# Smooth bundles and jets

Nothing we have done so far has required the spaces of a fiber bundle to be manifolds; if they are, then we require the bundle projections $${\pi}$$ to be (infinitely) differentiable and $${\pi^{-1}(x)}$$ to be diffeomorphic to $${F}$$, resulting in a smooth bundle. A smooth G-bundle then has a structure group $${G}$$ which is a Lie group, and whose elements correspond to diffeomorphisms of $${F}$$.

If we consider a local section $${\sigma}$$ of a smooth fiber bundle $${(E,M,\pi,F)}$$ with $${\sigma(x)=p}$$, the equivalence class of all local sections that have both $${\sigma(x)=p}$$ and also the same tangent space $${T_{p}\sigma}$$ is called the jet $${j_{p}\sigma}$$ with representative $${\sigma}$$. We can also require that further derivatives of the section match the representative, in which case the order of matching derivatives defines the order of the jet, which is also called a k-jet so that the above definition would be that of a 1-jet. $${x}$$ is called the source of the jet and $${p}$$ is called its target. With some work to transition between local sections, one can then form a jet manifold by considering jets with all sources and representative sections, which becomes a jet bundle by considering jets to be fibers over their source.

The above depicts a jet with representative $${\sigma}$$, source $${x}$$, and target $${p}$$.