{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:M4PSN3XOXCPCNT7NKDDWW2GXPD","short_pith_number":"pith:M4PSN3XO","canonical_record":{"source":{"id":"0911.3857","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-11-19T17:58:46Z","cross_cats_sorted":[],"title_canon_sha256":"c2011a795c4fc2dad85159e53a8eb3858da71daea4108f50b491e936019223a4","abstract_canon_sha256":"7b0f4efb588921a0b6ae869187f8430f7eebfa1b7dead925d0c86122fc652bc4"},"schema_version":"1.0"},"canonical_sha256":"671f26eeeeb89e26cfed50c76b68d778ceb3d15b12066fc6386c3a3c3e9709b0","source":{"kind":"arxiv","id":"0911.3857","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.3857","created_at":"2026-05-18T03:54:06Z"},{"alias_kind":"arxiv_version","alias_value":"0911.3857v3","created_at":"2026-05-18T03:54:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.3857","created_at":"2026-05-18T03:54:06Z"},{"alias_kind":"pith_short_12","alias_value":"M4PSN3XOXCPC","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"M4PSN3XOXCPCNT7N","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"M4PSN3XO","created_at":"2026-05-18T12:26:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:M4PSN3XOXCPCNT7NKDDWW2GXPD","target":"record","payload":{"canonical_record":{"source":{"id":"0911.3857","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-11-19T17:58:46Z","cross_cats_sorted":[],"title_canon_sha256":"c2011a795c4fc2dad85159e53a8eb3858da71daea4108f50b491e936019223a4","abstract_canon_sha256":"7b0f4efb588921a0b6ae869187f8430f7eebfa1b7dead925d0c86122fc652bc4"},"schema_version":"1.0"},"canonical_sha256":"671f26eeeeb89e26cfed50c76b68d778ceb3d15b12066fc6386c3a3c3e9709b0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:54:06.331658Z","signature_b64":"KptsPZdr7Zj1+RZ1c27vCO4yk9sVPldwlSRbx5eE8+sH5U08xDQBecaOcZJABkCPIeQY93vVn3fcmaikd9T0DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"671f26eeeeb89e26cfed50c76b68d778ceb3d15b12066fc6386c3a3c3e9709b0","last_reissued_at":"2026-05-18T03:54:06.330890Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:54:06.330890Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0911.3857","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:54:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K6T7kHDOB7NOvrvljFITj/Af6Ko1GIdKwlIV7xafkqY1XgiAURqpeU4QjWVC6FdArJA8SPknVw4SKx8oRJ83Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T06:09:29.386301Z"},"content_sha256":"c6619e072559a8bf7908b5f885c968a8e0d778502f132f73d33fc6c060ac7393","schema_version":"1.0","event_id":"sha256:c6619e072559a8bf7908b5f885c968a8e0d778502f132f73d33fc6c060ac7393"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:M4PSN3XOXCPCNT7NKDDWW2GXPD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Parallel submanifolds with an intrinsic product structure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Tillmann Jentsch","submitted_at":"2009-11-19T17:58:46Z","abstract_excerpt":"Let $M$ and $N$ be Riemannian symmetric spaces and $f:M\\to N$ be a parallel isometric immersion. We additionally assume that there exist simply connected, irreducible Riemannian symmetric spaces $M_i$ with $\\dim(M_i)\\geq 2$ for $i=1,...,r$ such that $M\\cong M_1\\times...\\times M_r$ . As a starting point, we describe how the intrinsic product structure of $M$ is reflected by a distinguished, fiberwise orthogonal direct sum decomposition of the corresponding first normal bundle. Then we consider the (second) osculating bundle $\\osc f$, which is a $\\nabla^N$-parallel vector subbundle of the pullba"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.3857","kind":"arxiv","version":3},"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":"2026-05-18T03:54:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AutLEKzKqFgl6c+Od9yyX3IQdbSNpOSoy+8+iqXuNF6V2A8Bp/kPQvtufKiVjl36Ww3BL1Y8UzDgALzrYEm5Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T06:09:29.386661Z"},"content_sha256":"3580418fe3f45b13072bfa72fc1150eef46a3393c4008085a9e7b84e76b7602c","schema_version":"1.0","event_id":"sha256:3580418fe3f45b13072bfa72fc1150eef46a3393c4008085a9e7b84e76b7602c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/M4PSN3XOXCPCNT7NKDDWW2GXPD/bundle.json","state_url":"https://pith.