Jet Bundles
Since the atlas on each Jr(π) defines a manifold, the triples (Jr(π), πr,k, Jk(π)), (Jr(π), πr,0, E) and (Jr(π), πr, M) all define fibered manifolds. In particular, if (E, π, M) is a fiber bundle, the triple (Jr(π), πr, 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 πr jrσ = idW, so jrσ really is a section. In local coordinates, jrσ is given by
We identify j0σ with σ.
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