Jet bundle

In differential geometry, the jet bundle is a certain construction which 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.

Historically, jet bundles are attributed to Ehresmann, and were an advance on the method (prolongation) of Elie 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.)

More recently, 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.

Jets

:"Main article: Jet (mathematics).

Let $(mathcal\{E\},\; pi,\; mathcal\{M\})$ be a fiber bundle in a category of manifolds and let $p\; in\; mathcal\{M\}$, with $dimmathcal\{M\}=m$. Let $Gamma(pi),$ denote the set of all local sections whose domain contains $p,$. Let $I=(I(1),I(2),ldots,I(m))$ be a multi-index (an ordered $m$-tuple of integers), then

:$|I|\; :=\; sum\_\{i=1\}^\{m\}\; I(i)$

:$frac\{partial^\; eta^\{alpha\{partial\; x^\{I\; ight|\_\{p\},\; quad\; 1\; leq\; |I|\; leq\; r$

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 $sigma,$ is denoted $j^\{r\}\_\{p\}sigma$. The integer $r$ is also called the order of the jet.

$p,$ is the source of $j^\{r\}\_\{p\}sigma$.

$sigma,(p)$ is the target of $j^\{r\}\_\{p\}sigma$.

Jet manifolds

The $r^\{th\},$ jet manifold of $pi,$ is the set

:$\{j^\{r\}\_\{p\}sigma:p\; in\; mathcal\{M\},\; sigma\; in\; Gamma(pi)\}$

and is denoted $J^\{r\}pi,$. We may define projections $pi\_\{r\},$ and $pi\_\{r,0\},$ called the source and target projections respectively, by

:

where

: sigma^{alpha{partial x^

for all $p\; in\; mathcal\{M\}$ and $sigma\; in\; Gamma\_\{p\}(pi),$. A general 1-form on $J^\{1\}pi,$ takes the form

:$heta\; =\; a(x,\; u,\; u\_\{1\})dx\; +\; b(x,\; u,\; u\_\{1\})du\; +\; c(x,\; u,u\_\{1\})du\_\{1\},$

A section $sigma\; in\; Gamma\_\{p\}(pi),$ has first prolongation $j^\{1\}sigma\; =\; (u,u\_\{1\})\; =\; left(sigma(p),\; left.frac\{partial\; sigma\}\{partial\; x\}\; ight|\_\{p\}\; ight),$.Hence, $(j^\{1\}sigma)^\{*\}\; heta,$ can be calculated as

:

is called a vector field on $mathcal\{E\}$ with $V\; =\; ho^\{i\}(x,u)\; frac\{partial\}\{partial\; x^\{i\; +\; phi^\{alpha\}(x,u)\; frac\{partial\}\{partial\; u^\{alpha,$ and $psi\; in\; Gamma(Tmathcal\{E\}),$.

The jet bundle $J^\{r\}pi,$ is coordinated by $(x,u,w)\; stackrel\{mathrm\{def\{=\}\; (x^\{i\},u^\{alpha\},w\_\{i\}^\{alpha\}),$. For fixed $(x,u,w),$, identify

:

and so $j^\{1\}\_\{p\}sigma\; in\; mathcal\{S\},$ for "every" $p\; in\; mathbb\{R\}^\{2\},$.

Jet Prolongation

A local diffeomorphism $psi:J^\{r\}pi\; longrightarrow\; J^\{r\}pi,$ defines a contact transformation of order $r,$ if it preserves the contact ideal, meaning that if $heta,$ is any contact form on $J^\{r\}pi,$, then $j^\{r\}psi^\{*\}\; heta,$ is also a contact form.

The flow generated by a vector field $V^\{r\},$ on the jet space $J^\{r\},$ forms a one-parameter group of contact transformations if and only if the Lie derivative $mathcal\{L\}\_\{V^\{r(\; heta)$ of any contact form $heta,$ preserves the contact ideal.

Let us begin with the first order case. Consider a general vector field $V^\{1\},$ on $J^\{1\}pi,$, given by

:$V^\{1\}\; stackrel\{mathrm\{def\{=\}\; ho^\{i\}(u^\{1\})frac\{partial\}\{partial\; x^\{i\; +\; phi^\{alpha\}(u^\{1\})frac\{partial\}\{partial\; u^\{alpha\; +\; chi^\{alpha\}\_\{i\}(u^\{1\})frac\{partial\}\{partial\; u^\{alpha\}\_\{i\; ,$

We now apply $mathcal\{L\}\_\{V^\{1$ to the basic contact forms $heta^\{alpha\}\; =\; du^\{alpha\}\; -\; u\_\{i\}^\{alpha\}dx^\{i\},$, and obtain

