{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:UPHDR2YKUZVQFFIXXPIZRUIFA6","short_pith_number":"pith:UPHDR2YK","canonical_record":{"source":{"id":"2602.15301","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.DG","submitted_at":"2026-02-17T01:53:29Z","cross_cats_sorted":[],"title_canon_sha256":"350075b43520ac71e393b43cfbc54d826e24d3a0c2c1d817639d4a665a4e6326","abstract_canon_sha256":"6b2fb30499065e474fc200d1011572c7f176d05bb980dea4e1abd6e8a3750fce"},"schema_version":"1.0"},"canonical_sha256":"a3ce38eb0aa66b029517bbd198d105078f908c1231bdb811419aebda9ae03508","source":{"kind":"arxiv","id":"2602.15301","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.15301","created_at":"2026-05-20T01:05:09Z"},{"alias_kind":"arxiv_version","alias_value":"2602.15301v2","created_at":"2026-05-20T01:05:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.15301","created_at":"2026-05-20T01:05:09Z"},{"alias_kind":"pith_short_12","alias_value":"UPHDR2YKUZVQ","created_at":"2026-05-20T01:05:09Z"},{"alias_kind":"pith_short_16","alias_value":"UPHDR2YKUZVQFFIX","created_at":"2026-05-20T01:05:09Z"},{"alias_kind":"pith_short_8","alias_value":"UPHDR2YK","created_at":"2026-05-20T01:05:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:UPHDR2YKUZVQFFIXXPIZRUIFA6","target":"record","payload":{"canonical_record":{"source":{"id":"2602.15301","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.DG","submitted_at":"2026-02-17T01:53:29Z","cross_cats_sorted":[],"title_canon_sha256":"350075b43520ac71e393b43cfbc54d826e24d3a0c2c1d817639d4a665a4e6326","abstract_canon_sha256":"6b2fb30499065e474fc200d1011572c7f176d05bb980dea4e1abd6e8a3750fce"},"schema_version":"1.0"},"canonical_sha256":"a3ce38eb0aa66b029517bbd198d105078f908c1231bdb811419aebda9ae03508","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T01:05:09.347984Z","signature_b64":"8SdE6hpiKTwkzVr4bU09k0wtLmTDU1sTVRpCuBwVkgmyAZt1u9zatonlqP+bW5KMebp8SnZegrCxNLzHUmcACQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3ce38eb0aa66b029517bbd198d105078f908c1231bdb811419aebda9ae03508","last_reissued_at":"2026-05-20T01:05:09.347308Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T01:05:09.347308Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2602.15301","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":"2026-05-20T01:05:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gf2OmXyPLmtuBeGQtGmCwJJ4fm97TDUshucgEwEeUhEx1HicMKuZYrssoVXtoZGLRPr/r947/U0h6JILrYtGBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T15:03:23.892055Z"},"content_sha256":"1d52c2ede998be4c2fbb3f743e1547c96da88cbadece7c525f587218396d1e8c","schema_version":"1.0","event_id":"sha256:1d52c2ede998be4c2fbb3f743e1547c96da88cbadece7c525f587218396d1e8c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:UPHDR2YKUZVQFFIXXPIZRUIFA6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"B.-Y. Chen's inequalities for Riemannian submersion and their applications","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Mukut Mani Tripathi, Ravindra Singh","submitted_at":"2026-02-17T01:53:29Z","abstract_excerpt":"In this paper, we introduce B.-Y. Chen inequalities for Riemannian submersions between Riemannian manifolds. We derive these inequalities for vertical, horizontal, and mixed distributions, establishing relationships between intrinsic invariants and extrinsic invariants. We also investigate the corresponding equality cases. As applications, the results are obtained for submersions whose total space is a real, complex, generalized Sasakian space form. Several examples are provided to illustrate both equality and strict inequality cases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.15301","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2602.15301/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":"2026-05-20T01:05:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zaz+pW/GkL1UEhjJT3Tbl8AHmN+/3SE0XO71M1cNHyxzacENKcj7e7MMQlM82CxlyyAYZDakxdQY5DxELKKLBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T15:03:23.892441Z"},"content_sha256":"fa7c77a8f874ada2d59607e3d0d5919386d964266aaed9c086b1a27b3b52b582","schema_version":"1.0","event_id":"sha256:fa7c77a8f874ada2d59607e3d0d5919386d964266aaed9c086b1a27b3b52b582"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UPHDR2YKUZVQFFIXXPIZRUIFA6/bundle.json","state_url":"https://pith.science/pith