Jet Bundle - Jet Bundles

Jet Bundles

Since the atlas on each Jr(π) defines a manifold, the triples (Jr(π), π_r,k, Jk(π)), (Jr(π), π_r,0, E) and (Jr(π), π_r, M) all define fibered manifolds. In particular, if (E, π, M) is a fiber bundle, the triple (Jr(π), π_r, M) defines the r-th jet bundle of π.

If W ⊂ M is an open submanifold, then

If p ∈ M, then the fiber is denoted .

Let σ be a local section of π with domain W ⊂ M. The r-th jet prolongation of σ is the map jrσ: W → Jr(π) defined by

Note that π_r jrσ = id_W, so jrσ really is a section. In local coordinates, jrσ is given by

We identify j0σ with σ.