A condition for the stability of ℝ-covered on foliations of 3-manifolds

We give a sufficient condition for a codimension one, transversely orientable foliation of a closed 3-manifold to have the property that any foliation sufficiently close to it be R R -covered. This condition can be readily verified for many examples. Further, if an R R -covered foliation has a compact leaf L L , then any transverse loop meeting L L lifts to a copy of the leaf space, and the ambient manifold fibers over S 1 S^1 with L L as fiber.