Tautness for riemannian foliations on non-compact manifolds

manuscripta mathematica (Impact Factor: 0.5). 05/2005; 126(2):177-200. DOI: 10.1007/s00229-008-0172-0

ABSTRACT For a riemannian foliation F{\mathcal{F}} on a closed manifold M, it is known that F{\mathcal{F}} is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form
km\kappa_\mu (relatively to a suitable riemannian metricμ) is zero (cf. Álvarez in Ann Global Anal Geom 10:179–194, 1992). In the transversally
orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group Hn(M/F)H^{n}\,(M\,/\,{\mathcal{F}}) , where n = codimFn = {\rm codim}\,{\mathcal{F}} (cf. Masa in Comment Math Helv 67:17–27, 1992). By the Poincaré Duality (cf. Kamber et and Tondeur in Astérisque 18:458–471,
1984) this last condition is equivalent to the non-vanishing of the basic twisted cohomology group H0km(M/F)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).

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Available from: Martintxo Saralegi-Aranguren, Jan 28, 2014
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    • "Perhaps one should approach the problem from a different angle, and consider some other cohomology theory. 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. "
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    ABSTRACT: We prove that the basic intersection cohomology $IH^*_\overline{p}(M/\mathcal{F}$, where $\mathcal{F}$ is the singular foliation determined by an isometric action of a Lie group $G$ on the compact manifold $M$, verifies the Poincaré Duality Property.
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    • "[26]), the relation between the above conditions is not straightforwardly exportable to the singular framework. As pointed out by the authors in [11], the singular nature of a SRF on a compact manifold prevents any global metric on it from making all the leaves minimal (see also [17]). "
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    ABSTRACT: It is known that, for a regular riemannian foliation on a compact manifold, the properties of its basic cohomology (non-vanishing of the top-dimensional group and Poincar\'e Duality) and the tautness of the foliation are closely related. If we consider singular riemannian foliations, there is little or no relation between these properties. We present an example of a singular isometric flow for which the top dimensional basic cohomology group is non-trivial, but its basic cohomology does not satisfy the Poincar\'e Duality property. We recover this property in the basic intersection cohomology. It is not by chance that the top dimensional basic intersection cohomology groups of the example are isomorphic to either 0 or $\mathbb{R}$. We prove in this Note that this holds for any singular riemannian foliation of a compact connected manifold. As a Corollary, we get that the tautness of the regular stratum of the singular riemannian foliation can be detected by the basic intersection cohomology.
    Bulletin of the Polish Academy of Sciences Mathematics 07/2006; 89:429-440. DOI:10.4064/ba53-4-8
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