{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JCLJQD56WYMFKQPXFUFMYMNYUS","short_pith_number":"pith:JCLJQD56","canonical_record":{"source":{"id":"1509.00994","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-09-03T09:02:20Z","cross_cats_sorted":[],"title_canon_sha256":"d7bbe0aa4caa4b8fac4a68273ec2dadce211f8dc3ecf870774863f1d2ccf2377","abstract_canon_sha256":"3fc5d4b9ef6e600bcbcc7ece26ca1ad651dc65777023b59918ad25419bf76412"},"schema_version":"1.0"},"canonical_sha256":"4896980fbeb6185541f72d0acc31b8a4ac2b156aaa5dcf837e0b175ca5fc944b","source":{"kind":"arxiv","id":"1509.00994","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.00994","created_at":"2026-05-18T01:34:04Z"},{"alias_kind":"arxiv_version","alias_value":"1509.00994v1","created_at":"2026-05-18T01:34:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.00994","created_at":"2026-05-18T01:34:04Z"},{"alias_kind":"pith_short_12","alias_value":"JCLJQD56WYMF","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JCLJQD56WYMFKQPX","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JCLJQD56","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JCLJQD56WYMFKQPXFUFMYMNYUS","target":"record","payload":{"canonical_record":{"source":{"id":"1509.00994","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-09-03T09:02:20Z","cross_cats_sorted":[],"title_canon_sha256":"d7bbe0aa4caa4b8fac4a68273ec2dadce211f8dc3ecf870774863f1d2ccf2377","abstract_canon_sha256":"3fc5d4b9ef6e600bcbcc7ece26ca1ad651dc65777023b59918ad25419bf76412"},"schema_version":"1.0"},"canonical_sha256":"4896980fbeb6185541f72d0acc31b8a4ac2b156aaa5dcf837e0b175ca5fc944b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:04.582688Z","signature_b64":"bTFdnO7lcY6PTuceqq6taJeUnSqX0upU4ETWr5iKTxxO19wX/8dHJW2XY+HePhgYXe2wz2eB0GlCSoaL1kQ2CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4896980fbeb6185541f72d0acc31b8a4ac2b156aaa5dcf837e0b175ca5fc944b","last_reissued_at":"2026-05-18T01:34:04.581944Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:04.581944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.00994","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":"2026-05-18T01:34:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NDC3km3POew9Juan2Ay8ibs7fZ20g5ZXDyUovc+RxMmHHEEaAsFDdxpGYo0L0R/CLlZc/wD///oNw0sxiIr3BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T07:37:11.634595Z"},"content_sha256":"19fa98c586dc892d3295c3300ac303471d5508ee1d950448a5fc3aa9cf4ee74d","schema_version":"1.0","event_id":"sha256:19fa98c586dc892d3295c3300ac303471d5508ee1d950448a5fc3aa9cf4ee74d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JCLJQD56WYMFKQPXFUFMYMNYUS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Invariant submanifolds of metric contact pairs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Amine Hadjar, Paola Piu","submitted_at":"2015-09-03T09:02:20Z","abstract_excerpt":"We show that $\\phi$-invariant submanifolds of metric contact pairs with orthogonal characteristic foliations make constant angles with the Reeb vector fields. Our main result is that for the normal case such submanifolds of dimension at least $2$ are all minimal. We prove that an odd-dimensional $\\phi$-invariant submanifold of a metric contact pair with orthogonal characteristic foliations inherits a contact form with an almost contact metric structure, and this induced structure is contact metric if and only if the submanifold is tangent to one Reeb vector field and orthogonal to the other on"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00994","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":"2026-05-18T01:34:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9O/t6TO86e8uIbEXGLoLuBQ6Pe2LT3iZYzYdICfbR45S45ug9WBgmr1NcGYbf+EqTF+11K2DO76ukUd5aI0eDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T07:37:11.635395Z"},"content_sha256":"8457e64f3898f72b8b1c0defbb4350685636e60243f77419708cde28ea192916","schema_version":"1.0","event_id":"sha256:8457e64f3898f72b8b1c0defbb4350685636e60243f77419708cde28ea192916"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JCLJQD56WYMFKQPXFUFMYMNYUS/bundle.json","state_url":"https://pith.science