{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:WAR4YETIZWPFBJ5E52Y3L27RIC","short_pith_number":"pith:WAR4YETI","canonical_record":{"source":{"id":"1908.09465","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-08-26T04:36:26Z","cross_cats_sorted":[],"title_canon_sha256":"38988c60508cd0cb27b6fdca206d25a7f6ff2e6b70fdac0d607dcb0fec2a594b","abstract_canon_sha256":"44362157bdbc383825403e1180b9a3ade40d46573139f04d9ff31512793354f2"},"schema_version":"1.0"},"canonical_sha256":"b023cc1268cd9e50a7a4eeb1b5ebf140996c6da7797b15b886be18040fd4d0b1","source":{"kind":"arxiv","id":"1908.09465","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.09465","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"arxiv_version","alias_value":"1908.09465v1","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.09465","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"pith_short_12","alias_value":"WAR4YETIZWPF","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"pith_short_16","alias_value":"WAR4YETIZWPFBJ5E","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"pith_short_8","alias_value":"WAR4YETI","created_at":"2026-07-04T23:59:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:WAR4YETIZWPFBJ5E52Y3L27RIC","target":"record","payload":{"canonical_record":{"source":{"id":"1908.09465","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-08-26T04:36:26Z","cross_cats_sorted":[],"title_canon_sha256":"38988c60508cd0cb27b6fdca206d25a7f6ff2e6b70fdac0d607dcb0fec2a594b","abstract_canon_sha256":"44362157bdbc383825403e1180b9a3ade40d46573139f04d9ff31512793354f2"},"schema_version":"1.0"},"canonical_sha256":"b023cc1268cd9e50a7a4eeb1b5ebf140996c6da7797b15b886be18040fd4d0b1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:59:40.376622Z","signature_b64":"/JDyn8D8IvL/68gKNIvr+b5W2WySunV/xqieHSXG5QWZigeCkxlnQjsJfZz2qiHLPCGklgLMnYRgLg0aXjONBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b023cc1268cd9e50a7a4eeb1b5ebf140996c6da7797b15b886be18040fd4d0b1","last_reissued_at":"2026-07-04T23:59:40.376085Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:59:40.376085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.09465","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":"2026-07-04T23:59:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KhBZWsTUBAarHWcgbQ/3fin1ipvrxGjdQWhXqvNJ++adEgNrPZspyDy+RPW2ndyFDvDJ1CcBkQb4BDL43drcCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T22:44:40.200478Z"},"content_sha256":"ed0ae0bc4a3e6b9fcaa949bed7ed228f1f0a782831cf560ceac9e9a783b120c9","schema_version":"1.0","event_id":"sha256:ed0ae0bc4a3e6b9fcaa949bed7ed228f1f0a782831cf560ceac9e9a783b120c9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:WAR4YETIZWPFBJ5E52Y3L27RIC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weighted Projective Ricci Curvature in Finsler Geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"A. Tayebi, B. Najafi, T. Tabatabaeifar","submitted_at":"2019-08-26T04:36:26Z","abstract_excerpt":"In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09465","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.09465/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T23:59:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Op84KrL0HQHwMwc5ZRH1V9Pb+KzFlM0lYdAJYXYFkVu3kqmkWaPYDz2Cv8Yv9MAa5gxTu0BdceMgFsTaFwefBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T22:44:40.201059Z"},"content_sha256":"4799eef1129bf3ec876dc5ba6859782c822eaa0bd5d2900ffebc1d80d90a92b6","schema_version":"1.0","event_id":"sha256:4799eef1129bf3ec876dc5ba6859782c822eaa0bd5d2900ffebc1d80d90a92b6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WAR4YETIZWPFBJ5E52Y3L27RIC/bundle.json","state_url":"https://pith.science/pith/WAR4YETIZWPFBJ5E52Y3L27RIC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WAR4YETIZWPFBJ5E52Y3L27RIC/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":"2026-08-16T22:44:40Z","links":{"resolver":"https://pith.science/pith/WAR4YETIZWPFBJ5E52Y3L27RIC","bundle":"https://pith.science/pith/WAR4YETIZWPFBJ5E52Y3L27RIC/bundle.json","state":"https://pith.science/pith/WAR4YETIZWPFBJ5E52Y3L27RIC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WAR4YETIZWPFBJ5E52Y3L27RIC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:WAR4YETIZWPFBJ5E52Y3L27RIC","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":"44362157bdbc383825403e1180b9a3ade40d46573139f04d9ff31512793354f2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-08-26T04:36:26Z","title_canon_sha256":"38988c60508cd0cb27b6fdca206d25a7f6ff2e6b70fdac0d607dcb0fec2a594b"},"schema_version":"1.0","source":{"id":"1908.09465","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.09465","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"arxiv_version","alias_value":"1908.09465v1","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.09465","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"pith_short_12","alias_value":"WAR4YETIZWPF","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"pith_short_16","alias_value":"WAR4YETIZWPFBJ5E","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"pith_short_8","alias_value":"WAR4YETI","created_at":"2026-07-04T23:59:40Z"}],"graph_snapshots":[{"event_id":"sha256:4799eef1129bf3ec876dc5ba6859782c822eaa0bd5d2900ffebc1d80d90a92b6","target":"graph","created_at":"2026-07-04T23:59:40Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.09465/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.","authors_text":"A. Tayebi, B. Najafi, T. Tabatabaeifar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-08-26T04:36:26Z","title":"Weighted Projective Ricci Curvature in Finsler Geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09465","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:ed0ae0bc4a3e6b9fcaa949bed7ed228f1f0a782831cf560ceac9e9a783b120c9","target":"record","created_at":"2026-07-04T23:59:40Z","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":"44362157bdbc383825403e1180b9a3ade40d46573139f04d9ff31512793354f2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-08-26T04:36:26Z","title_canon_sha256":"38988c60508cd0cb27b6fdca206d25a7f6ff2e6b70fdac0d607dcb0fec2a594b"},"schema_version":"1.0","source":{"id":"1908.09465","kind":"arxiv","version":1}},"canonical_sha256":"b023cc1268cd9e50a7a4eeb1b5ebf140996c6da7797b15b886be18040fd4d0b1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b023cc1268cd9e50a7a4eeb1b5ebf140996c6da7797b15b886be18040fd4d0b1","first_computed_at":"2026-07-04T23:59:40.376085Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:59:40.376085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/JDyn8D8IvL/68gKNIvr+b5W2WySunV/xqieHSXG5QWZigeCkxlnQjsJfZz2qiHLPCGklgLMnYRgLg0aXjONBg==","signature_status":"signed_v1","signed_at":"2026-07-04T23:59:40.376622Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.09465","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ed0ae0bc4a3e6b9fcaa949bed7ed228f1f0a782831cf560ceac9e9a783b120c9","sha256:4799eef1129bf3ec876dc5ba6859782c822eaa0bd5d2900ffebc1d80d90a92b6"],"state_sha256":"9d99448d9a5a7b776c3f015637d04f2d8a79ad61d88de49e956e2c6a48451d9b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"66W3XNsOMkI4ftQ6rALcYv9e67qDaSZ1/Ee4bsyUNfQlXvg3y/bY4AdgbinCJlASKTIOpDRT/oguCJJwzOqHCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T22:44:40.206729Z","bundle_sha256":"143258ab7f65a16274390ee4d0af70b67f93378e1d441fc12ee4622c042fb922"}}