Jet bundle

In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Jets may also be seen as the coordinate free versions of Taylor expansions. Historically, jet bundles are attributed to Charles Ehresmann, and were an advance on the method (prolongation) of Élie Cartan, of dealing geometrically with higher derivatives, by imposing differential form conditions on newly introduced formal variables. Jet bundles are sometimes called sprays, although sprays usually refer more specifically to the associated vector field induced on the corresponding bundle (e.g., the geodesic spray on Finsler manifolds.) Since the early 1980s, jet bundles have appeared as a concise way to describe phenomena associated with the derivatives of maps, particularly those associated with the calculus of variations. Consequently, the jet bundle is now recognized as the correct domain for a geometrical covariant field theory and much work is done in general relativistic formulations of fields using this approach. Jet (mathematics) Suppose M is an m-dimensional manifold and that (E, π, M) is a fiber bundle. For p ∈ M, let Γ(p) denote the set of all local sections whose domain contains p. Let I = (I(1), I(2), ..., I(m)) be a multi-index (an m-tuple of non-negative integers, not necessarily in ascending order), then define: Define the local sections σ, η ∈ Γ(p) to have the same r-jet at p if The relation that two maps have the same r-jet is an equivalence relation. An r-jet is an equivalence class under this relation, and the r-jet with representative σ is denoted . The integer r is also called the order of the jet, p is its source and σ(p) is its target. The r-th jet manifold of π is the set We may define projections πr and πr,0 called the source and target projections respectively, by If 1 ≤ k ≤ r, then the k-jet projection is the function πr,k defined by From this definition, it is clear that πr = π o πr,0 and that if 0 ≤ m ≤ k, then πr,m = πk,m o πr,k.