# 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.^{[1]} 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 *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*defined by

_{r,k}From this definition, it is clear that *π _{r}* =

*π*o

*π*

_{r,0}and that if 0 ≤

*m*≤

*k*, then

*π*=

_{r,m}*π*o

_{k,m}*π*. It is conventional to regard

_{r,k}*π*as the identity map on

_{r,r}*J*(

^{r}*π*) and to identify

*J*

^{0}(

*π*) with

*E*.

A coordinate system on *E* will generate a coordinate system on *J ^{r}*(

*π*). Let (

*U*,

*u*) be an adapted coordinate chart on

*E*, where

*u*= (

*x*,

^{i}*u*). The

^{α}**induced coordinate chart (**on

*U*,^{r}*u*)^{r}*J*(

^{r}*π*) is defined by

Given an atlas of adapted charts (*U*, *u*) on *E*, the corresponding collection of charts (*U ^{r}*,

*u*) is a finite-dimensional

^{r}*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 *(x ^{i}, u)* are coordinates on

*M*×

**R**, where

*u*= id

_{R}is the identity coordinate, then the derivative coordinates

*u*on

_{i}*J*correspond to the coordinates ∂

^{1}(π)_{i}on

*T*M*.

The space *J ^{r}*(π) carries a natural distribution, that is, a sub-bundle of the tangent bundle

*TJ*(π)), called the

^{r}*Cartan distribution*. The Cartan distribution is spanned by all tangent planes to graphs of holonomic sections; that is, sections of the form

*j*for

^{r}φ*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* ≃ **R**^{2} and *M* ≃ **R**. Then, *(J ^{1}(π), π, M)* defines the first jet bundle, and may be coordinated by

*(x, u, u*, where

_{1})for all *p* ∈ *M* and σ in Γ_{p}(π). A general 1-form on *J ^{1}(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, u _{1})θ_{0}* must necessarily be a multiple of the basic contact form θ

_{0}=

*du*−

*u*. Proceeding to the second jet space

_{1}dx*J*with additional coordinate

^{2}(π)*u*, such that

_{2}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 *J ^{r+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 *J ^{r+1}(π)* can be written as a linear combination

A vector field is called **vertical**, meaning that all the horizontal coefficients vanish, if *ρ ^{i}* = 0.

having coordinates *(x, u, ρ ^{i}, φ^{α})*, with an element in the fiber

*T*of

_{xu}E*TE*over

*(x, u)*in

*E*, called

**a tangent vector in**. A section

*TE*Let π be the trivial bundle (**R**^{2} × **R**, pr_{1}, **R**^{2}) with global coordinates (*x*^{1}, *x*^{2}, *u*^{1}). Then the map *F* : *J*^{1}(π) → **R** defined by

A local diffeomorphism *ψ* : *J ^{r}*(

*π*) →

*J*(

^{r}*π*) defines a contact transformation of order

*r*if it preserves the contact ideal, meaning that if θ is any contact form on

*J*(

^{r}*π*), then

*ψ*θ*is also a contact form.

Let us begin with the first order case. Consider a general vector field *V*^{1} on *J*^{1}(*π*), given by

The former requirements provide explicit formulae for the coefficients of the first derivative terms in *V ^{1}*:

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* ≅ **R**^{2} and *M* ≃ **R**. Then, *(J ^{1}(π), π, E)* defines the first jet bundle, and may be coordinated by

*(x, u, u*, where

_{1})Now, *θ* has no *u*_{2} dependency. Hence, from this equation we will pick up the formula for *ρ*, which will necessarily be the same result as we found for *V ^{1}*. Therefore, the problem is analogous to prolonging the vector field

*V*to

^{1}*J*

^{2}(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 *V ^{2}* is

Note that the first prolongation of *V* can be recovered by omitting the second derivative terms in *V ^{2}*, or by projecting back to

*J*.

^{1}(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

(*gr _{f}* is known as the

**graph of the function**) of the trivial bundle (

*f**M*×

*N*, π

_{1},

*M*). However, this restriction does not simplify the theory, as the global triviality of π does not imply the global triviality of π

_{1}.