Tautness for riemannian foliations on non-compact manifolds

Abstract

For a riemannian foliation $\mathcal{F}$ on a closed manifold $M$, it is known that $\mathcal{F}$ is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form $\kappa_\mu$ (relatively to a suitable riemannian metric $\mu$) is zero. In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group $H^{^{n}}(M/\mathcal{F})$, where $n = \codim \mathcal{F}$. By the Poincar\'e Duality, this last condition is equivalent to the non-vanishing of the basic twisted cohomology group $H^{^{0}}_{_{\kappa_\mu}}(M/\mathcal{F})$, when $M$ is oriented. When $M$ is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation on a compact manifold (CERF).Comment: 18 page

Tenseness of Riemannian flows

“…Remark 1.3. -For Riemannian foliations of any dimension which can be suitably embedded into a singular Riemannian foliation on a compact manifold, strongly tenseness was proved in [28,29] by the application of Domínguez's theorem. For any Riemannian foliation such that the space of leaf closures is compact, the strongly tenseness was proved in [25, Theorem 1.9].…”

“…Remark 1.8. -For Riemannian foliations of any dimension which can be suitably embedded into a singular Riemannian foliation on a compact manifold, the Álvarez class is well-defined as shown in [28,29] by the application of Domínguez's theorem. First, we state the twisted Poincaré duality of the basic cohomology, which is a consequence of a theorem of Sergiescu and the dichotomy theorem (Theorem 1.4).…”

“…Let κ be a representative of the Álvarez class of (M, F). Then, the following are equivalent: For Riemannian foliations of any dimension which can be suitably embedded into a singular Riemannian foliation on a compact manifold, the equivalence of the first three statements of Corollary 1.10 are shown in [28,29] by the application of Domínguez's theorem.…”

“…Here, recall that a foliated manifold (M, F) is called tense if M admits a metric such that the mean curvature form of F is basic. Theorem 1.1 can be regarded as a generalization of Masa's characterization of tautness, and has many applications in the study of geometrical and cohomological properties of Riemannian foliations (see, for example, [17,18,34,27,28,29]). …”

Finiteness of the basic intersection cohomology of a Killing foliation

“…An open covering {U, V} of M by saturated open subsets is a basic covering. It possesses a subordinated partition of the unity made up of basic functions defined on M (see [9]). For a such covering we have the Mayer-Vietoris short sequence…”