Confoliations ( University Lecture Series )

Publication series ：University Lecture Series

Author： Yakov M. Eliashberg;William P. Thurston  

Publisher： American Mathematical Society‎

Publication year： 1997

E-ISBN: 9781470421625

P-ISBN(Paperback): 9780821807767

Subject： O189.3 analytical topology

Keyword： Geometry and Topology

Language： ENG

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Confoliations

Description

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. Features: A unified approach to the topology of codimension-one foliations and contact geometry. Insight on the geometric nature of integrability. New results, in particular on the perturbation of confoliations into contact structures.

Chapter

Cover

Title

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Content

Introduction

Chapter 1. Geometric nature of integrability

1.1. Foliations, contact structures and confoliations

1.2. Dynamics of codimension one foliations

1.3. Plane fields transversal to 1-dimensional bundles

Chapter 2. Perturbation of confoliations into contact structures

2.1. Linear perturbations

2.2. Conformally-Anosov Flows

2.3. Non-linear deformations

2.4. Approximations of foliations by contact structures

2.5. Perturbation near holonomy curves

2.6. Alternative approaches to Proposition 2.5.1

2.7. Transitive confoliations

2.8. Propagation of the perturbation along the leaves

2.9. Discussion

Chapter 3. Taut vs. Tight

3.1. Tight contact structures and taut foliations

3.2. Symplectic filling

3.3. The inequality

3.4. Contact geometry of planes in the standard contact R[sup(3)]

3.5. Tight and Taut confoliations

3.6. Homotopy of confoliations

3.7. A few open problems about confoliations

Bibliography

Back Cover

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