Reeb vector field

In mathematics, the Reeb vector field, named after the French mathematician Georges Reeb, is a notion that appears in various domains of contact geometry including:


 * in a contact manifold, given a contact 1-form $$\alpha$$, the Reeb vector field satisfies $$R \in \mathrm{ker }\ d\alpha, \ \alpha (R) = 1 $$,
 * in particular, in the context of Sasakian manifold.

Definition
Let $$\xi$$ be a contact vector field on a manifold $$M$$ of dimension $$2n+1$$. Let $$\xi = Ker \; \alpha$$ for a 1-form $$\alpha$$ on $$M$$ such that $$\alpha \wedge (d \alpha)^n \neq 0$$. Given a contact form $$\alpha$$, there exists a unique field (the Reeb vector field) $$X_\alpha$$ on $$M$$ such that: .
 * $$i(X_\alpha)d \alpha = 0$$
 * $$i(X_\alpha) \alpha = 1$$