{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:AKT3GOOBGNKXJNSOWNCJ4XEUGP","short_pith_number":"pith:AKT3GOOB","canonical_record":{"source":{"id":"1806.11102","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-06-27T18:23:40Z","cross_cats_sorted":[],"title_canon_sha256":"180d5ae991c67e00afda940387b31e850a9324fc042b066efb85082e56774ddb","abstract_canon_sha256":"51e90725cc7980cbbbe6cd240ceea7a226b69387611520d1d765b63bc008e336"},"schema_version":"1.0"},"canonical_sha256":"02a7b339c1335574b64eb3449e5c9433c1cb4ea9a7e8a2a8af0a5ce9fdeb1018","source":{"kind":"arxiv","id":"1806.11102","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.11102","created_at":"2026-05-18T00:12:03Z"},{"alias_kind":"arxiv_version","alias_value":"1806.11102v1","created_at":"2026-05-18T00:12:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.11102","created_at":"2026-05-18T00:12:03Z"},{"alias_kind":"pith_short_12","alias_value":"AKT3GOOBGNKX","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"AKT3GOOBGNKXJNSO","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"AKT3GOOB","created_at":"2026-05-18T12:32:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:AKT3GOOBGNKXJNSOWNCJ4XEUGP","target":"record","payload":{"canonical_record":{"source":{"id":"1806.11102","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-06-27T18:23:40Z","cross_cats_sorted":[],"title_canon_sha256":"180d5ae991c67e00afda940387b31e850a9324fc042b066efb85082e56774ddb","abstract_canon_sha256":"51e90725cc7980cbbbe6cd240ceea7a226b69387611520d1d765b63bc008e336"},"schema_version":"1.0"},"canonical_sha256":"02a7b339c1335574b64eb3449e5c9433c1cb4ea9a7e8a2a8af0a5ce9fdeb1018","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:03.561623Z","signature_b64":"oUtKsFw3DTyTaOdgg0pQUEtePW68puO0SGcmQwLXNCBFsIV089FbmGu9c/clB3jNjxB4gUQ+bSwMC3CgwpMQAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02a7b339c1335574b64eb3449e5c9433c1cb4ea9a7e8a2a8af0a5ce9fdeb1018","last_reissued_at":"2026-05-18T00:12:03.561053Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:03.561053Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.11102","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":"2026-05-18T00:12:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2otK/FohTSIFBh6NCPvKRFzH7wPOXTijetnYtudKg/hpfnwk1duZ4BkQ4LbvKrzmRgeShcSlpZd9oLJDy/guCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T11:19:40.069105Z"},"content_sha256":"f0d0f11781ef244e2e76e9b88ffe444d0c5e39021251204e6aae64e6544ed0ce","schema_version":"1.0","event_id":"sha256:f0d0f11781ef244e2e76e9b88ffe444d0c5e39021251204e6aae64e6544ed0ce"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:AKT3GOOBGNKXJNSOWNCJ4XEUGP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Geometry of warped product and CR-warped product submanifolds in Kaehler manifolds: modified version","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Bang-Yen Chen","submitted_at":"2018-06-27T18:23:40Z","abstract_excerpt":"The warped product $N_1\\times_f N_2$ of two Riemannian manifolds $(N_1,g_1)$ and $(N_2,g_2)$ is the product manifold $N_1\\times N_2$ equipped with the warped product metric $g=g_1+f^2 g_2$, where $f$ is a positive function on $N_1$. Warped products play very important roles in differential geometry as well as in physics. A submanifold $M$ of a Kaehler manifold $\\tilde M$ is called a $CR$-warped product if it is a warped product $M_T\\times_f N_\\perp$ of a complex submanifold $M_T$ and a totally real submanifold $M_\\perp$ of $\\tilde M$.\n  In this article we survey recent results on warped produc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.11102","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":"2026-05-18T00:12:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kUfmfpXx8CLmzrU+DgBOKCfFKv5nYFtCPmMJjFZ/kGL2lPNV5HQgKNkRGDSqlrw0Xr+Pcpw9MGJGAJPbExTQDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T11:19:40.069477Z"},"content_sha256":"12f402ab11520eea54f40e224a6f8181488596a774eb4629e1c99f7633f221d4","schema_version":"1.0","event_id":"sha256:12f402ab11520eea54f40e224a6f8181488596a774eb4629e1c99f7633f221d4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AKT3GOOBGNKXJNSOWNCJ4XEUGP/bundle.json","state_url":"https://pith