{"paper":{"title":"Rigidity of the Alvarez class","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hiraku Nozawa","submitted_at":"2009-09-07T01:37:16Z","abstract_excerpt":"Let $(M,\\mathcal{F})$ be a closed manifold with a Riemannian foliation. The \\'{A}lvarez class is the cohomology class of degree 1 of $M$ whose triviality characterizes the minimizability of $(M,\\mathcal{F})$. We show that the integral of the \\'{A}lvarez class along every closed path in $M$ is the logarism of an algebraic integer if $\\pi_{1}M$ is polycyclic or $\\mathcal{F}$ is of polynomial growth."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.1125","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"}