Suppose we are given an open cover $\mathcal{U}=(U_{i})_{i \in I}$ of a smooth manifold $M$, a cocycle of smooth transition functions $g_{ij}: U_{ij} \to G$ where $G$ is a Lie group, and a (not necessarily faithful) action $\lambda: G \times S \to S$ on a smooth manifold $S$. Then this is the data of a fibre bundle $E \to M$ with typical fibre $S$ and with structure group $G$ over $M$. From this cocycle point of view, how can we describe the jet bundle $J^r E \to M$?
2 Answers
You start with the trivializations $S \times U_i \to U_i$ over your cover. Let me presume that you can take for granted the construction of the jet bundles $J^r(S \times U_i) \to U_i$, whose typical fiber I'll take to be $S^r$. The group action $(\lambda,\mathrm{id}) \colon G \times (S\times U_i) \to S \times U_i$ has the jet extension $$ (\lambda^r,\mathrm{id}) = j^r(\lambda,\mathrm{id}) \colon (G \times S^r) \times U_i = G \times J^r(S\times U_i) \to S^r \times U_i = J^r(S \times U_i) . $$ You already have your $G$cocycle on $M$, so you need only use the typical fiber $S^r$ and the action $\lambda^r \colon G \times S^r \to S^r$, instead of $S$ and $\lambda$, to get $J^rE \to M$.

$\begingroup$ I am afraid that this is not correct in general. The fact that a bundle $E$ is associated to a principal $G$bundle $P$ does not imply that the $k$th jet prolongation is associated to $P$. Just think of the case where $P$ is the first order frame bundle of $M$, and $E$ is any natural vector bundle. If $J^kE$ were associated to $P$ again, then any diffeomorphsim of $M$, which coincides with the identity to first order in a point $x\in M$ would act trivially on the fiber of $J^kE$ over $x$. $\endgroup$ Oct 20, 2016 at 8:33

$\begingroup$ You are right that the construction above doesn't give the right transition functions for higher order frame bundles, if one wants to construct them as jet prolongations of the first order frame bundle, when the latter is considered as a $GL(n)$ principalbundle. But I would also argue that it is not the right way to get the higher order frame bundle. In the intermediate bundles $S\times U_i$ above, implicitly, diffeomorphisms of the base act trivially on the $S$fibers. This is already not true for the frame bundle. $\endgroup$ Oct 20, 2016 at 10:12

$\begingroup$ I see, so you get the right bundle but not in a functorial way, right? $\endgroup$ Oct 21, 2016 at 6:59

$\begingroup$ I'm a bit confused right now. How would you write the cocycles, Andreas Cap? I'd be happy to upvote if you wouldn't mind posting an answer as well. $\endgroup$– ಠ_ಠOct 27, 2016 at 7:19

$\begingroup$ @AndreasCap < this is the right way to ping someone. $\endgroup$ Oct 27, 2016 at 8:45
I would not really call this an answer but rather an extended comment. I think that the answer by @IgorKhavkine is correct but at the same time misleading in a certain sense. The point about this is that if $E$ is a fiber bundle with fiber $S$ and structure group $G$, then the jet prolongation $J^kE$ does not naturally have structure group $E$. In contrast to my comment, I think that if $E=P\times_G S$, then $J^kE$ can be realized as an associated bundle to $P$ but this realization is not canonical. Otherwise put, if in the construction in Igor's answer, you start from an equivalent cocycle the result will not be an equivalent cocycle. The resulting bundles (i.e.~candidates for $J^kE$ will be isomorphic as fiber bundles but not as fiber bundles with structure group $G$).
If you want to do things in a natural way, the whole procedure becomes a bit tedious. There are concepts of jet prolongations of principal bundles, so to a principal bundle $P$ you can associate a $k$th jet prolongation $P^{(k)}$, which has a larger structure group (basically the group of $k$jets in a point of principal bundle isomorphisms $\mathbb R^n\times G\to\mathbb R^n\times G$ covering the identity, where $n$ is the dimension of the base). If $E$ is associated to $P$ then $J^kE$ is naturally associated to $P^{(k)}$. The corresponding transition functions basically can be viewed as the $k$jets of the initial transition functions, and this construction is compatible (more or less by construction) with equivalence of cocycles. Carrying these things out in detail unfortunately is quite tedious, a detailed discussion can be found in Chapter XII of the book on natural transformations in differential geometry by I. Kolar, P. Michor and J. Slovak, which is available via Peter Michor's homepage at http://www.mat.univie.ac.at/~michor/listpubl.html#books .