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.
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. It is conventional to regard πr,r as the identity map on J r(π) and to identify J 0(π) with E.
Given an atlas of adapted charts (U, u) on E, the corresponding collection of charts (U r, u r) is a finite-dimensional C∞ atlas on J r(π).
is well-defined and is clearly injective. Writing it out in coordinates shows that it is a diffeomorphism, because if (xi, u) are coordinates on M × R, where u = idR is the identity coordinate, then the derivative coordinates ui on J1(π) correspond to the coordinates ∂i on T*M.
The space Jr(π) carries a natural distribution, that is, a sub-bundle of the tangent bundle TJr(π)), called the Cartan distribution. The Cartan distribution is spanned by all tangent planes to graphs of holonomic sections; that is, sections of the form jrφ for φ a section of π.
The Cartan distribution is the main geometrical structure on jet spaces and plays an important role in the geometric theory of partial differential equations. The Cartan distributions are completely non-integrable. In particular, they are not involutive. The dimension of the Cartan distribution grows with the order of the jet space. However, on the space of infinite jets J∞ the Cartan distribution becomes involutive and finite-dimensional: its dimension coincides with the dimension of the base manifold M.
Consider the case (E, π, M), where E ≃ R2 and M ≃ R. Then, (J1(π), π, M) defines the first jet bundle, and may be coordinated by (x, u, u1), where
for all p ∈ M and σ in Γp(π). A general 1-form on J1(π) takes the form
This will vanish for all sections σ if and only if c = 0 and a = −bσ′(x). Hence, θ = b(x, u, u1)θ0 must necessarily be a multiple of the basic contact form θ0 = du − u1dx. Proceeding to the second jet space J2(π) with additional coordinate u2, such that
which implies that e = 0 and a = −bσ′(x) − cσ′′(x). Therefore, θ is a contact form if and only if
In general, providing x, u ∈ R, a contact form on Jr+1(π) can be written as a linear combination of the basic contact forms
Similar arguments lead to a complete characterization of all contact forms.
In local coordinates, every contact one-form on Jr+1(π) can be written as a linear combination
A vector field is called vertical, meaning that all the horizontal coefficients vanish, if ρi = 0.
having coordinates (x, u, ρi, φα), with an element in the fiber TxuE of TE over (x, u) in E, called a tangent vector in TE. A section
Let π be the trivial bundle (R2 × R, pr1, R2) with global coordinates (x1, x2, u1). Then the map F : J1(π) → R defined by
A local diffeomorphism ψ : Jr(π) → Jr(π) defines a contact transformation of order r if it preserves the contact ideal, meaning that if θ is any contact form on Jr(π), then ψ*θ is also a contact form.
Let us begin with the first order case. Consider a general vector field V1 on J1(π), given by
The former requirements provide explicit formulae for the coefficients of the first derivative terms in V1:
These results are best understood when applied to a particular example. Hence, let us examine the following.
Consider the case (E, π, M), where E ≅ R2 and M ≃ R. Then, (J1(π), π, E) defines the first jet bundle, and may be coordinated by (x, u, u1), where
Now, θ has no u2 dependency. Hence, from this equation we will pick up the formula for ρ, which will necessarily be the same result as we found for V1. Therefore, the problem is analogous to prolonging the vector field V1 to J2(π). That is to say, we may generate the r-th prolongation of a vector field by recursively applying the Lie derivative of the contact forms with respect to the prolonged vector fields, r times. So, we have
Therefore, the Lie derivative of the second contact form with respect to V2 is
Note that the first prolongation of V can be recovered by omitting the second derivative terms in V2, or by projecting back to J1(π).
This article has defined jets of local sections of a bundle, but it is possible to define jets of functions f: M → N, where M and N are manifolds; the jet of f then just corresponds to the jet of the section
(grf is known as the graph of the function f) of the trivial bundle (M × N, π1, M). However, this restriction does not simplify the theory, as the global triviality of π does not imply the global triviality of π1.