Contact form ($\alpha$)

A contact form on a $(2n+1)$-manifold is a one-form satisfying $\alpha \wedge (d\alpha {)}^n\ne 0$ everywhere.

The Reeb vector field of a contact form is uniquely characterized by

For ${\alpha }_s=\frac{1}{2}(r_1^{2}d{\theta }_1+s{r}_2^{2}d{\theta }_2)$ on $S^3$, the Reeb field is $2{\partial }_{{\theta }_1}+2s^{-1}{\partial }_{{\theta }_2}$. If $s$ is irrational, its only closed orbits are the two coordinate circles.