{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:7YBMZ2F67CRXN7MXC7PITI5WCM","short_pith_number":"pith:7YBMZ2F6","canonical_record":{"source":{"id":"1208.0976","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-08-05T02:05:35Z","cross_cats_sorted":[],"title_canon_sha256":"99f75eb358378e784ced28a4adfbf6cc3e1c8a985695eec2a2da5362b15223d4","abstract_canon_sha256":"250b5f80ced638a44f0c829524fa2cbc46fe7abe503ffb7a1170a59c76fa2cf8"},"schema_version":"1.0"},"canonical_sha256":"fe02cce8bef8a376fd9717de89a3b613255da75a01156f20519ac299376c3ee0","source":{"kind":"arxiv","id":"1208.0976","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.0976","created_at":"2026-05-18T03:45:55Z"},{"alias_kind":"arxiv_version","alias_value":"1208.0976v2","created_at":"2026-05-18T03:45:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.0976","created_at":"2026-05-18T03:45:55Z"},{"alias_kind":"pith_short_12","alias_value":"7YBMZ2F67CRX","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7YBMZ2F67CRXN7MX","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7YBMZ2F6","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:7YBMZ2F67CRXN7MXC7PITI5WCM","target":"record","payload":{"canonical_record":{"source":{"id":"1208.0976","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-08-05T02:05:35Z","cross_cats_sorted":[],"title_canon_sha256":"99f75eb358378e784ced28a4adfbf6cc3e1c8a985695eec2a2da5362b15223d4","abstract_canon_sha256":"250b5f80ced638a44f0c829524fa2cbc46fe7abe503ffb7a1170a59c76fa2cf8"},"schema_version":"1.0"},"canonical_sha256":"fe02cce8bef8a376fd9717de89a3b613255da75a01156f20519ac299376c3ee0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:55.011794Z","signature_b64":"/6WI3lK+GiCcEWuJPBX6lURPqcKNM1QDnq719Zn9mYZAUJKkdXZJtnCsVOy7PXTrLONSJW4kXBGGy9unf16ZAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe02cce8bef8a376fd9717de89a3b613255da75a01156f20519ac299376c3ee0","last_reissued_at":"2026-05-18T03:45:55.011149Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:55.011149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.0976","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":"2026-05-18T03:45:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RnBBj2DDDDpgHSCbhIHodGIpoqLO8E+8DrjEZv5mdiWtA38fKNcyeXbx1fLLHNTYx2esZ31AcmBEJJ2/DnpeDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T00:22:45.374219Z"},"content_sha256":"6bbe28b6d06e39ca4625b6d7884f5fa204b67d5387657dbd1b267dfdab7bebd2","schema_version":"1.0","event_id":"sha256:6bbe28b6d06e39ca4625b6d7884f5fa204b67d5387657dbd1b267dfdab7bebd2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:7YBMZ2F67CRXN7MXC7PITI5WCM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Polar manifolds and actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Karsten Grove, Wolfgang Ziller","submitted_at":"2012-08-05T02:05:35Z","abstract_excerpt":"A group action is called polar if there exists an immersed submanifold (a section) which intersects all orbits orthogonally. Such group actions have been studied extensively on symmetric spaces. We show how to construct a manifold admitting a polar group action by prescribing their isotropy groups along a fundamental domain, generalizing the classical construction for cohomogeneity manifolds. We give many examples showing the richness of this class of group actions. We also relate the topology of the section to the topology of the manifold.\n  This is a replacement of an earlier version. Small "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.0976","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":"2026-05-18T03:45:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RdBzLVEd/O5gDXzKlKLeRdju22bMnAJrl3Gm2agCgi4hiCo98AfvcH4vyHwK7eZ+Fuj+1BL2WNNM+mMnWaPyBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T00:22:45.374580Z"},"content_sha256":"abbb02691a01945ff659716e94f311cbdd72c5147f71e6c3bfb52b8c5970b621","schema_version":"1.0","event_id":"sha256:abbb02691a01945ff659716e94f311cbdd72c5147f71e6c3bfb52b8c5970b621"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7YBMZ2F67CRXN7MXC7PITI5WCM/bundle.json","state_url":"https://pith.science/pith