science/pith/M4PSN3XOXCPCNT7NKDDWW2GXPD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/M4PSN3XOXCPCNT7NKDDWW2GXPD/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":"2026-05-30T06:09:29Z","links":{"resolver":"https://pith.science/pith/M4PSN3XOXCPCNT7NKDDWW2GXPD","bundle":"https://pith.science/pith/M4PSN3XOXCPCNT7NKDDWW2GXPD/bundle.json","state":"https://pith.science/pith/M4PSN3XOXCPCNT7NKDDWW2GXPD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/M4PSN3XOXCPCNT7NKDDWW2GXPD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:M4PSN3XOXCPCNT7NKDDWW2GXPD","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":"7b0f4efb588921a0b6ae869187f8430f7eebfa1b7dead925d0c86122fc652bc4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-11-19T17:58:46Z","title_canon_sha256":"c2011a795c4fc2dad85159e53a8eb3858da71daea4108f50b491e936019223a4"},"schema_version":"1.0","source":{"id":"0911.3857","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.3857","created_at":"2026-05-18T03:54:06Z"},{"alias_kind":"arxiv_version","alias_value":"0911.3857v3","created_at":"2026-05-18T03:54:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.3857","created_at":"2026-05-18T03:54:06Z"},{"alias_kind":"pith_short_12","alias_value":"M4PSN3XOXCPC","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"M4PSN3XOXCPCNT7N","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"M4PSN3XO","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:3580418fe3f45b13072bfa72fc1150eef46a3393c4008085a9e7b84e76b7602c","target":"graph","created_at":"2026-05-18T03:54:06Z","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":"Let $M$ and $N$ be Riemannian symmetric spaces and $f:M\\to N$ be a parallel isometric immersion. We additionally assume that there exist simply connected, irreducible Riemannian symmetric spaces $M_i$ with $\\dim(M_i)\\geq 2$ for $i=1,...,r$ such that $M\\cong M_1\\times...\\times M_r$ . As a starting point, we describe how the intrinsic product structure of $M$ is reflected by a distinguished, fiberwise orthogonal direct sum decomposition of the corresponding first normal bundle. Then we consider the (second) osculating bundle $\\osc f$, which is a $\\nabla^N$-parallel vector subbundle of the pullba","authors_text":"Tillmann Jentsch","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-11-19T17:58:46Z","title":"Parallel submanifolds with an intrinsic product structure"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.3857","kind":"arxiv","version":3},"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:c6619e072559a8bf7908b5f885c968a8e0d778502f132f73d33fc6c060ac7393","target":"record","created_at":"2026-05-18T03:54:06Z","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":"7b0f4efb588921a0b6ae869187f8430f7eebfa1b7dead925d0c86122fc652bc4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-11-19T17:58:46Z","title_canon_sha256":"c2011a795c4fc2dad85159e53a8eb3858da71daea4108f50b491e936019223a4"},"schema_version":"1.0","source":{"id":"0911.3857","kind":"arxiv","version":3}},"canonical_sha256":"671f26eeeeb89e26cfed50c76b68d778ceb3d15b12066fc6386c3a3c3e9709b0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"671f26eeeeb89e26cfed50c76b68d778ceb3d15b12066fc6386c3a3c3e9709b0","first_computed_at":"2026-05-18T03:54:06.330890Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:54:06.330890Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KptsPZdr7Zj1+RZ1c27vCO4yk9sVPldwlSRbx5eE8+sH5U08xDQBecaOcZJABkCPIeQY93vVn3fcmaikd9T0DA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:54:06.331658Z","signed_message":"canonical_sha256_bytes"},"source_id":"0911.3857","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c6619e072559a8bf7908b5f885c968a8e0d778502f132f73d33fc6c060ac7393","sha256:3580418fe3f45b13072bfa72fc1150eef46a3393c4008085a9e7b84e76b7602c"],"state_sha256":"b5feff43263464b1cb9e5fcd2af4fd867d3261d8d58f30dafa439cb6c902ffef"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6piLo+3kXE5EbzFp5PTMbZtGFTKzn9vIypYHEz9F9ElatZjZvfm7kF6/5VqCrj/29YIe8lSgVYo5T3OkU/rPBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T06:09:29.388579Z","bundle_sha256":"29973c9536328301154abaaa49e466518682f7f8053a057b4a2d61bd03054c2c"}}