A codimension one foliation of a 3-manifold is said to be taut if for every leaf in the leaf space of , there is a circle transverse to (i.e., a closed loop transverse to the tangent field of ) which intersects .
Taut foliations play a significant role in various aspects of topology and are credited as being one of two major tools (along with incompressible surfaces) responsible for revealing significant topological and geometric information about 3-manifolds (Gabai and Oertel 1989). As such, a considerable amount of research has gone into the study of taut foliations on 3-manifolds. One well-known result is that every taut foliation is necessarily Reebless and that, for any non-taut Reebless foliation, the leaves which don't admit a closed transversal are necessarily tori. Additionally, the closed leaves of a taut foliation are homologically nontrivial.
Some classification results for taut foliations are also known. One such result, attributed by Eliashberg and Thurston to Novikov and Sullivan, says that a foliation on a closed 3-manifold is taut if it is different from the foliation on and satisfies any one of the following:
1. Each leaf of is intersected by a transversal closed curve.
2. There exists a vector field on which is transversal to and preserves a volume form on .
Moreover, a necessary and sufficient condition for tautness of a foliation is that contain no generalized Reeb components (Goodman 1975).