Editing Jet bundle

{{Short description|Construction in differential topology}}
{{Redirect-distinguish|Jet space|Space jet (disambiguation){{!}}space jet}}
{{technical|date=May 2025}}In [[differential topology]], the '''jet bundle''' is a certain construction that makes a new [[smooth manifold|smooth]] [[fiber bundle]] out of a given smooth fiber bundle. It makes it possible to write [[differential equation]]s on [[Fiber bundle#Sections|section]]s of a fiber bundle in an invariant form. [[Jet (mathematics)|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 ([[Cartan's equivalence method|prolongation]]) of [[Élie Cartan]], of dealing ''geometrically'' with [[derivative|higher derivatives]], by imposing [[differential form]] conditions on newly introduced formal variables.  Jet bundles are sometimes called '''sprays''', although [[spray (mathematics)|sprays]] usually refer more specifically to the associated [[vector field]] induced on the corresponding bundle (e.g., the [[geodesic spray]] on [[Finsler manifold]]s.)

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]].<ref>{{cite book
 | last = Krupka
 | first = Demeter
 |author-link= Demeter Krupka
 | title = Introduction to Global Variational Geometry
 | year=  2015
 | publisher = Atlantis Press
 | isbn = 978-94-6239-073-7
 |url= https://www.springer.com/it/book/9789462390720
 }}</ref> Consequently, the jet bundle is now recognized as the correct domain for a [[covariant classical field theory|geometrical covariant field theory]] and much work is done in [[general relativity|general relativistic]] formulations of fields using this approach.

==Jets==
{{main|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 {{tmath|1=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:

:<math>\begin{align}
                                   |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{align}</math>

Define the local sections σ, η ∈ Γ(p) to have the same '''''r''-jet''' at ''p'' if

:<math>
  \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.
</math>

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 <math>j^r_p\sigma</math>. The integer ''r'' is also called the '''order''' of the jet,  ''p'' is its '''source''' and σ(''p'') is its '''target'''.

==Jet manifolds==
The '''''r''-th jet manifold of π''' is the set

:<math>J^r (\pi) = \left \{j^r_p\sigma:p \in M, \sigma \in \Gamma(p) \right \}.</math>

We may define projections ''π<sub>r</sub>'' and ''π''<sub>''r'',0</sub> called the '''source and target projections''' respectively, by

:<math>\begin{cases}
  \pi_r: J^r(\pi) \to M \\
  j^r_p\sigma \mapsto p
\end{cases}, \qquad \begin{cases}
  \pi_{r, 0}: J^r(\pi) \to E \\
  j^r_p\sigma \mapsto \sigma(p)
\end{cases}</math>

If 1 ≤ ''k'' ≤ ''r'', then the '''''k''-jet projection''' is the function ''π<sub>r,k</sub>'' defined by

<math display=block>\begin{cases}
  \pi_{r, k}: J^r(\pi) \to J^{k}(\pi) \\
           j^r_p\sigma \mapsto j^{k}_p\sigma
\end{cases}</math>

From this definition, it is clear that ''π<sub>r</sub>'' = ''π'' <small> o </small> ''π''<sub>''r'',0</sub> and that if 0 ≤ ''m'' ≤ ''k'', then ''π<sub>r,m</sub>'' = ''π<sub>k,m</sub>'' <small> o </small> ''π<sub>r,k</sub>''. It is conventional to regard ''π<sub>r,r</sub>'' as the [[identity function|identity map]] on ''J&thinsp;<sup>r</sup>''(''π'') and to identify ''J''&thinsp;<sup>0</sup>(''π'') with ''E''.

The functions ''π<sub>r,k</sub>'', ''π''<sub>''r'',0</sub> and ''π<sub>r</sub>'' are [[Smooth function|smooth]] [[surjective]] [[submersion (mathematics)|submersion]]s.

[[File:Jet Bundle Image FbN.png|500px|center]]

A [[coordinate system]] on ''E'' will generate a coordinate system on ''J&thinsp;<sup>r</sup>''(''π''). Let (''U'', ''u'') be an adapted [[coordinate chart]] on ''E'', where ''u'' = (''x<sup>i</sup>'', ''u<sup>α</sup>''). The '''induced coordinate chart (''U<sup>r</sup>'', ''u<sup>r</sup>'')''' on ''J&thinsp;<sup>r</sup>''(''π'') is defined by

<math display=block>\begin{align}
  U^r &= \left\{j^r_p \sigma: p \in M, \sigma(p) \in U\right\} \\
  u^r &= \left(x^i, u^\alpha, u^\alpha_I\right)
\end{align}</math>

where

<math display=block>\begin{align}
       x^i\left(j^r_p\sigma\right) &= x^i(p) \\
  u^\alpha\left(j^r_p\sigma\right) &= u^\alpha(\sigma(p))
\end{align}</math>

and the <math>n \left(\binom{m+r}{r} - 1\right)</math> functions known as the '''derivative coordinates''':

<math display=block>\begin{cases}
  u^\alpha_I:U^k \to \mathbf{R} \\
  u^\alpha_I\left(j^r_p\sigma\right) = \left.\frac{\partial^{|I|} \sigma^\alpha}{\partial x^I}\right|_p
\end{cases}</math>

Given an atlas of adapted charts (''U'', ''u'') on ''E'', the corresponding collection of charts (''U&thinsp;<sup>r</sup>'', ''u&thinsp;<sup>r</sup>'') is a [[finite-dimensional]] ''C''<sup>∞</sup> atlas on ''J&thinsp;<sup>r</sup>''(''π'').

