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# 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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