.science/pith/AKT3GOOBGNKXJNSOWNCJ4XEUGP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AKT3GOOBGNKXJNSOWNCJ4XEUGP/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":"2026-06-22T11:19:40Z","links":{"resolver":"https://pith.science/pith/AKT3GOOBGNKXJNSOWNCJ4XEUGP","bundle":"https://pith.science/pith/AKT3GOOBGNKXJNSOWNCJ4XEUGP/bundle.json","state":"https://pith.science/pith/AKT3GOOBGNKXJNSOWNCJ4XEUGP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AKT3GOOBGNKXJNSOWNCJ4XEUGP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:AKT3GOOBGNKXJNSOWNCJ4XEUGP","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":"51e90725cc7980cbbbe6cd240ceea7a226b69387611520d1d765b63bc008e336","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-06-27T18:23:40Z","title_canon_sha256":"180d5ae991c67e00afda940387b31e850a9324fc042b066efb85082e56774ddb"},"schema_version":"1.0","source":{"id":"1806.11102","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.11102","created_at":"2026-05-18T00:12:03Z"},{"alias_kind":"arxiv_version","alias_value":"1806.11102v1","created_at":"2026-05-18T00:12:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.11102","created_at":"2026-05-18T00:12:03Z"},{"alias_kind":"pith_short_12","alias_value":"AKT3GOOBGNKX","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"AKT3GOOBGNKXJNSO","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"AKT3GOOB","created_at":"2026-05-18T12:32:13Z"}],"graph_snapshots":[{"event_id":"sha256:12f402ab11520eea54f40e224a6f8181488596a774eb4629e1c99f7633f221d4","target":"graph","created_at":"2026-05-18T00:12:03Z","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":"The warped product $N_1\\times_f N_2$ of two Riemannian manifolds $(N_1,g_1)$ and $(N_2,g_2)$ is the product manifold $N_1\\times N_2$ equipped with the warped product metric $g=g_1+f^2 g_2$, where $f$ is a positive function on $N_1$. Warped products play very important roles in differential geometry as well as in physics. A submanifold $M$ of a Kaehler manifold $\\tilde M$ is called a $CR$-warped product if it is a warped product $M_T\\times_f N_\\perp$ of a complex submanifold $M_T$ and a totally real submanifold $M_\\perp$ of $\\tilde M$.\n  In this article we survey recent results on warped produc","authors_text":"Bang-Yen Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-06-27T18:23:40Z","title":"Geometry of warped product and CR-warped product submanifolds in Kaehler manifolds: modified version"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.11102","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:f0d0f11781ef244e2e76e9b88ffe444d0c5e39021251204e6aae64e6544ed0ce","target":"record","created_at":"2026-05-18T00:12:03Z","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":"51e90725cc7980cbbbe6cd240ceea7a226b69387611520d1d765b63bc008e336","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-06-27T18:23:40Z","title_canon_sha256":"180d5ae991c67e00afda940387b31e850a9324fc042b066efb85082e56774ddb"},"schema_version":"1.0","source":{"id":"1806.11102","kind":"arxiv","version":1}},"canonical_sha256":"02a7b339c1335574b64eb3449e5c9433c1cb4ea9a7e8a2a8af0a5ce9fdeb1018","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"02a7b339c1335574b64eb3449e5c9433c1cb4ea9a7e8a2a8af0a5ce9fdeb1018","first_computed_at":"2026-05-18T00:12:03.561053Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:12:03.561053Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oUtKsFw3DTyTaOdgg0pQUEtePW68puO0SGcmQwLXNCBFsIV089FbmGu9c/clB3jNjxB4gUQ+bSwMC3CgwpMQAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:12:03.561623Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.11102","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f0d0f11781ef244e2e76e9b88ffe444d0c5e39021251204e6aae64e6544ed0ce","sha256:12f402ab11520eea54f40e224a6f8181488596a774eb4629e1c99f7633f221d4"],"state_sha256":"5b9a4440dcfd45e2da35b435a99a48fd946956c698f8b2817bb29d375a513781"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k9mh4wZtBe01yQYDwlLII5whB2eiCMOiSojx/tj9yiaEuKpeFPXkFn/J6AJTan9N3ISMZ7olB5Y47roJiRMPDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T11:19:40.071439Z","bundle_sha256":"4b2e6c81ae4ebfdacf48fc2aa59b09ada6833914b9e30deb0c6b2ccc37967571"}}