/7YBMZ2F67CRXN7MXC7PITI5WCM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7YBMZ2F67CRXN7MXC7PITI5WCM/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":"2026-06-21T00:22:45Z","links":{"resolver":"https://pith.science/pith/7YBMZ2F67CRXN7MXC7PITI5WCM","bundle":"https://pith.science/pith/7YBMZ2F67CRXN7MXC7PITI5WCM/bundle.json","state":"https://pith.science/pith/7YBMZ2F67CRXN7MXC7PITI5WCM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7YBMZ2F67CRXN7MXC7PITI5WCM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:7YBMZ2F67CRXN7MXC7PITI5WCM","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":"250b5f80ced638a44f0c829524fa2cbc46fe7abe503ffb7a1170a59c76fa2cf8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-08-05T02:05:35Z","title_canon_sha256":"99f75eb358378e784ced28a4adfbf6cc3e1c8a985695eec2a2da5362b15223d4"},"schema_version":"1.0","source":{"id":"1208.0976","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.0976","created_at":"2026-05-18T03:45:55Z"},{"alias_kind":"arxiv_version","alias_value":"1208.0976v2","created_at":"2026-05-18T03:45:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.0976","created_at":"2026-05-18T03:45:55Z"},{"alias_kind":"pith_short_12","alias_value":"7YBMZ2F67CRX","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7YBMZ2F67CRXN7MX","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7YBMZ2F6","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:abbb02691a01945ff659716e94f311cbdd72c5147f71e6c3bfb52b8c5970b621","target":"graph","created_at":"2026-05-18T03:45:55Z","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":"A group action is called polar if there exists an immersed submanifold (a section) which intersects all orbits orthogonally. Such group actions have been studied extensively on symmetric spaces. We show how to construct a manifold admitting a polar group action by prescribing their isotropy groups along a fundamental domain, generalizing the classical construction for cohomogeneity manifolds. We give many examples showing the richness of this class of group actions. We also relate the topology of the section to the topology of the manifold.\n  This is a replacement of an earlier version. Small ","authors_text":"Karsten Grove, Wolfgang Ziller","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-08-05T02:05:35Z","title":"Polar manifolds and actions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.0976","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":"sha256:6bbe28b6d06e39ca4625b6d7884f5fa204b67d5387657dbd1b267dfdab7bebd2","target":"record","created_at":"2026-05-18T03:45:55Z","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":"250b5f80ced638a44f0c829524fa2cbc46fe7abe503ffb7a1170a59c76fa2cf8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-08-05T02:05:35Z","title_canon_sha256":"99f75eb358378e784ced28a4adfbf6cc3e1c8a985695eec2a2da5362b15223d4"},"schema_version":"1.0","source":{"id":"1208.0976","kind":"arxiv","version":2}},"canonical_sha256":"fe02cce8bef8a376fd9717de89a3b613255da75a01156f20519ac299376c3ee0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe02cce8bef8a376fd9717de89a3b613255da75a01156f20519ac299376c3ee0","first_computed_at":"2026-05-18T03:45:55.011149Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:45:55.011149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/6WI3lK+GiCcEWuJPBX6lURPqcKNM1QDnq719Zn9mYZAUJKkdXZJtnCsVOy7PXTrLONSJW4kXBGGy9unf16ZAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:45:55.011794Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.0976","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6bbe28b6d06e39ca4625b6d7884f5fa204b67d5387657dbd1b267dfdab7bebd2","sha256:abbb02691a01945ff659716e94f311cbdd72c5147f71e6c3bfb52b8c5970b621"],"state_sha256":"d40dbe9b6a70dba0892691cb9100eec4cbf3062bb8d832b30b4589582a8d9554"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UOhXta6qfrAiOJbR7W+ar1i+RhsiGugQmwXTnpMOXcsRBr80wDd0GMK3i9u1qUBiO7jI31B6dKut46gKD0hDAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T00:22:45.376553Z","bundle_sha256":"fa83c52acb1fb87cef08a206a7dcdeb3da2fb2cd4b0bcbb4d2499ce81cfc6ad3"}}