Contact structure on a jet bundle

Remarks

In adapted coordinates $(x^i, u^\alpha, u^\alpha_I)$, the contact forms are the combinations with smooth coefficients of $$\theta^\alpha_I = du^\alpha_I - \sum_{i} u^\alpha_{I + i}\, dx^i, \qquad 0 \le \abs{I} \le k - 1,$$ where $u^\alpha_\emptyset = u^\alpha$ and $I + i$ is the multi-index $I$ with $i$ added once.

A local section $\sigma$ of $J^k\pi \to M$ is holonomic, that is, of the form $j^k s$, exactly when $\sigma^*\theta = 0$ for every contact form $\theta$. For $k = 1$, $M = \R$ and one fiber coordinate this is a direct check: if $\sigma(x) = (x, s(x), q(x))$ then $\sigma^*(du - q\, dx) = (s'(x) - q(x))\, dx$, which vanishes exactly when $q = s'$.

Example: for $\pi \colon \R \times \R \to \R$, $J^1\pi = \R^3$ with coordinates $(x, u, q)$ and $\theta = du - q\, dx$. Then $d\theta = dx \wedge dq$ and $\theta \wedge d\theta = du \wedge dx \wedge dq \ne 0$. So $\theta$ is a contact form in the sense of contact geometry, and $J^1\pi$ is a contact manifold.

Non-example: on the same space the form $du$ is not a contact form. For $s(x) = x$ one gets $(j^1 s)^* du = dx \ne 0$.

For $k = 1$ and one fiber coordinate, $J^1\pi$ is a contact manifold for every $\dim M$, with contact form $du - \sum_i u_i\, dx^i$. For $k > 1$ or more than one fiber coordinate, the contact distribution has codimension greater than $1$. So it is not a contact structure in the sense of contact geometry, though the name is the same.