{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:B4BXUQTQ2W2HDH5D56FZCP3E57","short_pith_number":"pith:B4BXUQTQ","canonical_record":{"source":{"id":"0910.3456","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-10-19T05:01:14Z","cross_cats_sorted":[],"title_canon_sha256":"cecde13a8bfb336d36b6eed0d8993e164126b63c41f4b4d8880fe083976a3ded","abstract_canon_sha256":"30b399071fecba045de095fda430855563737be1061bf2454af5b5fcd3ebaf99"},"schema_version":"1.0"},"canonical_sha256":"0f037a4270d5b4719fa3ef8b913f64efc4bbe6a96e3f7f3ca800b54932bc79dd","source":{"kind":"arxiv","id":"0910.3456","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.3456","created_at":"2026-05-18T04:36:51Z"},{"alias_kind":"arxiv_version","alias_value":"0910.3456v3","created_at":"2026-05-18T04:36:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.3456","created_at":"2026-05-18T04:36:51Z"},{"alias_kind":"pith_short_12","alias_value":"B4BXUQTQ2W2H","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"B4BXUQTQ2W2HDH5D","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"B4BXUQTQ","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:B4BXUQTQ2W2HDH5D56FZCP3E57","target":"record","payload":{"canonical_record":{"source":{"id":"0910.3456","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-10-19T05:01:14Z","cross_cats_sorted":[],"title_canon_sha256":"cecde13a8bfb336d36b6eed0d8993e164126b63c41f4b4d8880fe083976a3ded","abstract_canon_sha256":"30b399071fecba045de095fda430855563737be1061bf2454af5b5fcd3ebaf99"},"schema_version":"1.0"},"canonical_sha256":"0f037a4270d5b4719fa3ef8b913f64efc4bbe6a96e3f7f3ca800b54932bc79dd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:36:51.317457Z","signature_b64":"9H1lqvIlkJ/9kEFf2IOV50bmCuaJL4EyzKpHmPB77t6J8PVbGWmI0nlguc8TYvSJkJ6VYdypU8vDddbApPcBCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f037a4270d5b4719fa3ef8b913f64efc4bbe6a96e3f7f3ca800b54932bc79dd","last_reissued_at":"2026-05-18T04:36:51.317028Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:36:51.317028Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0910.3456","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:36:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NSqlYc2VznPntloRrEhoTJhsaIAesT4413EjKUwbOlQBhEigd9tAM6X0d1fLDykYZBWN4rKGzBiolwQHQ0Q1DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T12:20:44.031342Z"},"content_sha256":"52cb5281d215038afa56ea34697708e841d78cc70029630ced26c8e0aafceaf6","schema_version":"1.0","event_id":"sha256:52cb5281d215038afa56ea34697708e841d78cc70029630ced26c8e0aafceaf6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:B4BXUQTQ2W2HDH5D56FZCP3E57","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Kentaro Saji, Kotaro Yamada, Masaaki Umehara","submitted_at":"2009-10-19T05:01:14Z","abstract_excerpt":"We give a definition of `coherent tangent bundles', which is an intrinsic formulation of wave fronts. In our application of coherent tangent bundles for wave fronts, the first fundamental forms and the third fundamental forms are considered as induced metrics of certain homomorphisms between vector bundles. They satisfy the completely same conditions, and so can reverse roles with each other. For a given wave front of a 2-manifold, there are two Gauss-Bonnet formulas. By exchanging the roles of the fundamental forms, we get two new additional Gauss-Bonnet formulas for the third fundamental for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.3456","kind":"arxiv","version":3},"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":"2026-05-18T04:36:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fGd2hrvgHfYkZdWsXXRMrkn9H9Ew5UBgwpaDHKTiRYqNZy9xyIVNOCyMLZ1o79+N6Rmxw7AN8dPQMosLqee+CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T12:20:44.032006Z"},"content_sha256":"d27aa8ad2b8fcfbe788fdb6857943a57c9274d971e05f9e7e343a3f770d96e9d","schema_version":"1.0","event_id":"sha256:d27aa8ad2b8fcfbe788fdb6857943a57c9274d971e05f9e7e343a3f770d96e9d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B4BXUQTQ2W2HDH5D56FZCP3E57/bundle.json","state_url":"https://pith.science/pith/B4BXUQTQ2W2HDH5D56FZCP3E57/