==Jet bundles==
Since the atlas on each <math>J^r(\pi)</math> defines a manifold, the triples ''<math>(J^r(\pi), \pi_{r,k}, J^k(\pi))</math>'', ''<math>(J^r(\pi), \pi_{r,0}, E)</math>'' and ''<math>(J^r(\pi), \pi_{r}, M)</math>'' all define fibered manifolds. In particular, if ''<math>(E, \pi, M)</math>''is a fiber bundle, the triple  ''<math>(J^r(\pi), \pi_{r}, M)</math>'' defines the '''''r''-th jet bundle of π'''.

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

:<math> J^r \left(\pi|_{\pi^{-1}(W)}\right) \cong \pi^{-1}_r(W).\,</math>

If ''p'' ∈ ''M'', then the fiber <math>\pi^{-1}_r(p)\,</math> is denoted <math>J^r_p(\pi)</math>.

Let σ be a local section of π with domain ''W'' ⊂ ''M''. The '''''r''-th jet prolongation of σ''' is the map <math>j^r\sigma: W \rightarrow J^r(\pi)</math> defined by

:<math> (j^r \sigma)(p) = j^r_p \sigma. \,</math>

Note that <math>\pi_r \circ j^r \sigma =\mathbb{id}_W</math>, so <math>j^r\sigma</math> really is a section. In local coordinates,  <math>j^r\sigma</math> is given by

:<math> \left(\sigma^\alpha, \frac{\partial^{|I|} \sigma^\alpha}{\partial x^{I}}\right) \qquad 1 \leq |I| \leq r. \,</math>

We identify  ''<math>j^ 0\sigma</math>''  with  <math>\sigma</math> .

=== Algebro-geometric perspective ===
An independently motivated construction of the sheaf of sections <math>\Gamma J^k\left(\pi_{TM}\right)</math>'' is given''.''

Consider a diagonal map <math display="inline">\Delta_n: M \to \prod_{i=1}^{n+1} M</math>, where the smooth manifold <math>M</math> is a [[locally ringed space]] by <math>C^k(U)</math> for each open <math>U</math>. Let <math>\mathcal{I}</math> be the [[ideal sheaf]] of <math>\Delta_n(M)</math>, equivalently let <math>\mathcal{I}</math> be the [[Sheaf (mathematics)|sheaf]] of smooth [[Germ (mathematics)|germs]] which vanish on <math>\Delta_n(M)</math> for all <math>0 < n \leq k</math>.  The [[Inverse image functor|pullback]] of the [[quotient sheaf]] <math>{\Delta_n}^*\left(\mathcal{I}/\mathcal{I}^{n+1}\right)</math> from <math display="inline">\prod_{i=1}^{n+1} M</math> to <math>M</math> by <math>\Delta_n</math> is the sheaf of k-jets.<ref>{{Cite web|url=http://math.stanford.edu/~vakil/files/jets.pdf|title=A beginner's guide to jet bundles from the point of view of algebraic geometry|last=Vakil|first=Ravi|date=August 25, 1998|access-date=June 25, 2017}}</ref>

The [[direct limit]] of the sequence of injections given by the canonical inclusions <math>\mathcal{I}^{n+1} \hookrightarrow \mathcal{I}^n</math> of sheaves, gives rise to the '''infinite jet sheaf''' <math>\mathcal{J}^\infty(TM)</math>. Observe that by the direct limit construction it is a filtered ring.

===Example===
If π is the [[trivial bundle]] (''M'' × '''R''', pr<sub>1</sub>, ''M''), then there is a canonical [[diffeomorphism]] between the first jet bundle <math>J^1(\pi)</math> and ''T*M'' × '''R'''. To construct this diffeomorphism, for each σ in <math>\Gamma_M(\pi)</math> write <math>\bar{\sigma} = pr_2 \circ \sigma \in C^\infty(M)\,</math>.

Then, whenever ''p'' ∈ ''M''

:<math>j^1_p \sigma = \left\{ \psi : \psi \in \Gamma_p (\pi); \bar{\psi}(p) = \bar{\sigma}(p); d\bar{\psi}_p = d\bar{\sigma}_p \right\}. \,</math>

Consequently, the mapping

:<math>\begin{cases}
  J^1(\pi) \to T^*M \times \mathbf{R} \\
  j^1_p\sigma \mapsto \left(d\bar{\sigma}_p, \bar{\sigma}(p)\right)
\end{cases}</math>

is well-defined and is clearly [[injective]]. Writing it out in coordinates shows that it is a diffeomorphism, because if ''(x<sup>i</sup>, u)'' are coordinates on ''M'' × '''R''', where ''u'' = id<sub>'''R'''</sub> is the identity coordinate, then the derivative coordinates ''u<sub>i</sub>'' on ''J<sup>1</sup>(π)'' correspond to the coordinates ∂<sub>''i''</sub> on ''T*M''.

