Plane distribution

A plane distribution on a three-manifold is a rank-two distribution. Locally it is the kernel of a nowhere-zero differential 1-form $\alpha$.

The Frobenius theorem gives the local criterion

A contact structure on a three-manifold is a plane distribution locally of the form $\ker \alpha$ with $\alpha \wedge d\alpha$ nowhere zero. It is maximally nonintegrable; the standard example on ${\mathbb{R}}^3$ is $\ker (dz-xdy)$.

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.