{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:24ZNKRE4YR3QUVM2FT5DLS44U7","short_pith_number":"pith:24ZNKRE4","canonical_record":{"source":{"id":"0909.4957","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-09-27T19:31:05Z","cross_cats_sorted":[],"title_canon_sha256":"1837f33e87333d746ed14683d526017072cc9185f1393073e9716efbdf97f52e","abstract_canon_sha256":"ac6e87bc5facc429cc8cb39a9365608225d7db06f0aa833849b7e46106567a9d"},"schema_version":"1.0"},"canonical_sha256":"d732d5449cc4770a559a2cfa35cb9ca7f1f1e3ae10157df84a72d19eca52e8d9","source":{"kind":"arxiv","id":"0909.4957","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.4957","created_at":"2026-05-18T03:45:01Z"},{"alias_kind":"arxiv_version","alias_value":"0909.4957v2","created_at":"2026-05-18T03:45:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.4957","created_at":"2026-05-18T03:45:01Z"},{"alias_kind":"pith_short_12","alias_value":"24ZNKRE4YR3Q","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"24ZNKRE4YR3QUVM2","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"24ZNKRE4","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:24ZNKRE4YR3QUVM2FT5DLS44U7","target":"record","payload":{"canonical_record":{"source":{"id":"0909.4957","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-09-27T19:31:05Z","cross_cats_sorted":[],"title_canon_sha256":"1837f33e87333d746ed14683d526017072cc9185f1393073e9716efbdf97f52e","abstract_canon_sha256":"ac6e87bc5facc429cc8cb39a9365608225d7db06f0aa833849b7e46106567a9d"},"schema_version":"1.0"},"canonical_sha256":"d732d5449cc4770a559a2cfa35cb9ca7f1f1e3ae10157df84a72d19eca52e8d9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:01.396916Z","signature_b64":"YWTOYlwIciHwTuCqJzl6nmtYEBBN4BPSI9yPw3lnFaMATmI827rUrWbo+GfzUMFG+Z6JH67FaJAGsYn5JUHnDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d732d5449cc4770a559a2cfa35cb9ca7f1f1e3ae10157df84a72d19eca52e8d9","last_reissued_at":"2026-05-18T03:45:01.396131Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:01.396131Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0909.4957","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":"2026-05-18T03:45:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VAZG7c/SJlPNRWvnIRBgCs2aWYsrnQHIMJqyx9rkBs+GmRpQquEny8oBnsXvywXo1/iVur50YnuiuE/oMkgECw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T20:45:24.679997Z"},"content_sha256":"e4b4c59a9b2f26e6c0c7910952f99360d8de897e6cd1ab05638e01f512061a9d","schema_version":"1.0","event_id":"sha256:e4b4c59a9b2f26e6c0c7910952f99360d8de897e6cd1ab05638e01f512061a9d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:24ZNKRE4YR3QUVM2FT5DLS44U7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Two notes on harmonic distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Kamil Niedzialomski","submitted_at":"2009-09-27T19:31:05Z","abstract_excerpt":"We say that a distribution is harmonic if it is harmonic when considered as a section of a Grassmann bundle. We find new examples of harmonic distributions and show nonexistense of harmonic distrubutions on some Riemannian manifolds by two different approaches. Firstly, we lift distributions to the second tangent bundle equipped with the Sasaki metric. Secondly, we deform conformally the metric on a base manifold."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.4957","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":"2026-05-18T03:45:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TOQdFd2dIyjrKcNIwqbkc6PLcC2gsSLg5JojwEWtinKY9FwLiVJqOEd0MZZpEXpMpcv9qkLj8dWOI1U+9OljAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T20:45:24.680343Z"},"content_sha256":"15518fa41704bd593960dee8285fcc5782808d3f7bb0564c60f4e7c39dcedc38","schema_version":"1.0","event_id":"sha256:15518fa41704bd593960dee8285fcc5782808d3f7bb0564c60f4e7c39dcedc38"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/24ZNKRE4YR3QUVM2FT5DLS44U7/bundle.json","state_url":"https://pith.science/pith/24ZNKRE4YR3QUVM2FT5DLS44U7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/24ZNKRE4YR3QUVM2FT5DLS44U7/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":"2026-06-27T20:45:24Z","links":{"resolver":"https://pith.science/pith/24ZNKRE4YR3QUVM2FT5DLS44U7","bundle":"https://pith.science/pith/24ZNKRE4YR3QUVM2FT5DLS44U7/bundle.json","state":"https://pith.science/pith/24ZNKRE4YR3QUVM2FT5DLS44U7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/24ZNKRE4YR3QUVM2FT5DLS44U7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:24ZNKRE4YR3QUVM2FT5DLS44U7","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":"ac6e87bc5facc429cc8cb39a9365608225d7db06f0aa833849b7e46106567a9d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-09-27T19:31:05Z","title_canon_sha256":"1837f33e87333d746ed14683d526017072cc9185f1393073e9716efbdf97f52e"},"schema_version":"1.0","source":{"id":"0909.4957","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.4957","created_at":"2026-05-18T03:45:01Z"},{"alias_kind":"arxiv_version","alias_value":"0909.4957v2","created_at":"2026-05-18T03:45:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.4957","created_at":"2026-05-18T03:45:01Z"},{"alias_kind":"pith_short_12","alias_value":"24ZNKRE4YR3Q","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"24ZNKRE4YR3QUVM2","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"24ZNKRE4","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:15518fa41704bd593960dee8285fcc5782808d3f7bb0564c60f4e7c39dcedc38","target":"graph","created_at":"2026-05-18T03:45:01Z","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 say that a distribution is harmonic if it is harmonic when considered as a section of a Grassmann bundle. We find new examples of harmonic distributions and show nonexistense of harmonic distrubutions on some Riemannian manifolds by two different approaches. Firstly, we lift distributions to the second tangent bundle equipped with the Sasaki metric. Secondly, we deform conformally the metric on a base manifold.","authors_text":"Kamil Niedzialomski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-09-27T19:31:05Z","title":"Two notes on harmonic distributions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.4957","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:e4b4c59a9b2f26e6c0c7910952f99360d8de897e6cd1ab05638e01f512061a9d","target":"record","created_at":"2026-05-18T03:45:01Z","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":"ac6e87bc5facc429cc8cb39a9365608225d7db06f0aa833849b7e46106567a9d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-09-27T19:31:05Z","title_canon_sha256":"1837f33e87333d746ed14683d526017072cc9185f1393073e9716efbdf97f52e"},"schema_version":"1.0","source":{"id":"0909.4957","kind":"arxiv","version":2}},"canonical_sha256":"d732d5449cc4770a559a2cfa35cb9ca7f1f1e3ae10157df84a72d19eca52e8d9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d732d5449cc4770a559a2cfa35cb9ca7f1f1e3ae10157df84a72d19eca52e8d9","first_computed_at":"2026-05-18T03:45:01.396131Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:45:01.396131Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YWTOYlwIciHwTuCqJzl6nmtYEBBN4BPSI9yPw3lnFaMATmI827rUrWbo+GfzUMFG+Z6JH67FaJAGsYn5JUHnDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:45:01.396916Z","signed_message":"canonical_sha256_bytes"},"source_id":"0909.4957","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e4b4c59a9b2f26e6c0c7910952f99360d8de897e6cd1ab05638e01f512061a9d","sha256:15518fa41704bd593960dee8285fcc5782808d3f7bb0564c60f4e7c39dcedc38"],"state_sha256":"6d3f44a7cb3bfbbaf498f80720ce8b420d9e37680753879d90df1e87512c2995"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wo5OLRqUmiVnexXIPw+4V9zmAlZrhFGj6H9+lZdeTnwe+jyzfP+3pUDjPyOg2Uo+3rg+iGZ/Lk59untbdnHcCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T20:45:24.682152Z","bundle_sha256":"d6e6926c6b2cfbafe00e8c65b856abe0e1c9a9498db6170a17e8764e8831bc33"}}