Likewise, if π is the trivial bundle ('''R''' × ''M'', pr<sub>1</sub>, '''R'''), then there exists a canonical diffeomorphism between <math>J^1(\pi)</math>and '''R''' × ''TM''.

==Contact structure==
The space ''J<sup>r</sup>''(π) carries a natural [[distribution (differential geometry)|distribution]], that is, a sub-bundle of the [[tangent bundle]] ''TJ<sup>r</sup>''(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 ''j<sup>r</sup>φ'' for ''φ'' a section of π.

The annihilator of the Cartan distribution is a space of [[one-form|differential one-forms]] called [[contact form]]s, on  ''J<sup>r</sup>''(π).  The space of differential one-forms on ''J<sup>r</sup>''(π) is denoted by <math>\Lambda^1J^r(\pi)</math> and the space of contact forms is denoted by <math>\Lambda_C^r\pi</math>.  A one form is a contact form provided its [[pullback (differential geometry)|pullback]] along every prolongation is zero.  In other words, <math>\theta\in\Lambda^1J^r\pi</math> is a contact form if and only if
:<math>\left(j^{r+1}\sigma\right)^*\theta = 0</math>
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 equation]]s. The Cartan distributions are completely non-integrable.  In particular, they are not [[distribution (differential geometry)|involutive]].  The dimension of the Cartan distribution grows with the order of the jet space.  However, on the space of infinite jets ''J<sup>∞</sup>'' the Cartan distribution becomes involutive and finite-dimensional: its dimension coincides with the dimension of the base manifold ''M''.

===Example===
Consider the case ''(E, π, M)'', where ''E'' ≃ '''R'''<sup>2</sup> and ''M'' ≃ '''R'''. Then, ''(J<sup>1</sup>(π), π, M)'' defines the first jet bundle, and may be coordinated by ''(x, u, u<sub>1</sub>)'', where

:<math>\begin{align}
    x\left(j^1_p\sigma\right) &= x(p) = x \\
    u\left(j^1_p\sigma\right) &= u(\sigma(p)) = u(\sigma(x)) = \sigma(x) \\
  u_1\left(j^1_p\sigma\right) &= \left.\frac{\partial \sigma}{\partial x}\right|_p = \sigma'(x)
\end{align}</math>

for all ''p'' ∈ ''M'' and σ in Γ<sub>''p''</sub>(π). A general 1-form on ''J<sup>1</sup>(π)'' takes the form

:<math>\theta = a(x, u, u_1)dx + b(x, u, u_1)du + c(x, u, u_1)du_1\,</math>

A section σ in Γ<sub>''p''</sub>(π) has first prolongation

:<math>j^1\sigma = (u, u_1) = \left(\sigma(p), \left. \frac{\partial \sigma}{\partial x} \right|_p \right).</math>

Hence, ''(j<sup>1</sup>σ)*θ'' can be calculated as

:<math>\begin{align}
  \left(j^1_p\sigma\right)^* \theta
    &= \theta \circ j^1_p\sigma \\
    &= a(x, \sigma(x), \sigma'(x))dx + b(x, \sigma(x), \sigma'(x))d(\sigma(x)) + c(x, \sigma(x),\sigma'(x))d(\sigma'(x)) \\
    &= a(x, \sigma(x), \sigma'(x))dx + b(x, \sigma(x), \sigma'(x))\sigma'(x)dx + c(x, \sigma(x), \sigma'(x))\sigma''(x)dx \\
    &= [a(x, \sigma(x), \sigma'(x)) + b(x, \sigma(x), \sigma'(x))\sigma'(x) + c(x, \sigma(x), \sigma'(x))\sigma''(x) ]dx 
\end{align}</math>

This will vanish for all sections σ if and only if ''c'' = 0 and ''a'' = −''bσ′(x)''. Hence, θ = ''b(x, u, u<sub>1</sub>)θ<sub>0</sub>'' must necessarily be a multiple of the basic contact form θ<sub>0</sub> = ''du'' − ''u<sub>1</sub>dx''. Proceeding to the second jet space ''J<sup>2</sup>(π)'' with additional coordinate ''u<sub>2</sub>'', such that

:<math>u_2(j^2_p\sigma) = \left.\frac{\partial^2 \sigma}{\partial x^2}\right|_p = \sigma''(x)\,</math>

a general 1-form has the construction

:<math>\theta = a(x, u, u_1,u_2)dx + b(x, u, u_1,u_2)du + c(x, u, u_1,u_2)du_1 + e(x, u, u_1,u_2)du_2\,</math>

