Taut foliation information

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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 Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface. Furthermore, for compact manifolds the existence, for every leaf ${\displaystyle L}$, of a transverse circle meeting ${\displaystyle L}$, implies the existence of a single transverse circle meeting every leaf.

Taut foliations were brought to prominence by the work of William Thurston and David Gabai.