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B4BXUQTQ2W2HDH5D56FZCP3E57/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":"2026-05-26T12:20:44Z","links":{"resolver":"https://pith.science/pith/B4BXUQTQ2W2HDH5D56FZCP3E57","bundle":"https://pith.science/pith/B4BXUQTQ2W2HDH5D56FZCP3E57/bundle.json","state":"https://pith.science/pith/B4BXUQTQ2W2HDH5D56FZCP3E57/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B4BXUQTQ2W2HDH5D56FZCP3E57/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:B4BXUQTQ2W2HDH5D56FZCP3E57","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":"30b399071fecba045de095fda430855563737be1061bf2454af5b5fcd3ebaf99","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-10-19T05:01:14Z","title_canon_sha256":"cecde13a8bfb336d36b6eed0d8993e164126b63c41f4b4d8880fe083976a3ded"},"schema_version":"1.0","source":{"id":"0910.3456","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.3456","created_at":"2026-05-18T04:36:51Z"},{"alias_kind":"arxiv_version","alias_value":"0910.3456v3","created_at":"2026-05-18T04:36:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.3456","created_at":"2026-05-18T04:36:51Z"},{"alias_kind":"pith_short_12","alias_value":"B4BXUQTQ2W2H","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"B4BXUQTQ2W2HDH5D","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"B4BXUQTQ","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:d27aa8ad2b8fcfbe788fdb6857943a57c9274d971e05f9e7e343a3f770d96e9d","target":"graph","created_at":"2026-05-18T04:36:51Z","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 give a definition of `coherent tangent bundles', which is an intrinsic formulation of wave fronts. In our application of coherent tangent bundles for wave fronts, the first fundamental forms and the third fundamental forms are considered as induced metrics of certain homomorphisms between vector bundles. They satisfy the completely same conditions, and so can reverse roles with each other. For a given wave front of a 2-manifold, there are two Gauss-Bonnet formulas. By exchanging the roles of the fundamental forms, we get two new additional Gauss-Bonnet formulas for the third fundamental for","authors_text":"Kentaro Saji, Kotaro Yamada, Masaaki Umehara","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-10-19T05:01:14Z","title":"Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.3456","kind":"arxiv","version":3},"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":"sha256:52cb5281d215038afa56ea34697708e841d78cc70029630ced26c8e0aafceaf6","target":"record","created_at":"2026-05-18T04:36:51Z","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":"30b399071fecba045de095fda430855563737be1061bf2454af5b5fcd3ebaf99","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-10-19T05:01:14Z","title_canon_sha256":"cecde13a8bfb336d36b6eed0d8993e164126b63c41f4b4d8880fe083976a3ded"},"schema_version":"1.0","source":{"id":"0910.3456","kind":"arxiv","version":3}},"canonical_sha256":"0f037a4270d5b4719fa3ef8b913f64efc4bbe6a96e3f7f3ca800b54932bc79dd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0f037a4270d5b4719fa3ef8b913f64efc4bbe6a96e3f7f3ca800b54932bc79dd","first_computed_at":"2026-05-18T04:36:51.317028Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:36:51.317028Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9H1lqvIlkJ/9kEFf2IOV50bmCuaJL4EyzKpHmPB77t6J8PVbGWmI0nlguc8TYvSJkJ6VYdypU8vDddbApPcBCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:36:51.317457Z","signed_message":"canonical_sha256_bytes"},"source_id":"0910.3456","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:52cb5281d215038afa56ea34697708e841d78cc70029630ced26c8e0aafceaf6","sha256:d27aa8ad2b8fcfbe788fdb6857943a57c9274d971e05f9e7e343a3f770d96e9d"],"state_sha256":"b6f331dc365c7d7ea2884fbf8545cac99caad6530be7ae3beff39d211b260df4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fygoEFhmZV+qMq6dd+1pSDKYu/j9v0I3gxDClRMFkcep2A7fodPlnOHnKUW1fEmrFH/WRa62LYXtIFz63N5XDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T12:20:44.035441Z","bundle_sha256":"5bb37d7e568ee46f85c3814c5d5117e3e601be25b63a6b3a8a5220c86fa35f82"}}