Taut Foliation
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 
.
3. 
admits a Riemannian metric for which all leaves are minimal surfaces.
Moreover, a necessary and sufficient condition for tautness of a foliation 
is that 
contain no generalized Reeb components (Goodman 1975).
REFERENCES:
Calegari, D. Foliations and the Geometry of 3-Manifolds. Oxford, England: Clarendon Press, 2007.
Eliashberg, Y. M. and Thurston, W. P. Confoliations. Providence, RI: Amer. Math. Soc., 1998.
Gabai, D. and Oertel, U. "Essential Laminations in 3-Manifolds." Ann. Math. 130, 41-73, 1989.
Goodman, S. "Closed Leaves in Foliations of Codimension One." Comm. Math. Helv. 50, 383-388, 1975.
