{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:UZZZBFVWD4BMDTO76VGZ7UFPJQ","short_pith_number":"pith:UZZZBFVW","canonical_record":{"source":{"id":"1609.02099","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-09-07T18:30:37Z","cross_cats_sorted":[],"title_canon_sha256":"c230640b61e5524195639be9666877c6b23542f9af80b82b8e670af0aba7c4e6","abstract_canon_sha256":"576e78d8f69dc9ed28132e8e2cf78caec0efd661dc807c36de4a656fb7ee030f"},"schema_version":"1.0"},"canonical_sha256":"a6739096b61f02c1cddff54d9fd0af4c3be79c6350480a08b6070459c6b02629","source":{"kind":"arxiv","id":"1609.02099","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.02099","created_at":"2026-05-18T01:04:36Z"},{"alias_kind":"arxiv_version","alias_value":"1609.02099v2","created_at":"2026-05-18T01:04:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.02099","created_at":"2026-05-18T01:04:36Z"},{"alias_kind":"pith_short_12","alias_value":"UZZZBFVWD4BM","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UZZZBFVWD4BMDTO7","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UZZZBFVW","created_at":"2026-05-18T12:30:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:UZZZBFVWD4BMDTO76VGZ7UFPJQ","target":"record","payload":{"canonical_record":{"source":{"id":"1609.02099","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-09-07T18:30:37Z","cross_cats_sorted":[],"title_canon_sha256":"c230640b61e5524195639be9666877c6b23542f9af80b82b8e670af0aba7c4e6","abstract_canon_sha256":"576e78d8f69dc9ed28132e8e2cf78caec0efd661dc807c36de4a656fb7ee030f"},"schema_version":"1.0"},"canonical_sha256":"a6739096b61f02c1cddff54d9fd0af4c3be79c6350480a08b6070459c6b02629","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:36.931711Z","signature_b64":"lxLsToXD7aCxYaBcmz5mGHb8ExtXo9mwA4JnVkj0ktkeKzqTrI+DrA3oP+/UKc/kayLVY0R1RS4KJBecTKySBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a6739096b61f02c1cddff54d9fd0af4c3be79c6350480a08b6070459c6b02629","last_reissued_at":"2026-05-18T01:04:36.931008Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:36.931008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.02099","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":"2026-05-18T01:04:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J0qStO9gWf2hR6R+s3sTDidiqTUrna71VF0D5Snsavf4GVtuVVSS4MUNDncOVLgRuOkCS1be243kFJyB/4lsCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T12:56:11.465493Z"},"content_sha256":"a6350c3abd1f685c47d009dbb41cc6a242ed4759d7d729d0ca11c5265f5c3a34","schema_version":"1.0","event_id":"sha256:a6350c3abd1f685c47d009dbb41cc6a242ed4759d7d729d0ca11c5265f5c3a34"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:UZZZBFVWD4BMDTO76VGZ7UFPJQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Gauss map on translational Riemannian manifolds and the topology of hypersurfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Eduardo R. Longa, Jaime B. Ripoll","submitted_at":"2016-09-07T18:30:37Z","abstract_excerpt":"We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this Gauss map to prove that if $M^{n}$ is a compact, connected and oriented immersed hypersurface of the unit sphere $\\mathbb{S}^{n+1}$ ($n\\geq2$) contained in a geodesic ball of radius $R$ and whose principal curvatures are strictly bigger than $\\tan\\left( R/2 \\right)$, then $M$ is diffeomorphic to $\\mathbb{S}^{n}$. Additionally, we show that for any $\\varepsilon\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.02099","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":"2026-05-18T01:04:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WGUYxSZ8EhuZ9XGXzhUEcKDKrnwbHKNs27Z3AX9K/8IkgaRNEVbqhYfjm6+Rv0f2TNGQ9+qATU7r8jySUE9IBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T12:56:11.466136Z"},"content_sha256":"7068e5e6184bbe2570f2e2b4deaaff9f2d0cdebd1eeec316499f2c5c48767739","schema_version":"1.0","event_id":"sha256:7068e5e6184bbe2570f2e2b4deaaff9f2d0cdebd1eeec316499f2c5c48767739"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UZZZBFVWD4BMDTO76VGZ7UFPJQ/bundle.json","state_url":"https