Confoliations

Y. Eliashberg, William P. Thurston0821807765, 9780821807767

This book presents the first steps of a theory of confoliations designed to link geometry and topology of three-dimensional contact structures with the geometry and topology of codimension-one foliations on three-dimensional manifolds. Developing almost independently, these theories at first glance belonged to two different worlds: The theory of foliations is part of topology and dynamical systems, while contact geometry is the odd-dimensional “brother” of symplectic geometry.
However, both theories have developed a number of striking similarities. Confoliations–which interpolate between contact structures and codimension-one foliations–should help us to understand better links between the two theories. These links provide tools for transporting results from one field to the other.

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Table of contents :
Contents……Page 3
Introduction……Page 4
1.1. Foliations, contact structures and confoliations……Page 5
1.2. Dynamics of codimension one foliations……Page 13
1.3. Plane fields transversal to 1-dimensional bundles……Page 17
2.1. Linear perturbations……Page 26
2.2. Conformally-Anosov Flows……Page 27
2.3. Non-linear deformations……Page 33
2.5. Perturbation near holonomy curves……Page 34
2.6. Alternative approaches to Proposition 2.5.1……Page 39
2.7. TVansitive confoliations……Page 40
2.8. Propagation of the perturbation along the leaves……Page 43
2.9. Discussion……Page 46
3.1. Tight contact structures and taut foliations……Page 48
3.2. Symplectic filling……Page 49
3.3. The inequality……Page 51
3.4. Contact geometry of planes in the standard contact R3……Page 54
3.5. Tight and Taut confoliations……Page 58
3.6. Homotopy of confoliations……Page 65
3.7. A few open problems about confoliations……Page 66
Bibliography……Page 68