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Confoliations. (English) Zbl 0893.53001

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

Citations:

Zbl 0809.53030
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