High Quality Content by WIKIPEDIA articles! In mathematics, a taut foliation is a codimension 1 foliation of a 3-manifold with the property there is a single transverse circle intersecting every leaf. By transverse circle, it is meant a closed loop that is always transverse to the tangent field of the foliation. Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface. Taut foliations were brought to prominence by the work of William Thurston and David Gabai. It is closely related to the concept of Reebless foliation. A taut foliation cannot have a Reeb component, since the component would act like a "dead-end" from which a transverse curve could never escape; consequently, the boundary torus of the Reeb component has no transverse circle puncturing it. A Reebless foliation can fail to be taut but the only leaves of the foliation with no puncturing transverse circle must be compact, and in particular, homeomorphic to a torus.

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