This is a contact form if and only if

:<math>\begin{align}
  \left(j^2_p\sigma\right)^* \theta
    &= \theta \circ j^2_p\sigma \\
    &= a(x, \sigma(x), \sigma'(x), \sigma''(x))dx + b(x, \sigma(x), \sigma'(x),\sigma''(x))d(\sigma(x)) +{} \\
    &\qquad\qquad c(x, \sigma(x), \sigma'(x),\sigma''(x))d(\sigma'(x)) + e(x, \sigma(x), \sigma'(x),\sigma''(x))d(\sigma''(x)) \\
    &= adx + b\sigma'(x)dx + c\sigma''(x)dx + e\sigma'''(x)dx \\
    &= [a + b\sigma'(x) + c\sigma''(x) + e\sigma'''(x)]dx\\
    &= 0
\end{align}</math>

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

:<math>\theta = b(x, \sigma(x), \sigma'(x))\theta_{0} + c(x, \sigma(x), \sigma'(x))\theta_1,</math>

where θ<sub>1</sub> = ''du''<sub>1</sub> − ''u''<sub>2</sub>''dx'' is the next basic contact form (Note that here we are identifying the form θ<sub>0</sub> with its pull-back <math>\left(\pi_{2,1}\right)^{*}\theta_{0}</math> to ''J<sup>2</sup>(π)'').

In general, providing ''x, u'' ∈ '''R''', a contact form on ''J<sup>r+1</sup>(π)'' can be written as a [[linear combination]] of the basic contact forms

:<math>\theta_k = du_k - u_{k+1}dx \qquad k = 0, \ldots, r - 1\,</math>

where

:<math> u_k\left(j^k \sigma\right) = \left.\frac{\partial^k \sigma}{\partial x^k}\right|_p.</math>

Similar arguments lead to a complete characterization of all contact forms.

In local coordinates, every contact one-form on ''J<sup>r+1</sup>(π)'' can be written as a linear combination

:<math>\theta = \sum_{|I|=0}^r P_\alpha^I \theta_I^\alpha</math>

with smooth coefficients <math>P^\alpha_i(x^i, u^\alpha, u^\alpha_I)</math> of the basic contact forms

:<math>\theta_I^\alpha = du^\alpha_I - u^\alpha_{I,i} dx^i\,</math>

''|I|'' is known as the '''order''' of the contact form <math>\theta_i^\alpha</math>. Note that contact forms on ''J<sup>r+1</sup>(π)'' have orders at most ''r''. Contact forms provide a characterization of those local sections of ''π<sub>r+1</sub>'' which are prolongations of sections of π.

Let ψ ∈ Γ<sub>''W''</sub>(''π<sub>r+1</sub>''), then ''ψ'' = ''j<sup>r+1</sup>''σ where σ ∈ Γ<sub>''W''</sub>(π) if and only if <math>\psi^* (\theta|_{W}) = 0, \forall \theta \in \Lambda_C^1 \pi_{r+1,r}.\,</math>

==Vector fields==
A general [[vector field]] on the total space ''E'', coordinated by <math>(x, u) \mathrel\stackrel{\mathrm{def}}{=} \left(x^i, u^\alpha\right)\,</math>, is

:<math>V \mathrel\stackrel{\mathrm{def}}{=} \rho^i(x, u)\frac{\partial}{\partial x^i} + \phi^{\alpha}(x, u)\frac{\partial}{\partial u^\alpha}.\,</math>

A vector field is called '''horizontal''', meaning that all the vertical coefficients vanish, if <math>\phi^\alpha</math> = 0.

A vector field is called '''vertical''', meaning that all the horizontal coefficients vanish, if ''ρ<sup>i</sup>'' = 0.

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

:<math>V_{(x, u)} \mathrel\stackrel{\mathrm{def}}{=} \rho^i(x, u) \frac{\partial}{\partial x^i} + \phi^{\alpha}(x, u) \frac{\partial}{\partial u^\alpha}\,</math>

having coordinates ''(x, u, ρ<sup>i</sup>, φ<sup>α</sup>)'', with an element in the fiber ''T<sub>xu</sub>E'' of ''TE'' over ''(x, u)'' in ''E'', called '''a [[tangent vector]] in ''TE'''''. A section

:<math>\begin{cases}
  \psi : E \to TE \\
    (x, u) \mapsto \psi(x, u) = V
\end{cases}</math>

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

:<math>V = \rho^i(x, u) \frac{\partial}{\partial x^i} + \phi^\alpha(x, u) \frac{\partial}{\partial u^\alpha}</math>

and ψ in ''Γ(TE)''.

The jet bundle ''J<sup>r</sup>(π)'' is coordinated by <math>(x, u, w) \mathrel\stackrel{\mathrm{def}}{=} \left(x^i, u^\alpha, w_i^\alpha\right)\,</math>.  For fixed ''(x, u, w)'', identify

:<math>
  V_{(x, u, w)} \mathrel\stackrel{\mathrm{def}}{=}
    V^i(x, u, w) \frac{\partial}{\partial x^i} +
      V^\alpha(x, u, w) \frac{\partial}{\partial u^\alpha} +
      V^\alpha_i(x, u, w) \frac{\partial}{\partial w^\alpha_i} +
      V^\alpha_{i_1 i_2}(x, u, w) \frac{\partial}{\partial w^\alpha_{i_1 i_2}} + \cdots +
      V^\alpha_{i_1 \cdots i_r}(x, u, w) \frac{\partial}{\partial w^\alpha_{i_1 \cdots i_r}}
</math>

having coordinates

:<math>\left(x, u, w, v^\alpha_i, v^\alpha_{i_1 i_2}, \cdots, v^\alpha_{i_1 \cdots i_r}\right),</math>

with an element in the fiber <math>T_{xuw}(J^r\pi)</math> of ''TJ<sup>r</sup>(π)'' over ''(x, u, w)'' ∈ ''J<sup>r</sup>(π)'', called '''a tangent vector in ''TJ<sup>r</sup>(π)'''''. Here,

