Jet Bundle - Jet Manifolds

Jet Manifolds

The r-th jet manifold of π is the set

and is denoted Jr(π). We may define projections π_r and π_r,0 called the source and target projections respectively, by

If 1 ≤ k ≤ r, then the k-jet projection is the function π_r,k defined by

From this definition, it is clear that π_r = π π_r,0 and that if 0 ≤ m ≤ k, then π_r,m = π_k,m π_r,k. It is conventional to regard π_r,r = id_Jr(π), the identity map on Jr(π) and to identify J0(π) with E.

The functions π_r,k, π_r,0 and π_r are smooth surjective submersions.

A coordinate system on E will generate a coordinate system on Jr(π). Let (U, u) be an adapted coordinate chart on E, where u = (xi, uα). The induced coordinate chart (Ur, ur) on Jr(π) is defined by

where

and the functions

are specified by

and are known as the derivative coordinates.

Given an atlas of adapted charts (U, u) on E, the corresponding collection of charts (Ur, ur) is a finite-dimensional C∞ atlas on Jr(π).