Taut foliation
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In mathematics, tautness is a rigidity property of foliations. A taut foliation is a codimension 1 foliation of a closed manifold with the property that every leaf meets a transverse circle.[1]: 155 By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation.
If the foliated manifold has non-empty tangential boundary, then a codimension 1 foliation is taut if every leaf meets a transverse circle or a transverse arc with endpoints on the tangential boundary. Equivalently, by a result of
Taut foliations were brought to prominence by the work of William Thurston and David Gabai.
Relation to Reebless foliations
Taut foliations are closely related to the concept of
Properties
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
Rummler–Sullivan theorem
By a theorem of Hansklaus Rummler and Dennis Sullivan, the following conditions are equivalent for transversely orientable codimension one foliations of closed, orientable, smooth manifolds M:[2][1]: 158
- is taut;
- there is a flow transverse to which preserves some volume form on M;
- there is a Riemannian metric on M for which the leaves of are least area surfaces.
References
- ^ Clarendon Press.
- ^ Alvarez Lopez, Jesús A. (1990). "On riemannian foliations with minimal leaves". Annales de l'Institut Fourier. 40 (1): 163–176.