/UPHDR2YKUZVQFFIXXPIZRUIFA6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UPHDR2YKUZVQFFIXXPIZRUIFA6/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":"2026-07-03T15:03:23Z","links":{"resolver":"https://pith.science/pith/UPHDR2YKUZVQFFIXXPIZRUIFA6","bundle":"https://pith.science/pith/UPHDR2YKUZVQFFIXXPIZRUIFA6/bundle.json","state":"https://pith.science/pith/UPHDR2YKUZVQFFIXXPIZRUIFA6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UPHDR2YKUZVQFFIXXPIZRUIFA6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:UPHDR2YKUZVQFFIXXPIZRUIFA6","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":"6b2fb30499065e474fc200d1011572c7f176d05bb980dea4e1abd6e8a3750fce","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.DG","submitted_at":"2026-02-17T01:53:29Z","title_canon_sha256":"350075b43520ac71e393b43cfbc54d826e24d3a0c2c1d817639d4a665a4e6326"},"schema_version":"1.0","source":{"id":"2602.15301","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.15301","created_at":"2026-05-20T01:05:09Z"},{"alias_kind":"arxiv_version","alias_value":"2602.15301v2","created_at":"2026-05-20T01:05:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.15301","created_at":"2026-05-20T01:05:09Z"},{"alias_kind":"pith_short_12","alias_value":"UPHDR2YKUZVQ","created_at":"2026-05-20T01:05:09Z"},{"alias_kind":"pith_short_16","alias_value":"UPHDR2YKUZVQFFIX","created_at":"2026-05-20T01:05:09Z"},{"alias_kind":"pith_short_8","alias_value":"UPHDR2YK","created_at":"2026-05-20T01:05:09Z"}],"graph_snapshots":[{"event_id":"sha256:fa7c77a8f874ada2d59607e3d0d5919386d964266aaed9c086b1a27b3b52b582","target":"graph","created_at":"2026-05-20T01:05:09Z","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/2602.15301/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we introduce B.-Y. Chen inequalities for Riemannian submersions between Riemannian manifolds. We derive these inequalities for vertical, horizontal, and mixed distributions, establishing relationships between intrinsic invariants and extrinsic invariants. We also investigate the corresponding equality cases. As applications, the results are obtained for submersions whose total space is a real, complex, generalized Sasakian space form. Several examples are provided to illustrate both equality and strict inequality cases.","authors_text":"Mukut Mani Tripathi, Ravindra Singh","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.DG","submitted_at":"2026-02-17T01:53:29Z","title":"B.-Y. Chen's inequalities for Riemannian submersion and their applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.15301","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":"sha256:1d52c2ede998be4c2fbb3f743e1547c96da88cbadece7c525f587218396d1e8c","target":"record","created_at":"2026-05-20T01:05:09Z","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":"6b2fb30499065e474fc200d1011572c7f176d05bb980dea4e1abd6e8a3750fce","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.DG","submitted_at":"2026-02-17T01:53:29Z","title_canon_sha256":"350075b43520ac71e393b43cfbc54d826e24d3a0c2c1d817639d4a665a4e6326"},"schema_version":"1.0","source":{"id":"2602.15301","kind":"arxiv","version":2}},"canonical_sha256":"a3ce38eb0aa66b029517bbd198d105078f908c1231bdb811419aebda9ae03508","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a3ce38eb0aa66b029517bbd198d105078f908c1231bdb811419aebda9ae03508","first_computed_at":"2026-05-20T01:05:09.347308Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T01:05:09.347308Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8SdE6hpiKTwkzVr4bU09k0wtLmTDU1sTVRpCuBwVkgmyAZt1u9zatonlqP+bW5KMebp8SnZegrCxNLzHUmcACQ==","signature_status":"signed_v1","signed_at":"2026-05-20T01:05:09.347984Z","signed_message":"canonical_sha256_bytes"},"source_id":"2602.15301","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1d52c2ede998be4c2fbb3f743e1547c96da88cbadece7c525f587218396d1e8c","sha256:fa7c77a8f874ada2d59607e3d0d5919386d964266aaed9c086b1a27b3b52b582"],"state_sha256":"ae379fa311dd1825827c1786c0b255d57806aebcc07536ea74e7cc1ff3b9995f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DOBEYDJpIy7TFvxpFJrxc1oH4DPSRd2JsZCswqWNY5WT46fYGrx3axlOSJVPXiZc3EQkPH3xtRusUvLU3n4jAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T15:03:23.894944Z","bundle_sha256":"995e3fcc4f5426d7f85495228405cc3af34e6e5a0ba4c9ce0481a020bdcced58"}}