{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:FCF6TCNEXIECSSFVU5UELGHK2Z","short_pith_number":"pith:FCF6TCNE","canonical_record":{"source":{"id":"1611.07066","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-11-21T21:25:11Z","cross_cats_sorted":[],"title_canon_sha256":"5155a2835b56b0f3c3e8e4040d6d20f080ef2d19c471c0668ac376db81c68d04","abstract_canon_sha256":"0109bd6200784039950718cf59c293ff96737fb1f0c3544375ad205598b43933"},"schema_version":"1.0"},"canonical_sha256":"288be989a4ba082948b5a7684598ead656aab7f23de32c86119692121057d88d","source":{"kind":"arxiv","id":"1611.07066","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.07066","created_at":"2026-05-18T00:50:18Z"},{"alias_kind":"arxiv_version","alias_value":"1611.07066v2","created_at":"2026-05-18T00:50:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.07066","created_at":"2026-05-18T00:50:18Z"},{"alias_kind":"pith_short_12","alias_value":"FCF6TCNEXIEC","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FCF6TCNEXIECSSFV","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FCF6TCNE","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:FCF6TCNEXIECSSFVU5UELGHK2Z","target":"record","payload":{"canonical_record":{"source":{"id":"1611.07066","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-11-21T21:25:11Z","cross_cats_sorted":[],"title_canon_sha256":"5155a2835b56b0f3c3e8e4040d6d20f080ef2d19c471c0668ac376db81c68d04","abstract_canon_sha256":"0109bd6200784039950718cf59c293ff96737fb1f0c3544375ad205598b43933"},"schema_version":"1.0"},"canonical_sha256":"288be989a4ba082948b5a7684598ead656aab7f23de32c86119692121057d88d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:18.070036Z","signature_b64":"tRAxAKx7SvAPvT9OpvZ1Lf6DKhRplna8QzvCThzx+dXrF85qqHYu7jnOC41BSWjtm622DGGBz3TwtvvRnZ/cAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"288be989a4ba082948b5a7684598ead656aab7f23de32c86119692121057d88d","last_reissued_at":"2026-05-18T00:50:18.069549Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:18.069549Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.07066","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":"2026-05-18T00:50:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"utKrHdGcLcYPjiVykSHKcAglI41umyO3yzXCSozlnqBV6HWJyFGTIYSIwJCeJR6kZ7k1PHf9mf8EzWSViUz4CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T08:19:40.379386Z"},"content_sha256":"7c647470925bc24973c12c4e92a6a7aaf76f3599bdcf0c806dad254efc645cb7","schema_version":"1.0","event_id":"sha256:7c647470925bc24973c12c4e92a6a7aaf76f3599bdcf0c806dad254efc645cb7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:FCF6TCNEXIECSSFVU5UELGHK2Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the critical points of the energy functional on vector fields of a Riemannian manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Giovanni Nunes, Jaime Ripoll","submitted_at":"2016-11-21T21:25:11Z","abstract_excerpt":"Given a compact Lie subgroup $G$ of the isometry group of a compact Riemannian manifold $M$ with a Riemannian connection $\\nabla,$ it is introduced a $G-$symmetrization process of a vector field of $M$ and it is proved that the critical points of the energy functional \\[ F(X):=\\frac{\\int_{M}\\left\\Vert \\nabla X\\right\\Vert ^{2}dM}{\\int_{M}\\left\\Vert X\\right\\Vert ^{2}dM}% \\] on the space of $\\ G-$invariant vector fields are critical points of $F$ on the space of all vector fields of $M,$ and that this inclusion may be strict in general. One proves that the infimum of $F$ on $\\mathbb{S}^{3}$ is no"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.07066","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":"2026-05-18T00:50:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mzYy6nGJoepmwfKKS1FvRrkd55b+XWUvjEeeUsqJa0p5zLTQoEWNYldnUB2YhVObvS+Zldx6At8yn4kSIIxBDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T08:19:40.379743Z"},"content_sha256":"976259b63e038bb7adefb74608df206be7b231f73316301945eb698300cc617c","schema_version":"1.0","event_id":"sha256:976259b63e038bb7adefb74608df206be7b231f73316301945eb698300cc617c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FCF6TCNEXIECSSFVU5UELGHK2Z/bundle.json","state_url":"https://pith.science/pith/FCF6TCNEXIECSSFVU5UELGHK2Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FCF6TCNEXIECSSFVU5UELGHK2Z/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":"2026-06-26T08:19:40Z","links":{"resolver":"https://pith.science/pith/FCF6TCNEXIECSSFVU5UELGHK2Z","bundle":"https://pith.science/pith/FCF6TCNEXIECSSFVU5UELGHK2Z/bundle.json","state":"https://pith.science/pith/FCF6TCNEXIECSSFVU5UELGHK2Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FCF6TCNEXIECSSFVU5UELGHK2Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FCF6TCNEXIECSSFVU5UELGHK2Z","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":"0109bd6200784039950718cf59c293ff96737fb1f0c3544375ad205598b43933","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-11-21T21:25:11Z","title_canon_sha256":"5155a2835b56b0f3c3e8e4040d6d20f080ef2d19c471c0668ac376db81c68d04"},"schema_version":"1.0","source":{"id":"1611.07066","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.07066","created_at":"2026-05-18T00:50:18Z"},{"alias_kind":"arxiv_version","alias_value":"1611.07066v2","created_at":"2026-05-18T00:50:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.07066","created_at":"2026-05-18T00:50:18Z"},{"alias_kind":"pith_short_12","alias_value":"FCF6TCNEXIEC","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FCF6TCNEXIECSSFV","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FCF6TCNE","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:976259b63e038bb7adefb74608df206be7b231f73316301945eb698300cc617c","target":"graph","created_at":"2026-05-18T00:50:18Z","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":"Given a compact Lie subgroup $G$ of the isometry group of a compact Riemannian manifold $M$ with a Riemannian connection $\\nabla,$ it is introduced a $G-$symmetrization process of a vector field of $M$ and it is proved that the critical points of the energy functional \\[ F(X):=\\frac{\\int_{M}\\left\\Vert \\nabla X\\right\\Vert ^{2}dM}{\\int_{M}\\left\\Vert X\\right\\Vert ^{2}dM}% \\] on the space of $\\ G-$invariant vector fields are critical points of $F$ on the space of all vector fields of $M,$ and that this inclusion may be strict in general. One proves that the infimum of $F$ on $\\mathbb{S}^{3}$ is no","authors_text":"Giovanni Nunes, Jaime Ripoll","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-11-21T21:25:11Z","title":"On the critical points of the energy functional on vector fields of a Riemannian manifold"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.07066","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":"sha256:7c647470925bc24973c12c4e92a6a7aaf76f3599bdcf0c806dad254efc645cb7","target":"record","created_at":"2026-05-18T00:50:18Z","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":"0109bd6200784039950718cf59c293ff96737fb1f0c3544375ad205598b43933","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-11-21T21:25:11Z","title_canon_sha256":"5155a2835b56b0f3c3e8e4040d6d20f080ef2d19c471c0668ac376db81c68d04"},"schema_version":"1.0","source":{"id":"1611.07066","kind":"arxiv","version":2}},"canonical_sha256":"288be989a4ba082948b5a7684598ead656aab7f23de32c86119692121057d88d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"288be989a4ba082948b5a7684598ead656aab7f23de32c86119692121057d88d","first_computed_at":"2026-05-18T00:50:18.069549Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:50:18.069549Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tRAxAKx7SvAPvT9OpvZ1Lf6DKhRplna8QzvCThzx+dXrF85qqHYu7jnOC41BSWjtm622DGGBz3TwtvvRnZ/cAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:50:18.070036Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.07066","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7c647470925bc24973c12c4e92a6a7aaf76f3599bdcf0c806dad254efc645cb7","sha256:976259b63e038bb7adefb74608df206be7b231f73316301945eb698300cc617c"],"state_sha256":"d34a704bb4930036e9c29a83ffbbeb13e7f388701e3c6da83eacdf4eecd03e00"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LbrR8X/Rvs0qJPX7L65MBYc8nJJWrF6BSkVjI+NWw3kYzMIITscLguq/bsK04jFdmN5/Sobvs3V5n80ycIQLAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T08:19:40.381647Z","bundle_sha256":"a75cace6c7c4fa4186afef84d6f03d8f4066cfe3a794105e983b39b9d9d4237e"}}