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 Poincaré Duality, this last condition is equivalent to the non-vanishing of the basic twisted cohomology group $H_{{}_{{\kappa }_{\mu }}}^{{}^0}(M/\mathcal{F})$, 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).

18 pages