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.
Jets
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 ) ) {\displaystyle 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:
| I | := ∑ i = 1 m I ( i ) ∂ | I | ∂ x I := ∏ i = 1 m ( ∂ ∂ x i ) I ( i ) . {\displaystyle {\begin{aligned}|I|&:=\sum _{i=1}^{m}I(i)\\{\frac {\partial ^{|I|}}{\partial x^{I}}}&:=\prod _{i=1}^{m}\left({\frac {\partial }{\partial x^{i}}}\right)^{I(i)}.\end{aligned}}}
Define the local sections σ, η ∈ Γ(p) to have the same r-jet at p if
∂ | I | σ α ∂ x I | p = ∂ | I | η α ∂ x I | p , 0 ≤ | I | ≤ r . {\displaystyle \left.{\frac {\partial ^{|I|}\sigma ^{\alpha }}{\partial x^{I}}}\right|_{p}=\left.{\frac {\partial ^{|I|}\eta ^{\alpha }}{\partial x^{I}}}\right|_{p},\quad 0\leq |I|\leq r.}
… excerpt ends here. Continue reading the full article.
