Taut foliation

Taut foliation

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.

The existence of a taut foliation implies various useful properties about a closed 3-manifold. For example, a closed, orientable 3-manifold, which admits a taut foliation with no sphere leaf, must be irreducible, covered by mathbb R^3, and have negatively curved fundamental group.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Foliation — In mathematics, a foliation is a geometric device used to study manifolds. Informally speaking, a foliation is a kind of clothing worn on a manifold, cut from a striped fabric. On each sufficiently small piece of the manifold, these stripes give… …   Wikipedia

  • List of mathematics articles (T) — NOTOC T T duality T group T group (mathematics) T integration T norm T norm fuzzy logics T schema T square (fractal) T symmetry T table T theory T.C. Mits T1 space Table of bases Table of Clebsch Gordan coefficients Table of divisors Table of Lie …   Wikipedia

  • 3-manifold — In mathematics, a 3 manifold is a 3 dimensional manifold. The topological, piecewise linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3… …   Wikipedia

  • David Gabai — Nationality  United States Fields …   Wikipedia

  • Western sculpture — ▪ art Introduction       three dimensional artistic forms produced in what is now Europe and later in non European areas dominated by European culture (such as North America) from the Metal Ages (Europe, history of) to the present.       Like… …   Universalium

  • List of differential geometry topics — This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. Contents 1 Differential geometry of curves and surfaces 1.1 Differential geometry of curves 1.2 Differential… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”