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In dimensions $4m+2$ the examples are constructed on $S^{4m-1}\\times S^3$; in dimensions $4m$ they are obtained as totally geodesic fixed point submanifolds $S^{4m-1}\\times S^1$. Since the normal rank is odd in all examples, the Ferus--Adams bound predicted by Toponogov's question would force the leaf dimension to be zero. Thus positive mixed sectional curvature "},"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":"2606.21610","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-19T17:14:23Z","cross_cats_sorted":[],"title_canon_sha256":"7785f7c757b5a372cac4cb8e5be00c86723c22b1149aac97e12301f0449ad719","abstract_canon_sha256":"f68e744e4cc9938ce5fb48f6c814cc8246c27dc6d538296a8aa6167a6eba1b4c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T01:13:16.208229Z","signature_b64":"cTnOjlL5yLuYP/PJ/7y34bgmTgXwFSpE+rMZmflp8LxY4cWyAQTSpxcw7cW8qHIwJ+IMenshIiXaYl73vEe4DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9e31921f4b27a0326d2f6d347156844cd4358bd6e918e6084d7837802f1b527d","last_reissued_at":"2026-06-23T01:13:16.207771Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T01:13:16.207771Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Closed counterexamples to Toponogov's question on mixed curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Yaroslav Bazaikin","submitted_at":"2026-06-19T17:14:23Z","abstract_excerpt":"We construct explicit closed Riemannian manifolds in every even dimension carrying one-dimensional totally geodesic foliations with positive mixed sectional curvature. The leaves are closed geodesics and form smooth circle fibrations. In dimensions $4m+2$ the examples are constructed on $S^{4m-1}\\times S^3$; in dimensions $4m$ they are obtained as totally geodesic fixed point submanifolds $S^{4m-1}\\times S^1$. Since the normal rank is odd in all examples, the Ferus--Adams bound predicted by Toponogov's question would force the leaf dimension to be zero. 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