{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:DCQWFGF2NCXHMKWEOOYE7NO4LR","short_pith_number":"pith:DCQWFGF2","canonical_record":{"source":{"id":"1609.01008","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-09-05T01:00:36Z","cross_cats_sorted":[],"title_canon_sha256":"dc52c99b4a1a8963e2859196cd3997f73afb40bd84aa75a4cb4523ffa8ef86bd","abstract_canon_sha256":"a00aeb05656f96fd774adad8c401543bc3bab145c7f6c13da93a3a9d2a961283"},"schema_version":"1.0"},"canonical_sha256":"18a16298ba68ae762ac473b04fb5dc5c6bedbf0b9199d2ec429e274f7d9979b7","source":{"kind":"arxiv","id":"1609.01008","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.01008","created_at":"2026-05-18T00:43:35Z"},{"alias_kind":"arxiv_version","alias_value":"1609.01008v2","created_at":"2026-05-18T00:43:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.01008","created_at":"2026-05-18T00:43:35Z"},{"alias_kind":"pith_short_12","alias_value":"DCQWFGF2NCXH","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"DCQWFGF2NCXHMKWE","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"DCQWFGF2","created_at":"2026-05-18T12:30:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:DCQWFGF2NCXHMKWEOOYE7NO4LR","target":"record","payload":{"canonical_record":{"source":{"id":"1609.01008","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-09-05T01:00:36Z","cross_cats_sorted":[],"title_canon_sha256":"dc52c99b4a1a8963e2859196cd3997f73afb40bd84aa75a4cb4523ffa8ef86bd","abstract_canon_sha256":"a00aeb05656f96fd774adad8c401543bc3bab145c7f6c13da93a3a9d2a961283"},"schema_version":"1.0"},"canonical_sha256":"18a16298ba68ae762ac473b04fb5dc5c6bedbf0b9199d2ec429e274f7d9979b7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:35.646753Z","signature_b64":"86WKDaec4ww1My6/nfNzHV/4Lip2D7H9MlFuXe5JT8fVRofbx2sDusYxNz3bTOo/wSsaTxnFQ5InMjcCBR4oCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"18a16298ba68ae762ac473b04fb5dc5c6bedbf0b9199d2ec429e274f7d9979b7","last_reissued_at":"2026-05-18T00:43:35.646328Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:35.646328Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.01008","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":"2026-05-18T00:43:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wj/OL952ArFJPpg3qQh7WIsIYbMVyMy07h5PKNCIKf/29TYstjeH+/nGNOdPs4//3wrzZxwvuzJsh7xnORcdAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:22:15.740192Z"},"content_sha256":"f713c0f3147ab3d737e18c261f567dd7244953f743cb789e37d2d8c2f4e89909","schema_version":"1.0","event_id":"sha256:f713c0f3147ab3d737e18c261f567dd7244953f743cb789e37d2d8c2f4e89909"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:DCQWFGF2NCXHMKWEOOYE7NO4LR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An integral formula for affine connections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Chao Xia, Junfang Li","submitted_at":"2016-09-05T01:00:36Z","abstract_excerpt":"In this article, we introduce a $2$-parameter family of affine connections and derive the Ricci curvature. We first establish an integral Bochner technique. On one hand, this technique yields a new proof to our recent work in \\cite{LX} for substatic manifolds. On the other hand, this technique leads to various geometric inequalities and eigenvalue estimates under a much more general Ricci curvature conditions. The new Ricci curvature condition interpolates between static Ricci tensor and $1$-Bakry-Emery Ricci, and also includes the conformal Ricci as an intermediate case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.01008","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":"2026-05-18T00:43:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"euNTKAuEj7CtQSRUw4TMCTfpOB5f1aMmtS/Ca8hVLG0ojfrcxEIngWzswGSge+Ze0rqHvaS182NjhFZlBtQaBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:22:15.740557Z"},"content_sha256":"530970c738281c5243aae19f3c537b08fd987faaa01f16b60e0dabaeff410b8d","schema_version":"1.0","event_id":"sha256:530970c738281c5243aae19f3c537b08fd987faaa01f16b60e0dabaeff410b8d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DCQWFGF2NCXHMKWEOOYE7NO4LR/bundle.json","state_url":"https://pith.