Confoliations. (English) Zbl 0893.53001

Eliashberg, Yakov M.; Thurston, William P.

University Lecture Series. 13. Providence, RI: American Mathematical Society (AMS). ix, 66 p. (1998).

A transversely oriented plane field \(\xi\) on an oriented 3-manifold can be defined as the kernel of some nowhere vanishing 1-form \(\alpha\). \(\xi\) is called a confoliation if either \(\alpha \wedge d \alpha \geq 0\) or \(\alpha \wedge d \alpha \leq 0\). Thus confoliations provide a link between codimension 1 foliations (the case \(\alpha \wedge d \alpha \equiv 0\)) and contact structures (the case that \(\alpha \wedge d \alpha\) never vanishes).
This monograph develops the foundations of confoliations on 3-manifolds. The authors prove that \(C^2\)-taut foliations can be \(C^0\)-perturbed into a tight contact structure.
The concept of a confoliation, without the name, first appeared in the work of S. J. Altschuler [Ill. J. Math 39, 98-118 (1995; Zbl 0809.53030)].

Reviewer: J.Hebda (St.Louis)

MSC:

53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
57-02	Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
57N10	Topology of general \(3\)-manifolds (MSC2010)
53C12	Foliations (differential geometric aspects)
57R15	Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57R30	Foliations in differential topology; geometric theory

Keywords:

confoliation; contact structure; foliation; 3-manifold

Citations:

Zbl 0809.53030