/pith/JCLJQD56WYMFKQPXFUFMYMNYUS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JCLJQD56WYMFKQPXFUFMYMNYUS/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":"2026-05-31T07:37:11Z","links":{"resolver":"https://pith.science/pith/JCLJQD56WYMFKQPXFUFMYMNYUS","bundle":"https://pith.science/pith/JCLJQD56WYMFKQPXFUFMYMNYUS/bundle.json","state":"https://pith.science/pith/JCLJQD56WYMFKQPXFUFMYMNYUS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JCLJQD56WYMFKQPXFUFMYMNYUS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JCLJQD56WYMFKQPXFUFMYMNYUS","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":"3fc5d4b9ef6e600bcbcc7ece26ca1ad651dc65777023b59918ad25419bf76412","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-09-03T09:02:20Z","title_canon_sha256":"d7bbe0aa4caa4b8fac4a68273ec2dadce211f8dc3ecf870774863f1d2ccf2377"},"schema_version":"1.0","source":{"id":"1509.00994","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.00994","created_at":"2026-05-18T01:34:04Z"},{"alias_kind":"arxiv_version","alias_value":"1509.00994v1","created_at":"2026-05-18T01:34:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.00994","created_at":"2026-05-18T01:34:04Z"},{"alias_kind":"pith_short_12","alias_value":"JCLJQD56WYMF","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JCLJQD56WYMFKQPX","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JCLJQD56","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:8457e64f3898f72b8b1c0defbb4350685636e60243f77419708cde28ea192916","target":"graph","created_at":"2026-05-18T01:34:04Z","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 show that $\\phi$-invariant submanifolds of metric contact pairs with orthogonal characteristic foliations make constant angles with the Reeb vector fields. Our main result is that for the normal case such submanifolds of dimension at least $2$ are all minimal. We prove that an odd-dimensional $\\phi$-invariant submanifold of a metric contact pair with orthogonal characteristic foliations inherits a contact form with an almost contact metric structure, and this induced structure is contact metric if and only if the submanifold is tangent to one Reeb vector field and orthogonal to the other on","authors_text":"Amine Hadjar, Paola Piu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-09-03T09:02:20Z","title":"Invariant submanifolds of metric contact pairs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00994","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":"sha256:19fa98c586dc892d3295c3300ac303471d5508ee1d950448a5fc3aa9cf4ee74d","target":"record","created_at":"2026-05-18T01:34:04Z","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":"3fc5d4b9ef6e600bcbcc7ece26ca1ad651dc65777023b59918ad25419bf76412","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-09-03T09:02:20Z","title_canon_sha256":"d7bbe0aa4caa4b8fac4a68273ec2dadce211f8dc3ecf870774863f1d2ccf2377"},"schema_version":"1.0","source":{"id":"1509.00994","kind":"arxiv","version":1}},"canonical_sha256":"4896980fbeb6185541f72d0acc31b8a4ac2b156aaa5dcf837e0b175ca5fc944b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4896980fbeb6185541f72d0acc31b8a4ac2b156aaa5dcf837e0b175ca5fc944b","first_computed_at":"2026-05-18T01:34:04.581944Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:04.581944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bTFdnO7lcY6PTuceqq6taJeUnSqX0upU4ETWr5iKTxxO19wX/8dHJW2XY+HePhgYXe2wz2eB0GlCSoaL1kQ2CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:04.582688Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.00994","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:19fa98c586dc892d3295c3300ac303471d5508ee1d950448a5fc3aa9cf4ee74d","sha256:8457e64f3898f72b8b1c0defbb4350685636e60243f77419708cde28ea192916"],"state_sha256":"366881057c6428063a1ca0b6f2838963aa4551508d2cc607df6dfe9326a8056b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/miw3Q3EtZDcHf9qxh8BtXwaDetLmMW0helEHUzKd0d1N7oOktpv4YRGvV76Usbkk9Bfjnlbg3fcB6yPBVuuBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T07:37:11.639712Z","bundle_sha256":"a7e78c1190cf5d82ee6cbaeecf46aa049f9b793574b44bbe3b0cb79f601ac17f"}}