{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:JSCFYDALUP4HQVFOOL6K6XRY6V","short_pith_number":"pith:JSCFYDAL","canonical_record":{"source":{"id":"1307.6040","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-07-23T12:29:46Z","cross_cats_sorted":[],"title_canon_sha256":"9323c5c9a6dfd48ae7858d95b77079e5f8148fb8abdd85fb02c681b02c413536","abstract_canon_sha256":"9e1ea85cc1df12cdf5feb4cfbc1ea29740685518c14db31b7cb8dd6784c7b337"},"schema_version":"1.0"},"canonical_sha256":"4c845c0c0ba3f87854ae72fcaf5e38f5677a0be78524c2e8f28bd90c00d1afd5","source":{"kind":"arxiv","id":"1307.6040","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.6040","created_at":"2026-05-18T03:17:45Z"},{"alias_kind":"arxiv_version","alias_value":"1307.6040v1","created_at":"2026-05-18T03:17:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.6040","created_at":"2026-05-18T03:17:45Z"},{"alias_kind":"pith_short_12","alias_value":"JSCFYDALUP4H","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"JSCFYDALUP4HQVFO","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"JSCFYDAL","created_at":"2026-05-18T12:27:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:JSCFYDALUP4HQVFOOL6K6XRY6V","target":"record","payload":{"canonical_record":{"source":{"id":"1307.6040","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-07-23T12:29:46Z","cross_cats_sorted":[],"title_canon_sha256":"9323c5c9a6dfd48ae7858d95b77079e5f8148fb8abdd85fb02c681b02c413536","abstract_canon_sha256":"9e1ea85cc1df12cdf5feb4cfbc1ea29740685518c14db31b7cb8dd6784c7b337"},"schema_version":"1.0"},"canonical_sha256":"4c845c0c0ba3f87854ae72fcaf5e38f5677a0be78524c2e8f28bd90c00d1afd5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:17:45.500981Z","signature_b64":"8j1uOZpFkXecxCgUNuEKJZGLWIzCUUcdKzjzqy2TXWd8g32eIxmOIEYbLAXoyNR/CbohbjuovgTiBXvLMUGSBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c845c0c0ba3f87854ae72fcaf5e38f5677a0be78524c2e8f28bd90c00d1afd5","last_reissued_at":"2026-05-18T03:17:45.500506Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:17:45.500506Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.6040","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:17:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"++9NR3O881iFIivOlCab+YMg3Le8hMpSut1a/k+pmJtW399VhNtNc6PTe0SE/kvHI909EBrw6AXFJO2EkNhjCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T03:55:42.132119Z"},"content_sha256":"b97fa72416bee1942c598e8ffbfa14bb5f86e19062cc062a0d7e9fd4b05ca619","schema_version":"1.0","event_id":"sha256:b97fa72416bee1942c598e8ffbfa14bb5f86e19062cc062a0d7e9fd4b05ca619"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:JSCFYDALUP4HQVFOOL6K6XRY6V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Height functions on compact symmetric spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"E. Mac\\'ias-Virg\\'os, M.J. Pereira-S\\'aez","submitted_at":"2013-07-23T12:29:46Z","abstract_excerpt":"We consider height functions on symmetric spaces $M\\cong G/K$ embedded in the associated matrix Lie group $G$. In particular we study the relationship between the critical sets of the height function on $G$ and its restriction to $M$. Also we prove that the gradient flow on $M$ can be integrated by means of a generalized Cayley transform. This allows to obtain explicit local charts for the critical submanifolds. Finally, we discuss how to reduce the generic case to a height function whose ground hyperplane is orhogonal to a real diagonal matrix. This result requires to prove the existence of a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.6040","kind":"arxiv","version":1},"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":"2026-05-18T03:17:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JXJEe0Za1RhzV4cUntURoZSPv+tmN5X008HYolSYuUKWeKnOoWd4uGA7cn4N/MPV07hhSKCdt05y12bRTRLZBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T03:55:42.132492Z"},"content_sha256":"a49951f0409239f4dc1c50495c182494a4e1f394f5cbee5bde35b07e06df5b01","schema_version":"1.0","event_id":"sha256:a49951f0409239f4dc1c50495c182494a4e1f394f5cbee5bde35b07e06df5b01"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JSCFYDALUP4HQVFOOL6K6XRY6V/bundle.json","state_url":"https://