://pith.science/pith/UZZZBFVWD4BMDTO76VGZ7UFPJQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UZZZBFVWD4BMDTO76VGZ7UFPJQ/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":"2026-05-27T12:56:11Z","links":{"resolver":"https://pith.science/pith/UZZZBFVWD4BMDTO76VGZ7UFPJQ","bundle":"https://pith.science/pith/UZZZBFVWD4BMDTO76VGZ7UFPJQ/bundle.json","state":"https://pith.science/pith/UZZZBFVWD4BMDTO76VGZ7UFPJQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UZZZBFVWD4BMDTO76VGZ7UFPJQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:UZZZBFVWD4BMDTO76VGZ7UFPJQ","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":"576e78d8f69dc9ed28132e8e2cf78caec0efd661dc807c36de4a656fb7ee030f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-09-07T18:30:37Z","title_canon_sha256":"c230640b61e5524195639be9666877c6b23542f9af80b82b8e670af0aba7c4e6"},"schema_version":"1.0","source":{"id":"1609.02099","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.02099","created_at":"2026-05-18T01:04:36Z"},{"alias_kind":"arxiv_version","alias_value":"1609.02099v2","created_at":"2026-05-18T01:04:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.02099","created_at":"2026-05-18T01:04:36Z"},{"alias_kind":"pith_short_12","alias_value":"UZZZBFVWD4BM","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UZZZBFVWD4BMDTO7","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UZZZBFVW","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:7068e5e6184bbe2570f2e2b4deaaff9f2d0cdebd1eeec316499f2c5c48767739","target":"graph","created_at":"2026-05-18T01:04:36Z","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 introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this Gauss map to prove that if $M^{n}$ is a compact, connected and oriented immersed hypersurface of the unit sphere $\\mathbb{S}^{n+1}$ ($n\\geq2$) contained in a geodesic ball of radius $R$ and whose principal curvatures are strictly bigger than $\\tan\\left( R/2 \\right)$, then $M$ is diffeomorphic to $\\mathbb{S}^{n}$. Additionally, we show that for any $\\varepsilon\\","authors_text":"Eduardo R. Longa, Jaime B. Ripoll","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-09-07T18:30:37Z","title":"The Gauss map on translational Riemannian manifolds and the topology of hypersurfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.02099","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:a6350c3abd1f685c47d009dbb41cc6a242ed4759d7d729d0ca11c5265f5c3a34","target":"record","created_at":"2026-05-18T01:04:36Z","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":"576e78d8f69dc9ed28132e8e2cf78caec0efd661dc807c36de4a656fb7ee030f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-09-07T18:30:37Z","title_canon_sha256":"c230640b61e5524195639be9666877c6b23542f9af80b82b8e670af0aba7c4e6"},"schema_version":"1.0","source":{"id":"1609.02099","kind":"arxiv","version":2}},"canonical_sha256":"a6739096b61f02c1cddff54d9fd0af4c3be79c6350480a08b6070459c6b02629","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a6739096b61f02c1cddff54d9fd0af4c3be79c6350480a08b6070459c6b02629","first_computed_at":"2026-05-18T01:04:36.931008Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:36.931008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lxLsToXD7aCxYaBcmz5mGHb8ExtXo9mwA4JnVkj0ktkeKzqTrI+DrA3oP+/UKc/kayLVY0R1RS4KJBecTKySBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:36.931711Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.02099","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a6350c3abd1f685c47d009dbb41cc6a242ed4759d7d729d0ca11c5265f5c3a34","sha256:7068e5e6184bbe2570f2e2b4deaaff9f2d0cdebd1eeec316499f2c5c48767739"],"state_sha256":"7fab60304f95af5b80703cf1a02760255d47b938e8a4ddace03e1e7e87bdc1e1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zlmbB5TtJzmu0CPr6kZ+sBOC5i/u46/93hguwrqBe6ClFOwFgTskHi9mBlvHbtSMYoqTrT5HqHG2m4OwSqHVAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T12:56:11.469848Z","bundle_sha256":"4fe1c427c3f1cac22708ffd615472813a50182d13ed8de019cb1a64076d07be2"}}