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 μ) 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 $H^{n}\,(M\,/\,{\mathcal{F}})$ , where $n = {\rm codim}\,{\mathcal{F}}$ (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 $H^{0}_{\kappa_\mu}(M\,/\,{\mathcal{F}})$ , 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).