:<math>v^\alpha_i, v^\alpha_{i_1 i_2}, \ldots, v^\alpha_{i_1 \cdots i_r}</math>

are real-valued functions on ''J<sup>r</sup>(π)''. A section

:<math>\begin{cases}
  \Psi : J^r(\pi) \to  TJ^r(\pi) \\
        (x, u, w) \mapsto \Psi(u, w) = V
\end{cases}</math>

is '''a vector field on ''J<sup>r</sup>(π)''''', and we say <math>\Psi \in \Gamma(T\left(J^r\pi\right)).</math>

==Partial differential equations==
Let ''(E, π, M)'' be a fiber bundle. An '''''r''-th order [[partial differential equation]]''' on π is a [[closed manifold|closed]] [[embedding|embedded]] submanifold ''S'' of the jet manifold ''J<sup>r</sup>(π)''. A solution is a local section σ ∈ Γ<sub>''W''</sub>(π) satisfying <math>j^{r}_p\sigma \in S</math>, for all ''p'' in ''M''.

Consider an example of a first order partial differential equation.

===Example===
Let π be the trivial bundle ('''R'''<sup>2</sup> × '''R''', pr<sub>1</sub>, '''R'''<sup>2</sup>) with global coordinates (''x''<sup>1</sup>, ''x''<sup>2</sup>, ''u''<sup>1</sup>). Then the map ''F'' : ''J''<sup>1</sup>(π) → '''R''' defined by

:<math>F = u^1_1 u^1_2 - 2x^2 u^1</math>

gives rise to the differential equation

:<math>S = \left\{j^1_p\sigma \in J^1\pi\ :\ \left(u^1_1u^1_2 - 2x^2u^1\right)\left(j^1_p\sigma\right) = 0\right\}</math>

which can be written

:<math>\frac{\partial \sigma}{\partial x^1}\frac{\partial \sigma}{\partial x^2} - 2x^2\sigma = 0.</math>

The particular

:<math>\begin{cases}
  \sigma : \mathbf{R}^2 \to \mathbf{R}^2 \times \mathbf{R} \\
  \sigma(p_1, p_2) = \left( p^1, p^2, p^1(p^2)^2 \right)
\end{cases}</math>

has first prolongation given by

:<math>j^1\sigma\left(p_1, p_2\right) = \left( p^1, p^2, p^1\left(p^2\right)^2, \left(p^2\right)^2, 2p^1 p^2 \right) </math>

and is a solution of this differential equation, because

:<math>\begin{align}
  \left(u^1_1 u^1_2 - 2x^2 u^1 \right)\left(j^1_p\sigma\right)
    &= u^1_1\left(j^1_p\sigma\right)u^1_2\left(j^1_p\sigma\right) - 2x^2\left(j^1_p\sigma\right)u^1\left(j^1_p\sigma\right) \\
    &= \left(p^2\right)^2 \cdot 2p^1 p^2 - 2 \cdot p^2 \cdot p^1\left(p^2\right)^2 \\
    &= 2p^1\left(p^2\right)^3 - 2p^1 \left(p^2\right)^3 \\
    &= 0
\end{align}</math>

and so <math>j^1_p\sigma \in S</math> for ''every'' ''p'' ∈ '''R'''<sup>2</sup>.

==Jet prolongation==
A local diffeomorphism ''ψ'' : ''J<sup>r</sup>''(''π'') → ''J<sup>r</sup>''(''π'') defines a contact transformation of order ''r'' if it preserves the contact ideal, meaning that if θ is any contact form on ''J<sup>r</sup>''(''π''), then ''ψ*θ'' is also a contact form.

The flow generated by a vector field ''V<sup>r</sup>'' on the jet space ''J<sup>r</sup>(π)'' forms a one-parameter group of contact transformations if and only if the [[Lie derivative]] <math>\mathcal{L}_{V^r}(\theta)</math> of any contact form θ preserves the contact ideal.

