Confoliations

Overview

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 ...

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Overview

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.

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Product Details

  • ISBN-13: 9780821807767
  • Publisher: American Mathematical Society
  • Publication date: 11/26/1997
  • Series: University Lecture Series , #13
  • Edition number: 1
  • Pages: 66
  • Product dimensions: 9.70 (w) x 6.80 (h) x 0.20 (d)

Table of Contents

Introduction
Ch. 1 Geometric nature of integrability 1
1.1 Foliations, contact structures and confoliations 1
1.2 Dynamics of codimension one foliations 9
1.3 Plane fields transversal to 1-dimensional bundles 13
Ch. 2 Perturbation of confoliations into contact structures 23
2.1 Linear perturbations 23
2.2 Conformally-Anosov Flows 24
2.3 Non-linear deformations 30
2.4 Approximations of foliations by contact structures 31
2.5 Perturbation near holonomy curves 31
2.6 Alternative approaches to Proposition 2.5.1 36
2.7 Transitive confoliations 37
2.8 Propagation of the perturbation along the leaves 40
2.9 Discussion 43
Ch. 3 Taut vs. Tight 45
3.1 Tight contact structures and taut foliations 45
3.2 Symplectic filling 46
3.3 The inequality 48
3.4 Contact geometry of planes in the standard contact R[superscript 3] 51
3.5 Tight and Taut confoliations 55
3.6 Homotopy of confoliations 62
3.7 A few open problems about confoliations 63
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