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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.

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 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

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 *π _{r}* and

If 1 ≤ *k* ≤ *r*, then the ** k-jet projection** is the function

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

The functions *π _{r,k}*,

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

where

and the functions known as the **derivative coordinates**:

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

Since the atlas on each defines a manifold, the triples * *, * * and * * all define fibered manifolds. In particular, if * *is a fiber bundle, the triple * * defines the ** r-th jet bundle of π**.

If *W* ⊂ *M* is an open submanifold, then

If *p* ∈ *M*, then the fiber is denoted .

Let σ be a local section of π with domain *W* ⊂ *M*. The ** r-th jet prolongation of σ** is the map defined by

Note that , so really is a section. In local coordinates, is given by

We identify * * with .

An independently motivated construction of the sheaf of sections * is given*.

Consider a diagonal map , where the smooth manifold is a locally ringed space by for each open . Let be the ideal sheaf of , equivalently let be the sheaf of smooth germs which vanish on for all . The pullback of the quotient sheaf from to by is the sheaf of k-jets.^{[2]}

The direct limit of the sequence of injections given by the canonical inclusions of sheaves, gives rise to the **infinite jet sheaf** . Observe that by the direct limit construction it is a filtered ring.

If π is the trivial bundle (*M* × **R**, pr_{1}, *M*), then there is a canonical diffeomorphism between the first jet bundle and *T*M* × **R**. To construct this diffeomorphism, for each σ in write .

Then, whenever *p* ∈ *M*

Consequently, the mapping

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

Likewise, if π is the trivial bundle (**R** × *M*, pr_{1}, **R**), then there exists a canonical diffeomorphism between and **R** × *TM*.

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

The annihilator of the Cartan distribution is a space of differential one-forms called contact forms, on *J ^{r}*(π). The space of differential one-forms on

for all local sections σ of π over *M*.

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

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

for all *p* ∈ *M* and σ in Γ_{p}(π). A general 1-form on *J ^{1}(π)* takes the form

A section σ in Γ_{p}(π) has first prolongation

Hence, *(j ^{1}σ)*θ* can be calculated as

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

a general 1-form has the construction

This is a contact form if and only if

which implies that *e* = 0 and *a* = −*bσ′(x)* − *cσ′′(x)*. Therefore, θ is a contact form if and only if

where θ_{1} = *du*_{1} − *u*_{2}*dx* is the next basic contact form (Note that here we are identifying the form θ_{0} with its pull-back to *J ^{2}(π)*).

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

where

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

with smooth coefficients of the basic contact forms

*|I|* is known as the **order** of the contact form . Note that contact forms on *J ^{r+1}(π)* have orders at most

Let ψ ∈ Γ_{W}(*π _{r+1}*), then

A general vector field on the total space *E*, coordinated by , is

A vector field is called **horizontal**, meaning that all the vertical coefficients vanish, if = 0.

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

For fixed *(x, u)*, we identify

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

is called **a vector field on E** with

and ψ in *Γ(TE)*.

The jet bundle *J ^{r}(π)* is coordinated by . For fixed

having coordinates

with an element in the fiber of *TJ ^{r}(π)* over

are real-valued functions on *J ^{r}(π)*. A section

is **a vector field on J^{r}(π)**, and we say

Let *(E, π, M)* be a fiber bundle. An ** r-th order partial differential equation** on π is a closed embedded submanifold

Consider an example of a first order partial differential equation.

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

gives rise to the differential equation

which can be written

The particular

has first prolongation given by

and is a solution of this differential equation, because

and so for *every* *p* ∈ **R**^{2}.

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

The flow generated by a vector field *V ^{r}* on the jet space

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

We now apply to the basic contact forms and expand the exterior derivative of the functions in terms of their coordinates to obtain:

Therefore, *V ^{1}* determines a contact transformation if and only if the coefficients of

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

where

denotes the zeroth order truncation of the total derivative *D _{i}*.

Thus, the contact conditions uniquely prescribe the prolongation of any point or contact vector field. That is, if satisfies these equations, *V ^{r}* is called the

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

for all *p* ∈ *M* and *σ* in Γ_{p}(*π*). A contact form on *J ^{1}(π)* has the form

Consider a vector *V* on *E*, having the form

Then, the first prolongation of this vector field to *J ^{1}(π)* is

If we now take the Lie derivative of the contact form with respect to this prolonged vector field, we obtain

Hence, for preservation of the contact ideal, we require

And so the first prolongation of *V* to a vector field on *J ^{1}(π)* is

Let us also calculate the second prolongation of *V* to a vector field on *J ^{2}(π)*. We have as coordinates on

The contact forms are

To preserve the contact ideal, we require

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

and so

Therefore, the Lie derivative of the second contact form with respect to *V ^{2}* is

Hence, for to preserve the contact ideal, we require

And so the second prolongation of *V* to a vector field on *J*^{2}(π) 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

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

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

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

Given a *k*-th order system of PDEs *E* ⊆ *J ^{k}(π)*, the collection

Enhance *I(E)* by adding all the possible compositions of total derivatives applied to all its elements. This way we get a new ideal *I* of which is now closed under the operation of taking total derivative. The submanifold *E*_{(∞)} of *J*^{∞}(π) cut out by *I* is called the **infinite prolongation** of *E*.

Geometrically, *E*_{(∞)} is the manifold of **formal solutions** of *E*. A point of *E*_{(∞)} can be easily seen to be represented by a section σ whose *k*-jet's graph is tangent to *E* at the point with arbitrarily high order of tangency.

Analytically, if *E* is given by φ = 0, a formal solution can be understood as the set of Taylor coefficients of a section σ in a point *p* that make vanish the Taylor series of at the point *p*.

Most importantly, the closure properties of *I* imply that *E*_{(∞)} is tangent to the **infinite-order contact structure** on *J ^{∞}(π)*, so that by restricting to

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}: M*M*×*N**gr*=_{f}(p)*(p, f(p))*

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

- Ehresmann, C., "Introduction à la théorie des structures infinitésimales et des pseudo-groupes de Lie."
*Geometrie Differentielle,*Colloq. Inter. du Centre Nat. de la Recherche Scientifique, Strasbourg, 1953, 97-127. - Kolář, I., Michor, P., Slovák, J.,
*Natural operations in differential geometry.*Springer-Verlag: Berlin Heidelberg, 1993. ISBN 3-540-56235-4, ISBN 0-387-56235-4. - Saunders, D. J., "The Geometry of Jet Bundles", Cambridge University Press, 1989, ISBN 0-521-36948-7
- Krasil'shchik, I. S., Vinogradov, A. M., [et al.], "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, ISBN 0-8218-0958-X.
- Olver, P. J., "Equivalence, Invariants and Symmetry", Cambridge University Press, 1995, ISBN 0-521-47811-1