Jet Bundle - Infinite Jet Spaces

Infinite Jet Spaces

The inverse limit of the sequence of projections gives rise to the infinite jet space J∞(π). A point is the equivalence class of sections of π that have the same k-jet in p as σ for all values of k. The natural projection π_∞ maps into p.

Just by thinking in terms of coordinates, J∞(π) appears to be an infinite-dimensional geometric object. In fact, the simplest way of introducing a differentiable structure on J∞(π), not relying on differentiable charts, is given by the differential calculus over commutative algebras. Dual to the sequence of projections of manifolds is the sequence of injections of commutative algebras. Let's denote simply by . Take now the direct limit of the 's. It will be a commutative algebra, which can be assumed to be the smooth functions algebra over the geometric object J∞(π). Observe that, being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.

Roughly speaking, a concrete element will always belong to some, so it is a smooth function on the finite-dimensional manifold Jk(π) in the usual sense.