---
tag: 2GF2675154
kind: definition
name: Reeb vector field
msc: [53D10]
defines: GGM63KXCF2
uses: [HBQ8BZ7XB5, 0299VCW1TH]
sources:
  - url: https://en.wikipedia.org/wiki/Reeb_vector_field
---

## Definition

Let $M$ be a smooth manifold of dimension $2n + 1$ and let $\alpha$ be a contact form on $M$.
So $\alpha \wedge (d\alpha)^n$ vanishes nowhere and $(M, \ker \alpha)$ is a [[contact manifold|HBQ8BZ7XB5]].
The *Reeb vector field* of $\alpha$ is the unique [[vector field|0299VCW1TH]] $R_\alpha$ on $M$ such that
$$\alpha(R_\alpha) = 1 \qquad \text{and} \qquad \iota_{R_\alpha} d\alpha = 0.$$

## Remarks

Existence and uniqueness: $d\alpha$ has rank $2n$ at each point, so its kernel is a line, and $\alpha$ is nonzero on this line because $\alpha \wedge (d\alpha)^n \ne 0$.
The flow of $R_\alpha$ preserves $\alpha$, by Cartan's formula $\mathcal{L}_{R_\alpha} \alpha = d(\alpha(R_\alpha)) + \iota_{R_\alpha} d\alpha = 0$.
The Reeb field depends on $\alpha$, not only on the contact structure: replacing $\alpha$ by $h\alpha$ changes it in general.

Example: on $\R^3$ with $\alpha = dz - y\,dx$, one has $d\alpha = dx \wedge dy$, and $R_\alpha = \partial_z$, since $\alpha(\partial_z) = 1$ and $\iota_{\partial_z}(dx \wedge dy) = 0$.
On $S^3 \subset \C^2$ with $\alpha = \sum_j (x_j\, dy_j - y_j\, dx_j)$, the Reeb field is $R_\alpha = \sum_j (x_j \partial_{y_j} - y_j \partial_{x_j})$, whose flow is $z \mapsto e^{it} z$.

Non-example: on $\R^3$ with $\alpha = dz - y\,dx$, the field $\partial_y$ satisfies $\iota_{\partial_y} d\alpha = -dx \ne 0$ and $\alpha(\partial_y) = 0$, so it is not the Reeb field.