Let us begin with the first order case. Consider a general vector field ''V''<sup>1</sup> on ''J''<sup>1</sup>(''π''), given by

:<math>V^1\ \stackrel{\mathrm{def}}{=}\ \rho^i\left(x^i, u^\alpha, u_I^\alpha\right)\frac{\partial}{\partial x^i} + \phi^{\alpha}\left(x^i, u^\alpha, u_I^\alpha\right)\frac{\partial}{\partial u^{\alpha}} + \chi^{\alpha}_i\left(x^i, u^\alpha, u_I^\alpha\right)\frac{\partial}{\partial u^{\alpha}_i}.</math>

We now apply <math>\mathcal{L}_{V^1}</math> to the basic contact forms <math>\theta^{\alpha}_0 = du^{\alpha} - u_i^{\alpha}dx^i,</math> and expand the [[exterior derivative]] of the functions in terms of their coordinates to obtain:

:<math>\begin{align}
  \mathcal{L}_{V^1}\left(\theta^{\alpha}_0\right)
    &= \mathcal{L}_{V^1}\left(du^{\alpha} - u_i^{\alpha}dx^i\right) \\
    &= \mathcal{L}_{V^1}du^{\alpha} - \left(\mathcal{L}_{V^1}u_i^{\alpha}\right)dx^i - u_i^{\alpha}\left(\mathcal{L}_{V^1}dx^i\right) \\
    &= d\left(V^1u^{\alpha}\right) - V^1u_i^{\alpha}dx^i - u_i^{\alpha}d\left(V^1 x^i\right) \\
    &= d\phi^{\alpha} - \chi^{\alpha}_idx^i - u_i^{\alpha}d\rho^i \\
    &= \frac{\partial \phi^{\alpha}}{\partial x^i} dx^i + \frac{\partial \phi^{\alpha}}{\partial u^k} du^k + \frac{\partial \phi^{\alpha}}{\partial u^k_i} du^k_i - \chi^{\alpha}_i dx^i - u_i^{\alpha}\left[ \frac{\partial \rho^i}{\partial x^m} dx^m + \frac{\partial \rho^i}{\partial u^k} du^k + \frac{\partial \rho^i}{\partial u^k_m} du^k_m \right] \\
    &= \frac{\partial \phi^{\alpha}}{\partial x^i} dx^i +
       \frac{\partial \phi^{\alpha}}{\partial u^k} \left(\theta^k + u_i^k dx^i\right) +
       \frac{\partial \phi^{\alpha}}{\partial u^k_i} du^k_i -
       \chi^{\alpha}_i dx^i - u_l^{\alpha} \left[
         \frac{\partial \rho^l}{\partial x^i} dx^i +
         \frac{\partial \rho^l}{\partial u^k} \left(\theta^k + u_i^k dx^i\right) +
         \frac{\partial \rho^l}{\partial u^k_i} du^k_i
       \right]  \\
    &= \left[
         \frac{\partial \phi^{\alpha}}{\partial x^i} +
         \frac{\partial \phi^{\alpha}}{\partial u^k}u_i^k -
         u_l^\alpha \left(\frac{\partial \rho^l}{\partial x^i} + \frac{\partial \rho^l}{\partial u^k}u_i^k\right) -
         \chi^{\alpha}_i
       \right] dx^i +
       \left[ \frac{\partial \phi^{\alpha}}{\partial u^k_i} - u_l^{\alpha}\frac{\partial \rho^l}{\partial u^k_i}\right] du^k_i +
       \left( \frac{\partial \phi^{\alpha}}{\partial u^k} - u_l^{\alpha}\frac{\partial \rho^l}{\partial u^k} \right)\theta^k
\end{align}</math>

Therefore, ''V<sup>1</sup>'' determines a contact transformation if and only if the coefficients of ''dx<sup>i</sup>'' and <math>du^k_i</math> in the formula vanish. The latter requirements imply the '''contact conditions'''

:<math>\frac{\partial \phi^{\alpha}}{\partial u^k_i} - u^{\alpha}_l \frac{\partial \rho^l}{\partial u^k_i} = 0</math>

The former requirements provide explicit formulae for the coefficients of the first derivative terms in ''V<sup>1</sup>'':

:<math>\chi^{\alpha}_i = \widehat{D}_i \phi^{\alpha} - u^{\alpha}_l\left(\widehat{D}_i\rho^l\right)</math>

where

:<math>\widehat{D}_i = \frac{\partial}{\partial x^i} + u^k_i\frac{\partial}{\partial u^k}</math>

denotes the zeroth order truncation of the total derivative ''D<sub>i</sub>''.

Thus, the contact conditions uniquely prescribe the prolongation of any point or contact vector field. That is, if <math>\mathcal{L}_{V^r}</math> satisfies these equations, ''V<sup>r</sup>'' is called the '''''r''-th prolongation of ''V'' to a vector field on ''J<sup>r</sup>(π)'''''.

These results are best understood when applied to a particular example. Hence, let us examine the following.

===Example===
Consider the case ''(E, π, M)'', where ''E'' ≅ '''R'''<sup>2</sup> and ''M'' ≃ '''R'''. Then, ''(J<sup>1</sup>(π), π, E)'' defines the first jet bundle, and may be coordinated by ''(x, u, u<sub>1</sub>)'', where

:<math>\begin{align}
    x(j^1_{p}\sigma) &= x(p) = x  \\
    u(j^1_{p}\sigma) &= u(\sigma(p)) = u(\sigma(x)) = \sigma(x) \\
  u_1(j^1_{p}\sigma) &=  \left.\frac{\partial \sigma}{\partial x}\right|_{p} = \dot{\sigma}(x)
\end{align}</math>

for all ''p'' ∈ ''M'' and ''σ'' in Γ<sub>''p''</sub>(''π''). A contact form on ''J<sup>1</sup>(π)'' has the form

