Haefliger cohomology of complete Riemannian foliations
arXiv:1209.3817
Abstract
Haefliger cohomology characterizes taut foliated manifolds by Haefliger's theorem. We show that Haefliger cohomology characterizes strongly tense foliated manifolds, namely, foliated manifolds which admit a Riemannian metric such that the mean curvature form of the leaves is closed and basic. We show that Haefliger cohomology is dual to invariant cohomology for complete Riemannian foliations. As an application of these results, we prove that any complete Riemannian foliation is strongly tense, which is a generalization of Domínguez's tenseness theorem for Riemannian foliations on closed manifolds.
31 pages. v4: The definition of twisted Haefliger cohomology of foliated manifolds (Section 1.2, Definition 3.1) was corrected. In addition, several minor errors are corrected in the proof of Proposition 2.5, Theorem 3.7, Lemma 5.2, Section 5.3 and Lemma 6.6. v3: The main result is more general than that of the last version