Jet bundle

Definition

Let $\pi \colon E \to M$ be a smooth fiber bundle over a smooth manifold $M$, and let $k \ge 0$ be an integer. A local section of $\pi$ is a smooth map $s \colon U \to E$, with $U \subseteq M$ open, such that $\pi \circ s = \id_U$. For $p \in U$, let $j^k_p s$ be the $k$-jet of $s$ at $p$, viewed as a map from $M$ to $E$. The $k$-th jet manifold of $\pi$ is the set $$J^k \pi = \set{ j^k_p s : p \in M,\ s \text{ a local section of } \pi \text{ defined near } p }.$$ It comes with the maps $\pi_k \colon J^k \pi \to M$, $j^k_p s \mapsto p$, and $\pi_{k,0} \colon J^k\pi \to E$, $j^k_p s \mapsto s(p)$.

Let $(x^1, \dots, x^n, u^1, \dots, u^m)$ be adapted coordinates on an open set of $E$, so that $x^i$ are coordinates on $M$ and $u^\alpha$ are fiber coordinates. They induce coordinates $(x^i, u^\alpha, u^\alpha_I)$ on $J^k\pi$, where $I$ runs over multi-indices with $1 \le \abs{I} \le k$ and $$u^\alpha_I(j^k_p s) = \frac{\partial^{\abs{I}} s^\alpha}{\partial x^I}(p).$$ These charts make $J^k\pi$ a smooth manifold. With this structure, $\pi_k \colon J^k\pi \to M$ is called the $k$-th jet bundle of $\pi$.