:<math>\theta = du - u_1 dx</math>

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

:<math>V = x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x}</math>

Then, the first prolongation of this vector field to ''J<sup>1</sup>(π)'' is

:<math>\begin{align}
  V^1 &= V + Z \\
      &= x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + Z \\
      &= x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + \rho(x, u, u_1) \frac{\partial}{\partial u_1}
\end{align}</math>

If we now take the Lie derivative of the contact form with respect to this prolonged vector field, <math>\mathcal{L}_{V^1}(\theta),</math> we obtain

:<math>\begin{align}
  \mathcal{L}_{V^1}(\theta)
    &= \mathcal{L}_{V^1}(du - u_1dx) \\
    &= \mathcal{L}_{V^1}du - \left(\mathcal{L}_{V^1}u_1\right)dx - u_1\left(\mathcal{L}_{V^1}dx\right) \\
    &= d\left(V^1u\right) - V^1 u_1 dx - u_1 d\left(V^1x\right) \\
    &= dx - \rho(x, u, u_1)dx + u_1 du \\
    &= (1 - \rho(x, u, u_1))dx + u_1 du \\
    &= [1 - \rho(x, u, u_1)]dx + u_1(\theta + u_1 dx) && du = \theta + u_1 dx \\
    &= [1 + u_1u_1 - \rho(x, u, u_1)]dx + u_1\theta  
\end{align}</math>

Hence, for preservation of the contact ideal, we require

:<math>1 + u_1 u_1 - \rho(x, u, u_1) = 0 \quad \Leftrightarrow \quad \rho(x, u, u_1) = 1 + u_1 u_1.</math>

And so the first prolongation of ''V'' to a vector field on ''J<sup>1</sup>(π)'' is

:<math>V^1 = x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + (1 + u_1u_1)\frac{\partial}{\partial u_1}.</math>

Let us also calculate the second prolongation of ''V'' to a vector field on ''J<sup>2</sup>(π)''. We have <math>\{x, u, u_1, u_2\}</math> as coordinates on ''J<sup>2</sup>(π)''. Hence, the prolonged vector has the form

:<math> V^2 = x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + \rho(x, u, u_1, u_2)\frac{\partial}{\partial u_1} + \phi(x, u, u_1, u_2)\frac{\partial}{\partial u_2}.</math>

The contact forms are

:<math>\begin{align}
    \theta &= du - u_1dx \\
  \theta_1 &= du_1 - u_2dx
\end{align}</math>

To preserve the contact ideal, we require

:<math>\begin{align}
    \mathcal{L}_{V^2}(\theta) &= 0 \\
  \mathcal{L}_{V^2}(\theta_1) &= 0
\end{align}</math>

Now, ''θ'' has no ''u''<sub>2</sub> 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<sup>1</sup>''. Therefore, the problem is analogous to prolonging the vector field ''V<sup>1</sup>'' to ''J''<sup>2</sup>(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

:<math>\rho(x, u, u_1) = 1 + u_1 u_1</math>

and so

:<math>\begin{align}
  V^2 &= V^1 + \phi(x, u, u_1, u_2)\frac{\partial}{\partial u_2} \\
      &= x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + (1 + u_1 u_1)\frac{\partial}{\partial u_1} + \phi(x, u, u_1, u_2)\frac{\partial}{\partial u_2}
\end{align}</math>

Therefore, the Lie derivative of the second contact form with respect to ''V<sup>2</sup>'' is

:<math>\begin{align}
  \mathcal{L}_{V^2}(\theta_1)
    &= \mathcal{L}_{V^2}(du_1 - u_2dx) \\
    &=  \mathcal{L}_{V^2}du_1 - \left(\mathcal{L}_{V^2}u_2\right)dx - u_2\left(\mathcal{L}_{V^2}dx\right) \\
    &= d(V^2 u_1) - V^2u_2dx - u_2d(V^2x) \\
    &= d(1 + u_1 u_1) - \phi(x, u, u_1, u_2)dx + u_2du \\
    &= 2u_1du_1 - \phi(x, u, u_1, u_2)dx + u_2du \\
    &= 2u_1du_1 - \phi(x, u, u_1, u_2)dx + u_2 (\theta + u_1dx)              &   du &= \theta + u_1 dx \\
    &= 2u_1(\theta_1 + u_2dx) - \phi(x, u, u_1, u_2)dx + u_2(\theta + u_1dx) & du_1 &= \theta_1 + u_2 dx \\
    &= [3u_1u_2 - \phi(x, u, u_1, u_2)]dx + u_2\theta + 2u_1\theta_1 
\end{align}</math>

Hence, for <math>\mathcal{L}_{V^2}(\theta_1)</math> to preserve the contact ideal, we require

:<math>3u_1 u_2 - \phi(x, u, u_1, u_2) = 0 \quad \Leftrightarrow \quad \phi(x, u, u_1, u_2) = 3u_1 u_2.</math>

And so the second prolongation of ''V'' to a vector field on ''J''<sup>2</sup>(π) is

:<math> V^2 = x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + (1 + u_1 u_1)\frac{\partial}{\partial u_1} + 3u_1 u_2\frac{\partial}{\partial u_2}.</math>

Note that the first prolongation of ''V'' can be recovered by omitting the second derivative terms in ''V<sup>2</sup>'', or by projecting back to ''J<sup>1</sup>(π)''.

