Cohomological tautness for Riemannian foliations

Abstract

In this paper we present some new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation can be characterized cohomologically. We extend this cohomological characterization to a class of foliations which includes the foliated strata of any singular Riemannian foliation of a closed manifold

Cohomological tautness of singular Riemannian foliations

“…Nevertheless, in [22] and [23] we proved that, for a certain class of foliations called CERFs, the Álvarez class is well defined and the characterizations of tautness described above hold. 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.…”

“…where q is the codimension of F and the κ-twisted basic cohomology H * κ (M/F) stands for the cohomology of the basic de Rham complex with the twisted differential d κ ω = dω −κ ∧ω. It follows that, under the assumptions of Masa's theorem, F is taut if and only if H 0 κ (M/F) ∼ = R. For an account of the history of tautness and cohomology of Riemannian foliations see [23] and V. Sergiescu's Appendix in [15].…”

“…We shall say that a tense metric μ is strongly tense if κ μ is also closed (see [19,Def. 2.4]; in [22] and [23] it is called a D-metric).…”

Poincaré duality of the basic intersection cohomology of a Killing foliation

“…We introduced the intersection basic cohomology in [10] and the examples and results obtained indicate that this cohomology theory is suitable for the study of topology and geometry of singular Riemannian foliations, [8,9,11,12]. In the present paper we demonstrate that under suitable orientation assumptions the basic intersection cohomology of a Killing foliation satisfies the Poincaré duality property.…”

Deforming metrics of foliations

“…Proof. Since D is g 0 -umbilical, we have b = 1 n Hĝ at t = 0, where H is the mean curvature vector field of D. Applying to (13) the theorem on existence/uniqueness of a solution of ODEs, we conclude that b t = 1 nH tĝt for all t, for someH t ∈ Γ(D ⊥ ). Tracing this, we see thatH t is the mean curvature vector of b t , hence D is umbilical for any g t .…”