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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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. 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