|
|
|
|
|
Taut FoliationIn mathematics, a taut foliation is a co-dimension 1 foliation of a 3-manifold with the property that every leaf has a transverse circle intersecting it. 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 co-dimension 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. The existence of a taut foliation implies various useful properties about a closed 3-manifold. It must be irreducible, covered by , and have negatively curved fundamental group.
|
 |
|
| Copyright 2005-2009 OnPedia.com. All Rights Reserved |
|
|