Editing Jet bundle (section)

==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>