---
tag: HBQ8BZ7XB5
kind: concept
name: Contact manifold
msc: [53D10, 57R17]
specializes: [F45VQ47GPV]
sources:
  - url: https://en.wikipedia.org/wiki/Contact_geometry
---

## Summary

A contact manifold is an odd-dimensional [[smooth manifold|F45VQ47GPV]] with a field of hyperplanes.
This field is as far as possible from being tangent to a family of hypersurfaces.
Contact geometry is the odd-dimensional sister of symplectic geometry.
Level sets of energy in phase space, unit cotangent bundles and spaces of first-order jets carry natural contact structures.
It is the setting of geometric optics, thermodynamics and Legendrian knot theory.

## Explanations

### L1 human

At each point of a three-dimensional space, pick a plane through that point.
If the planes fit together as the tangent planes of a stack of surfaces, the field is integrable.
A contact structure is the opposite, a field of planes that twists everywhere.
In $\R^3$ the planes spanned by $\partial_y$ and $\partial_x + y\,\partial_z$ form such a field: moving in the $y$ direction, the planes turn about the $y$-axis.
No surface can be tangent to these planes on an open set.