  In this article, we study the geometric properties of codimension one
foliations on Riemannian manifolds equipped with vector fields that are closed
and conformal. Apart from its singularities, these vector fields define
codimension one foliations with nice geometric features, which we call Montiel
Foliations. We investigate conditions for which a foliation transverse to this
structure has totally geodesic leaves, as well as how the ambient space and the
geometry of the leaves forces a given foliation into a Montiel Foliation. Our
main results concern minimal leaves and constant mean curvature foliations,
having compact or complete noncompact leaves. Finally, we characterize totally
geodesic foliations by means of its relation to a prior Montiel Foliation.