==Infinite jet spaces==
The [[inverse limit]] of the sequence of projections <math>\pi_{k+1,k}:J^{k+1}(\pi)\to J^k(\pi)</math> gives rise to the '''infinite jet space''' ''J<sup>∞</sup>(π)''. A point <math>j_p^\infty(\sigma)</math> is the equivalence class of sections of π that have  the same ''k''-jet in ''p'' as σ  for all values of ''k''. The natural projection π<sub>∞</sub> maps <math>j_p^\infty(\sigma)</math> into ''p''.

Just by thinking in terms of coordinates, ''J<sup>∞</sup>(π)'' appears to be an infinite-dimensional geometric object. In fact, the simplest way of introducing a differentiable structure on ''J<sup>∞</sup>(π)'', not relying on differentiable charts, is given by the [[differential calculus over commutative algebras]]. Dual to the sequence of projections <math>\pi_{k+1,k}: J^{k+1}(\pi) \to J^k(\pi)</math> of manifolds is the sequence of injections <math>\pi_{k+1,k}^*: C^\infty(J^{k}(\pi)) \to C^\infty\left(J^{k+1}(\pi)\right)</math> of commutative algebras. Let's denote <math>C^\infty(J^{k}(\pi))</math> simply by <math>\mathcal{F}_k(\pi)</math>. Take now the [[direct limit]] <math>\mathcal{F}(\pi)</math> of the <math>\mathcal{F}_k(\pi)</math>'s. It will be a commutative algebra, which can be assumed to be the smooth functions algebra over the geometric object ''J<sup>∞</sup>(π)''. Observe that <math>\mathcal{F}(\pi)</math>, being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.

Roughly speaking, a concrete element <math>\varphi\in\mathcal{F}(\pi)</math> will always belong to some <math>\mathcal{F}_k(\pi)</math>, so it is a smooth function on the finite-dimensional manifold ''J<sup>k</sup>''(π) in the usual sense.

===Infinitely prolonged PDEs===
Given a ''k''-th order system of PDEs ''E'' ⊆ ''J<sup>k</sup>(π)'', the collection ''I(E)'' of vanishing on ''E'' smooth functions on ''J<sup>∞</sup>(π)'' is an [[ideal (ring theory)|ideal]] in the algebra <math>\mathcal{F}_k(\pi)</math>, and hence in the direct limit <math>\mathcal{F}(\pi)</math> too.

Enhance ''I(E)'' by adding all the possible compositions of [[total derivative]]s applied to all its elements. This way we get a new ideal ''I'' of <math>\mathcal{F}(\pi)</math> which is now closed under the operation of taking total derivative. The submanifold ''E''<sub>(∞)</sub> of ''J''<sup>∞</sup>(π) cut out by ''I'' is called the '''infinite prolongation''' of ''E''.

Geometrically, ''E''<sub>(∞)</sub> is the manifold of '''formal solutions''' of ''E''. A point <math>j_p^\infty(\sigma)</math> of ''E''<sub>(∞)</sub> can be easily seen to be represented by a section σ whose ''k''-jet's graph is tangent to ''E'' at the point <math>j_p^k(\sigma)</math> 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 <math>\varphi\circ j^k(\sigma)</math> at the point ''p''.

Most importantly, the closure properties of ''I'' imply that ''E''<sub>(∞)</sub> is tangent to the '''infinite-order contact structure''' <math>\mathcal{C}</math> on ''J<sup>∞</sup>(π)'', so that by restricting <math>\mathcal{C}</math> to ''E''<sub>(∞)</sub> one gets the [[diffiety]] <math>(E_{(\infty)}, \mathcal{C}|_{E_{(\infty)}})</math>, and can study the associated [[Diffiety#Vinogradov sequence|Vinogradov (C-spectral) sequence]].

==Remark==
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<sub>f</sub>: M'' → ''M'' × ''N''
:''gr<sub>f</sub>(p)'' = ''(p, f(p))''

(''gr<sub>f</sub>'' is known as the '''graph of the function ''f''''') of the trivial bundle (''M'' × ''N'', π<sub>1</sub>, ''M''). However, this restriction does not simplify the theory, as the global triviality of π does not imply the global triviality of π<sub>1</sub>.

==See also==
* [[Jet group]]
* [[Jet (mathematics)]]
* [[Lagrangian system]]
* [[Variational bicomplex]]

==References==
{{Reflist}}

==Further reading==
* 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., ''[http://www.emis.de/monographs/KSM/ 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}}.
* [[Peter J. Olver|Olver, P. J.]], "Equivalence, Invariants and Symmetry", Cambridge University Press, 1995, {{ISBN|0-521-47811-1}}


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[[Category:Differential topology]]
[[Category:Differential equations]]
[[Category:Fiber bundles]]