:

for all $p\; in\; mathcal\{M\}$ and $sigma\; in\; Gamma\_\{p\}(pi),$. A contact form on $J^\{1\}pi,$ has the form

:$heta\; =\; du\; -\; u\_\{1\}dx\; ,$

Let us consider a vector $V,$ on $mathcal\{E\}$, having the form

:$V\; =\; x\; frac\{partial\}\{partial\; u\}\; -\; u\; frac\{partial\}\{partial\; x\}\; ,$

Then, the first prolongation of this vector field to $J^\{1\}pi,$ is

:

Hence, for $mathcal\{L\}\_\{V^\{1(\; heta),$ to preserve the contact ideal, we require

:

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

:$ho(x,u,u\_\{1\})\; =\; 1\; +\; u\_\{1\}u\_\{1\}\; ,$

and so

:

Hence, for $mathcal\{L\}\_\{V^\{2(\; heta\_\{1\}),$ to preserve the contact ideal, we require

:

And so the second prolongation of $V,$ to a vector field on $J^\{2\}pi,$ is

:$V^\{2\}\; =\; x\; frac\{partial\}\{partial\; u\}\; -\; u\; frac\{partial\}\{partial\; x\}\; +\; (1\; +\; u\_\{1\}u\_\{1\})frac\{partial\}\{partial\; u\_\{1\; +\; 3u\_\{1\}u\_\{2\}frac\{partial\}\{partial\; u\_\{2\; ,$

Note that the first prolongation of $V,$ can be recovered by omitting the second derivative terms in $V^\{2\},$, or by projecting back to $J^\{1\}pi,$.

Infinite Jet Spaces

The inverse limit of the sequence of projections $pi\_\{k+1,k\}:J^\{k+1\}(pi)\; o\; J^k(pi)$ gives rise to the infinite jet space $J^infty(pi)$. A point $j\_p^infty(sigma)$ is the equivalence class of sections of $pi$ that have the same $k$-jet in $p$ as $sigma$ for all values of $k$. The natural projection $pi\_infty$ maps $j\_p^infty(sigma)$ into $p$.

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

Roughly speaking, a concrete element $varphiinmathcal\{F\}(pi)$ will always belong to some $mathcal\{F\}\_k(pi)$, so it is a smooth function on the finite-dimensional manifold $J^k(pi)$ in the usual sense.

Infinitely prolonged PDE's

Given a $k$-th order system of PDE's $mathcal\{E\}subseteq\; J^k(pi)$, the collection $I(mathcal\{E\})$ of vanishing on $mathcal\{E\}$ smooth functions on $J^infty(pi)$ is an ideal in the algebra $mathcal\{F\}\_k(pi)$, and hence in the direct limit $mathcal\{F\}(pi)$ too.

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

Geometrically, $mathcal\{E\}\_\{(infty)\}$ is the manifold of formal solutions of $mathcal\{E\}$. A point $j\_p^infty(sigma)$ of $mathcal\{E\}\_\{(infty)\}$ can be easily seen to be represented by a section $sigma$ whose $k$-jet's graph is tangent to $mathcal\{E\}$ at the point $j\_p^k(sigma)$ with arbitrarily high order of tangency.

Analytically, if $mathcal\{E\}$ is given by $varphi=0$, a formal solution can be understood as the set of Taylor coefficients of a section $sigma$ in a point $p$ that make vanish the Taylor series of $varphicirc\; j^k(sigma)$ at the point $p$.

Most importantly, the closure properties of $I$ imply that $mathcal\{E\}\_\{(infty)\}$ is tangent to the infinite-order contact structure $mathcal\{C\}$ on $J^infty(pi)$, so that by restricting $mathcal\{C\}$ to $mathcal\{E\}\_\{(infty)\}$ one gets the diffiety $(mathcal\{E\}\_\{(infty)\},mathcal\{C\}|\_\{mathcal\{E\}\_\{(infty))$, and can study the associated C-spectral sequence.

Remark

This article has defined jets of local sections of a bundle, but it is possible to define jets of functions $f:mathcal\{M\}\; longrightarrow\; mathcal\{N\},$, where $mathcal\{M\}$ and $mathcal\{N\}$ 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 $f,$) of the trivial bundle $(mathcal\{M\}\; imes\; mathcal\{N\},\; pi\_\{1\},\; mathcal\{F\})$. However, this restriction does not simplify the theory, as the global triviality of $pi,$ does not imply the global triviality of $pi\_\{1\},$.

References

* Ehresmann, C., "Introduction a 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.
* Kolár, 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.
* Olver, P.J., "Equivalence, Invariants and Symmetry", Cambridge University Press, 1995, ISBN 0-521-47811-1