---
tag: GGM63KXCF2
kind: concept
name: Reeb vector field
msc: [53D10, 37J55]
specializes: [0299VCW1TH]
sources:
  - url: https://en.wikipedia.org/wiki/Reeb_vector_field
same_as:
  wikidata: Q7306558
  wikipedia: Reeb vector field
---

## Summary

The Reeb vector field is the [[vector field|0299VCW1TH]] determined by a contact form on a [[contact manifold|HBQ8BZ7XB5]].
It is transverse to the contact hyperplanes and its flow preserves the contact form.
Reeb flows include geodesic flows on unit cotangent bundles and Hamiltonian flows on energy surfaces of contact type.
Their periodic orbits are the subject of the [[Weinstein conjecture|MP6B64DS31]] and of contact homology.

## Explanations

### L1 human

A contact form picks out a plane at each point, and it also measures how far a vector sticks out of that plane.
The Reeb vector field is the unique direction that sticks out by exactly one unit and along which the twisting of the planes does not change.
On the three-sphere with its standard contact form, the Reeb flow moves every point around a great circle, and these circles are the fibers of the Hopf fibration.