---
tag: WQQX6SAQ5A
kind: concept
name: Tight and overtwisted contact structures
msc: [57K33, 53D10]
sources:
  - url: https://en.wikipedia.org/wiki/Contact_geometry
---

## Summary

Contact structures on $3$-manifolds split into two classes.
An overtwisted contact structure contains an overtwisted disc, a disc whose boundary circle is tangent to the contact planes in a special twisted way.
A tight one contains no such disc.
Eliashberg showed that overtwisted structures are classified by homotopy theory alone, so they are flexible.
Tight structures are rigid and carry the deep information of three-dimensional [[contact topology|HBQ8BZ7XB5]].

## Explanations

### L1 human

A contact structure is a field of planes on a three-dimensional space that twists everywhere.
In an overtwisted one, the planes twist so much that one can find a special disc.
Its rim runs along the planes, and all along the rim the planes lie flat on the disc.
A tight contact structure never twists that much.
The standard structure on $\R^3$ is tight, by a theorem of Bennequin, while for example the planes $\ker(\cos r\, dz + r \sin r\, d\theta)$ in cylindrical coordinates are overtwisted.