832900Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:43.832900Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.03800","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:11:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3abeviXMtjm8JsnUYVhUkxbxG0CVlpKIDqW4tr8wYP62juk2cDs09vrUR/f7d+zsFYYLIMEISqKjuoDmJQH9CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T22:30:08.789522Z"},"content_sha256":"be070f9aa8f06e331e6cf7f90705347ed9a0e16b35551d8395a4ce2e6e0fa76d","schema_version":"1.0","event_id":"sha256:be070f9aa8f06e331e6cf7f90705347ed9a0e16b35551d8395a4ce2e6e0fa76d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:DN6EX3VA2FXFF2YMKHQCSW7T56","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A warped product version of the Cheeger-Gromoll splitting theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"William Wylie","submitted_at":"2015-06-11T19:51:04Z","abstract_excerpt":"We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form $CD(0,1)$. Even though we have to allow warping in our splitting, we are able to recover topological applications. In particular, for a smooth compact Riemannian manifold admitting a density which is $CD(0,1)$, we show that the fundamental group of $M$ is the fundamental group of a compact manifold with nonnegative sectional curvature. If the space is also locally"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.03800","kind":"arxiv","version":2},"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":"2026-05-18T01:11:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7GVm/WS7XDJvetfqHT/Bx8OTg7lAvMs69UYbc8xfR9YKi3T7LVetSvA+JbYkS3zioCvbH0Ix11FQ0xm98R4aAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T22:30:08.789862Z"},"content_sha256":"5c29cf93a09cb47287d3be4414d3a1e0a0e02baae61ded944debd87d32476631","schema_version":"1.0","event_id":"sha256:5c29cf93a09cb47287d3be4414d3a1e0a0e02baae61ded944debd87d32476631"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DN6EX3VA2FXFF2YMKHQCSW7T56/bundle.json","state_url":"https://pith.science/pith/DN6EX3VA2FXFF2YMKHQCSW7T56/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DN6EX3VA2FXFF2YMKHQCSW7T56/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":"2026-06-29T22:30:08Z","links":{"resolver":"https://pith.science/pith/DN6EX3VA2FXFF2YMKHQCSW7T56","bundle":"https://pith.science/pith/DN6EX3VA2FXFF2YMKHQCSW7T56/bundle.json","state":"https://pith.science/pith/DN6EX3VA2FXFF2YMKHQCSW7T56/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DN6EX3VA2FXFF2YMKHQCSW7T56/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:DN6EX3VA2FXFF2YMKHQCSW7T56","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":"f92ad9e269340de525fc719710740ce623c425f53373aca18afecdb26a9c06ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-11T19:51:04Z","title_canon_sha256":"b550d948438dcbf85dc57c799e93bd62609ba3b5396d07ac29e2a82fc1ec2a49"},"schema_version":"1.0","source":{"id":"1506.03800","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.03800","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"arxiv_version","alias_value":"1506.03800v2","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.03800","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"pith_short_12","alias_value":"DN6EX3VA2FXF","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"DN6EX3VA2FXFF2YM","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"DN6EX3VA","created_at":"2026-05-18T12:29:17Z"}],"graph_snapshots":[{"event_id":"sha256:5c29cf93a09cb47287d3be4414d3a1e0a0e02baae61ded944debd87d32476631","target":"graph","created_at":"2026-05-18T01:11:43Z","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":"We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form $CD(0,1)$. Even though we have to allow warping in our splitting, we are able to recover topological applications. In particular, for a smooth compact Riemannian manifold admitting a density which is $CD(0,1)$, we show that the fundamental group of $M$ is the fundamental group of a compact manifold with nonnegative sectional curvature. If the space is also locally","authors_text":"William Wylie","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-11T19:51:04Z","title":"A warped product version of the Cheeger-Gromoll splitting theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.03800","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:be070f9aa8f06e331e6cf7f90705347ed9a0e16b35551d8395a4ce2e6e0fa76d","target":"record","created_at":"2026-05-18T01:11:43Z","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":"f92ad9e269340de525fc719710740ce623c425f53373aca18afecdb26a9c06ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-11T19:51:04Z","title_canon_sha256":"b550d948438dcbf85dc57c799e93bd62609ba3b5396d07ac29e2a82fc1ec2a49"},"schema_version":"1.0","source":{"id":"1506.03800","kind":"arxiv","version":2}},"canonical_sha256":"1b7c4beea0d16e52eb0c51e0295bf3ef8a233d4ee1be368cc81b483e33720e7a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1b7c4beea0d16e52eb0c51e0295bf3ef8a233d4ee1be368cc81b483e33720e7a","first_computed_at":"2026-05-18T01:11:43.832900Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:43.832900Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"d7oEmIIj1fMX3lZG7u+uHYYlOWS5Qgi+cVSFYxI8xi//5LdV6GEs5X6H7bPSOb6GLIDQu6wmgzskTmH5Z1k+Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:43.833222Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.03800","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:be070f9aa8f06e331e6cf7f90705347ed9a0e16b35551d8395a4ce2e6e0fa76d","sha256:5c29cf93a09cb47287d3be4414d3a1e0a0e02baae61ded944debd87d32476631"],"state_sha256":"7593aa3cdc833728fde84fd9452f9f55add72454ac1b5febc5f3a4f5a5ed49fb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LmDoj2siNiQ6ASFLLzV/xgEavQJcfDPf1dfBk50Q8u92NS/hTM1uE9p/eKp9cIE9ox6EWrq+6xHzqzz2dJ3nBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T22:30:08.791663Z","bundle_sha256":"bd25e7c12a18b69a4c81f3bb29151b868d760ddfcb35f3bf0f6c6ef8e96d0413"}}