pith.science/pith/JSCFYDALUP4HQVFOOL6K6XRY6V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JSCFYDALUP4HQVFOOL6K6XRY6V/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":"2026-06-24T03:55:42Z","links":{"resolver":"https://pith.science/pith/JSCFYDALUP4HQVFOOL6K6XRY6V","bundle":"https://pith.science/pith/JSCFYDALUP4HQVFOOL6K6XRY6V/bundle.json","state":"https://pith.science/pith/JSCFYDALUP4HQVFOOL6K6XRY6V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JSCFYDALUP4HQVFOOL6K6XRY6V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:JSCFYDALUP4HQVFOOL6K6XRY6V","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":"9e1ea85cc1df12cdf5feb4cfbc1ea29740685518c14db31b7cb8dd6784c7b337","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-07-23T12:29:46Z","title_canon_sha256":"9323c5c9a6dfd48ae7858d95b77079e5f8148fb8abdd85fb02c681b02c413536"},"schema_version":"1.0","source":{"id":"1307.6040","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.6040","created_at":"2026-05-18T03:17:45Z"},{"alias_kind":"arxiv_version","alias_value":"1307.6040v1","created_at":"2026-05-18T03:17:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.6040","created_at":"2026-05-18T03:17:45Z"},{"alias_kind":"pith_short_12","alias_value":"JSCFYDALUP4H","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"JSCFYDALUP4HQVFO","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"JSCFYDAL","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:a49951f0409239f4dc1c50495c182494a4e1f394f5cbee5bde35b07e06df5b01","target":"graph","created_at":"2026-05-18T03:17:45Z","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 consider height functions on symmetric spaces $M\\cong G/K$ embedded in the associated matrix Lie group $G$. In particular we study the relationship between the critical sets of the height function on $G$ and its restriction to $M$. Also we prove that the gradient flow on $M$ can be integrated by means of a generalized Cayley transform. This allows to obtain explicit local charts for the critical submanifolds. Finally, we discuss how to reduce the generic case to a height function whose ground hyperplane is orhogonal to a real diagonal matrix. This result requires to prove the existence of a","authors_text":"E. Mac\\'ias-Virg\\'os, M.J. Pereira-S\\'aez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-07-23T12:29:46Z","title":"Height functions on compact symmetric spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.6040","kind":"arxiv","version":1},"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:b97fa72416bee1942c598e8ffbfa14bb5f86e19062cc062a0d7e9fd4b05ca619","target":"record","created_at":"2026-05-18T03:17:45Z","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":"9e1ea85cc1df12cdf5feb4cfbc1ea29740685518c14db31b7cb8dd6784c7b337","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-07-23T12:29:46Z","title_canon_sha256":"9323c5c9a6dfd48ae7858d95b77079e5f8148fb8abdd85fb02c681b02c413536"},"schema_version":"1.0","source":{"id":"1307.6040","kind":"arxiv","version":1}},"canonical_sha256":"4c845c0c0ba3f87854ae72fcaf5e38f5677a0be78524c2e8f28bd90c00d1afd5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4c845c0c0ba3f87854ae72fcaf5e38f5677a0be78524c2e8f28bd90c00d1afd5","first_computed_at":"2026-05-18T03:17:45.500506Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:17:45.500506Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8j1uOZpFkXecxCgUNuEKJZGLWIzCUUcdKzjzqy2TXWd8g32eIxmOIEYbLAXoyNR/CbohbjuovgTiBXvLMUGSBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:17:45.500981Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.6040","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b97fa72416bee1942c598e8ffbfa14bb5f86e19062cc062a0d7e9fd4b05ca619","sha256:a49951f0409239f4dc1c50495c182494a4e1f394f5cbee5bde35b07e06df5b01"],"state_sha256":"593dec5a69f9b35ba0f59ebc093a72c63a88b6bfed11bdacdfbcafab03762eff"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rXUEzYDLENL6Y+R2g+3XUY9ZBUMC2ebdon2/ONpeDgchAGExNN/fdTlqWAof9DU0zYK6NXvojUUttb+caXRUAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T03:55:42.134475Z","bundle_sha256":"faf4dc3e39370cff45eb15709b3557c1231bdfceb895293866fbe2b881ef23a7"}}