science/pith/DCQWFGF2NCXHMKWEOOYE7NO4LR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DCQWFGF2NCXHMKWEOOYE7NO4LR/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":"2026-06-22T23:22:15Z","links":{"resolver":"https://pith.science/pith/DCQWFGF2NCXHMKWEOOYE7NO4LR","bundle":"https://pith.science/pith/DCQWFGF2NCXHMKWEOOYE7NO4LR/bundle.json","state":"https://pith.science/pith/DCQWFGF2NCXHMKWEOOYE7NO4LR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DCQWFGF2NCXHMKWEOOYE7NO4LR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:DCQWFGF2NCXHMKWEOOYE7NO4LR","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":"a00aeb05656f96fd774adad8c401543bc3bab145c7f6c13da93a3a9d2a961283","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-09-05T01:00:36Z","title_canon_sha256":"dc52c99b4a1a8963e2859196cd3997f73afb40bd84aa75a4cb4523ffa8ef86bd"},"schema_version":"1.0","source":{"id":"1609.01008","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.01008","created_at":"2026-05-18T00:43:35Z"},{"alias_kind":"arxiv_version","alias_value":"1609.01008v2","created_at":"2026-05-18T00:43:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.01008","created_at":"2026-05-18T00:43:35Z"},{"alias_kind":"pith_short_12","alias_value":"DCQWFGF2NCXH","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"DCQWFGF2NCXHMKWE","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"DCQWFGF2","created_at":"2026-05-18T12:30:12Z"}],"graph_snapshots":[{"event_id":"sha256:530970c738281c5243aae19f3c537b08fd987faaa01f16b60e0dabaeff410b8d","target":"graph","created_at":"2026-05-18T00:43:35Z","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 article, we introduce a $2$-parameter family of affine connections and derive the Ricci curvature. We first establish an integral Bochner technique. On one hand, this technique yields a new proof to our recent work in \\cite{LX} for substatic manifolds. On the other hand, this technique leads to various geometric inequalities and eigenvalue estimates under a much more general Ricci curvature conditions. The new Ricci curvature condition interpolates between static Ricci tensor and $1$-Bakry-Emery Ricci, and also includes the conformal Ricci as an intermediate case.","authors_text":"Chao Xia, Junfang Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-09-05T01:00:36Z","title":"An integral formula for affine connections"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.01008","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:f713c0f3147ab3d737e18c261f567dd7244953f743cb789e37d2d8c2f4e89909","target":"record","created_at":"2026-05-18T00:43:35Z","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":"a00aeb05656f96fd774adad8c401543bc3bab145c7f6c13da93a3a9d2a961283","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-09-05T01:00:36Z","title_canon_sha256":"dc52c99b4a1a8963e2859196cd3997f73afb40bd84aa75a4cb4523ffa8ef86bd"},"schema_version":"1.0","source":{"id":"1609.01008","kind":"arxiv","version":2}},"canonical_sha256":"18a16298ba68ae762ac473b04fb5dc5c6bedbf0b9199d2ec429e274f7d9979b7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"18a16298ba68ae762ac473b04fb5dc5c6bedbf0b9199d2ec429e274f7d9979b7","first_computed_at":"2026-05-18T00:43:35.646328Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:43:35.646328Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"86WKDaec4ww1My6/nfNzHV/4Lip2D7H9MlFuXe5JT8fVRofbx2sDusYxNz3bTOo/wSsaTxnFQ5InMjcCBR4oCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:43:35.646753Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.01008","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f713c0f3147ab3d737e18c261f567dd7244953f743cb789e37d2d8c2f4e89909","sha256:530970c738281c5243aae19f3c537b08fd987faaa01f16b60e0dabaeff410b8d"],"state_sha256":"55f10bf1e74a3bb9b2594e140d3addacf498c6964a7289cda983a7004af6db4f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KXKya5XP1+e3q9dm+jEB1AmtZkn4ZNBSHNAHkqw0/3OaEgjsjx5yktCr9qzppi9fttittrc1bHSc9qvh/cnwAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T23:22:15.742449Z","bundle_sha256":"1e93abc8516e5e5786ef2ece985bb5b1a12405bd2f2bcf42dd211d2dd8d80582"}}