Abstract
For a Riemannian foliation {mathcal {F}} on a compact manifold M, J. A. Álvarez López proved that the geometrical tautness of {mathcal {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 {varvec{kappa }}_Min H^1(M/{mathcal {F}}) (the Álvarez class). In this work we generalize this result to the case of a singular Riemannian foliation {mathcal {K}} on a compact manifold X. In the singular case, no bundle-like metric on X can make all the leaves of {mathcal {K}} minimal. In this work, we prove that the Álvarez classes of the strata can be glued in a unique global Álvarez class {varvec{kappa }}_Xin H^1(X/{mathcal {K}}). As a corollary, if X is simply connected, then the restriction of {mathcal {K}} to each stratum is geometrically taut, thus generalizing a celebrated result of E. Ghys for the regular case.
Highlights
In this work we generalize this result to the case of a singular Riemannian foliation K on a compact manifold X
CERFs are regular Riemannian foliations on possibly non-compact manifolds that can be suitably embedded in a regular Riemannian foliation on a compact manifold called zipper, and whose basic cohomology is computed by a compact saturated subset called reppiz
The compactness of X does not imply the compactness of the strata, each stratum will inherit the cohomological behavior of tautness from its zipper
Summary
In this work we intend to understand the tautness character of all strata globally, by showing that the topology of X has, a strong influence on the tautness of each stratum, individually. Our main result is the following: Theorem 1.1 Let K be an SRF on a closed manifold X. To prove Theorem 1.1 we shall need to exploit the local description of the neighbourhood of a stratum of an SRF, and use strongly the fact that its associated sphere bundle admits a compact structure group. 2 and 3 we recall some known facts about SRFs and CERFs, respectively, and prove that certain bundles of singular strata are CERFs. In Sect. 4 we introduce the notion of thick foliated bundle and study the interplay between the Álvarez classes of their components. As thick foliated bundles appear in the local structure of an SRF, which we shall use to prove the main results in Sect. The Appendix is devoted to the reduction of the structure group of the sphere bundle of a stratum
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