Haefliger cohomology of complete Riemannian foliations

TL;DR

This paper extends the understanding of Haefliger cohomology to characterize strongly tense Riemannian foliations, demonstrating its duality with invariant cohomology and generalizing existing tenseness theorems.

Contribution

It shows Haefliger cohomology characterizes strongly tense foliations and proves that all complete Riemannian foliations are strongly tense, broadening prior results.

Findings

Haefliger cohomology characterizes strongly tense foliated manifolds.

Haefliger cohomology is dual to invariant cohomology for complete Riemannian foliations.

Any complete Riemannian foliation is strongly tense.

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 Dom\'{i}nguez's tenseness theorem for Riemannian foliations on closed manifolds.