Fiber of jet bundle of a fiber bundle

Question

Given a fiber bundle $p:E\to B$ with fiber $V$ and structure group $G$, one can define the corresponding $k$-jet bundle $E^k\subset J^k(B,E)$ of jets of local sections of $E$. On Wikipedia there is a hint of such a construction (http://en.wikipedia.org/wiki/Jet_bundle) but nowhere have I found literature which defines the fiber of this bundle explicitly, and with fiber I mean the analog of $V$ i.e. just like in Steenrod's "Topology of fiber bundles". To clarify once more: if $E^k$ looks locally like $B\times F$, then what is F exactly? I'm grateful for any answers/comments/hints/help.

Answer 1

A good motivating discussion is given in Eliashberg and Mishachev's Introduction to the h-principle as mentioned by Dan Ramras.

A more detailed description can be found in Kolar, Michor, and Slovak's Natural operations in differential geometry (which by the way, has a legally available electronic version). Using the local trivialisation of the fibre bundle you have that $E^k|_x$ for $x\in B$ can be identified with $J^k_x(B,V)$. One particular characterisation of this fibre is that $J^k_x(B,V) = \cup_{y\in V} J^k_x(B,V)_y$ and $J^k_x(B,V)_y$ is canonically identified with $Hom(T^{k*}_yV,T^{k*}_xB)$ where $T^{r*}_zZ = J^r_z(Z,\mathbb{R})_0$ is the set of $r$-covelocities.

Another way to get a picture of the jet bundle maybe through the following:

Thm The natural projection $\pi^r_{r-1}: J^r(B,V) \to J^{r-1}(B,V)$ is an affine bundle modelled on the pullback vector bundle $(\pi^{r-1}_0)^{-1}\left( TV\otimes S^rT^*B\right) \to J^{r-1}(B,V)$ where $S^r$ denotes the $r$-fold symmetric tensor product.

So the fibres of the jet bundle is something like a tower of affine bundles on top of each other.

A lot of this is also developed in Hubert Goldschmidt's 1967 JDG paper "Integrability criteria for systems of non-linear partial differential equations". The paper is based on his PhD work and is exceptionally clearly written.