Cohomological tautness of singular Riemannian foliations

Abstract

For a Riemannian foliation F on a compact manifold M, J. A. Alvarez Lopez proved that the geometrical tautness of F, that is, the existence of a Riemannian metric making all the leaves minimal submanifolds of M, can be characterized by the vanishing of a basic cohomology class kappa(M) is an element of H-1(M/F) (the Alvarez class). In thisworkwe generalize this result to the case of a singular Riemannian foliation K on a compact manifold X. In the singular case, no bundlelikemetric on X can make all the leaves ofK minimal. In this work, we prove that the Alvarez classes of the strata can be glued in a unique global Alvarez class kappa(X) is an element of H-1(X/K). As a corollary, if X is simply connected, then the restriction of K to each stratum is geometrically taut, thus generalizing a celebrated result of E. Ghys for the regular case.