Standard contact forms on the torus

Question

Is there a standard or "simple" contact structure on the $3$-dimensional torus $T^3$, like there are for example for the Eucliden space and the $3$-sphere?

My first thought was to consider a structure descending to $T^3$ from the standard structure defined by the form $$ \omega_{std} = dz + x\wedge dy$$ on $\mathbb{R^3}$, but this doesn't work because $\omega_{std}$ is not invariant under the action of $\mathbb{Z}^3$. Also, if such a structure were to descend to $T^3$, the Reeb flowlines would consist of periodic curves parallel to the $\partial_z$ direction, and in particular it would be transversal to a $2$-dimensional torus $T^2$. That, on the other hand, can not happen, because the contact condition would imply the existence of an exact volume form on $T^2$, contradicting Stokes theorem.

I know there are contact structures on $T^3$ because of a theorem of Thurston's that states that every closed smooth $3$ manifold supports a contact structure, but I couldn't find any explicit examples for the torus. I am particularly interested in explicit descriptions of the associated Reeb field and its flow.

Answer 1

Consider the $1$-form on $\mathbb{R}^3$ given by $$ \alpha = \cos(2\pi z) dx + \sin(2\pi z) dy, $$

where $(x,y,z)$ are the standard coordinates on $\mathbb{R}^3$. This is a contact form on $\mathbb{R}^3$ with Reeb vector field given by $$R_{\alpha} = \cos(2\pi z) \partial_x + \sin(2\pi z) \partial_y.$$ This vector field can be integrated explicitly and its integral curve passing through $(x_0,y_0,z_0)$ at time $t=0$ is given by $$\phi(t) = (x_0 + \cos(2\pi z_0)t, y_0 + \sin(2\pi z_0)t, z_0). $$

Consider the standard action of $\mathbb{Z}^3$ on $\mathbb{R}^3$ by translations. The $1$-form $\alpha$ is invariant under this action and, therefore, descends to a $1$-form on the quotient $\mathbb{R}^3/\mathbb{Z}^3 \simeq T^3$. This gives an example of a contact structure on $T^3$.

More generally, given any $k \in \mathbb{Z} \smallsetminus \{0\}$, one can define a contact form on $\mathbb{R}^3$ by $$ \cos(2\pi k z) dx + \sin(2\pi k z) dy. $$ This contact form is invariant under the above action of $\mathbb{Z}^3$ and, therefore, descends to a contact form on $T^3$.