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Moreover, we prove that the Euler characteristic of a compact Riemannian manifold $M^{4l+4}$ or $M^{4l+2}$ with positive sectional curvature is positive if $M$ admits an effective isometric action of a torus $T^l$, i.e., if the symmetry rank of $M$ is $\\ge l$."},"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":"1207.4086","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-07-17T18:55:12Z","cross_cats_sorted":[],"title_canon_sha256":"c2ff074b07df8dfd086045fedd4c1a5956864fda524544d16e9829e44f172c27","abstract_canon_sha256":"e013d53d3e4947c2b3c197084527e04001d9facc229e806515a1439753fe19c7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:49.058249Z","signature_b64":"/U1NP3Haqp5a7xRg7l57Xvmp/u0Z/3OuROa/y3DYmYfnAUAXBwQ18BoY4DukzXU4sx/JOIP5HMp2OQQjqOezBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"920b5eb01dfd599919534f99b3a3ce9e89a3622d6cd02f94eecf8efc49d19033","last_reissued_at":"2026-05-18T03:50:49.057634Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:49.057634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Catherine Searle, Thomas Puettmann","submitted_at":"2012-07-17T18:55:12Z","abstract_excerpt":"We prove that the Euler characteristic of an even-dimensional compact manifold with positive (nonnegative) sectional curvature is positive (nonnegative) provided that the manifold admits an isometric action of a compact Lie group $G$ with principal isotropy group $H$ and cohomogeneity $k$ such that $k - (\\rank G - \\rank H)\\le 5$. 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