Jet space or jet bundle

Note. In some texts the notation $J^{n} (R^{p}, R^{q})$ is used to mean what here we denote by $J^{n} (R^{p} \times R^{q})$ being $R^{p} \times R^{q} \to R^{p}$ the trivial bundle.

First approach

(@saunders1989geometry)
Let $π : E \mapsto M$ be a smooth vector bundle, with $d i m (E_{x}) = q$ . We can interpret it as $M$ being a manifold of independent variables and sections of $π$ as functions on $M$ . The set of sections forms a $C^{\infty} (M)$ -module.

Two sections $s, \tilde{s}$ are said to be $k$ -equivalent at a point $x \in M$ if their graphs are tangent to each other with order $k$ at $s (x) = \tilde{s} (x) \in E$ . By "their graphs are tangent" we mean that if we express the sections as functions $f$ and $\tilde{f}$ in a local trivialization, the partial derivatives of $f$ and $\tilde{f}$ at $x$ coincide up to order $k$ . This fact does not depend on the chosen trivialization.

The equivalence class of $s$ , denoted by $[s]_{x}^{k}$ is called the $k$ -jet of $s$ at $x$ , and the collection of all the classes, $J^{k} (E)_{x}$ , is the jet space at $x$ . Finally, we can construct a bundle

J^{k} (E) = ⋃_{x \in M} J^{k} (E)_{x} .

In this bundle we have special coordinates called the derivative coordinates. Let $(U, u)$ be adapted coordinates on $E$ , with $u = (x^{i}, u^{α})$ . The derivative coordinates induced by $(U, u)$ are $(\tilde{U}, \tilde{u})$ where

$\tilde{U} = {[s]_{x}^{k} : x \in M, s (x) \in U}$
$\tilde{u} = (x^{i}, u^{α}, u_{I}^{α})$ where
- $x^{i} ([s]_{x}^{k}) = x^{i} (s (x))$
- $u^{α} ([s]_{x}^{k}) = u^{α} (s (x))$
- $u_{I}^{α} ([s]_{x}^{k}) = \frac{\partial f}{\partial x_{I}}$ being $f$ the expression of $s$ in $(U, u)$ and $I$ a multiindex expressing partial derivatives of order up to $k$ .
  (@saunders1989geometry page 94).

The dimension of $J^{k} (E)$ is

p + q (\binom{p + k}{k}) .

If $P \in J^{k} (E)_{x}$ , with $P = [s]_{x}^{k}$ , we define $π_{0} (P) = s (x) \in E$ . See example 2.24 in @olver86.

The sections of $J^{k} (E)$ of the form

x \mapsto [s]_{x}^{k},

being $s : M \to E$ a section, are called holonomic sections or $k$ -graphs (see @bryant2013exterior page 23 Theorem 3.2.), or $k$ th-order jet prolongation of $s$ , and will be denoted by $j^{k} s$ .

The jet space or jet bundle $J^{k} (E)$ has a natural distribution called the Cartan distribution.

Infinite jet bundle

The infinite jet space $J^{\infty} (E)$ is defined as the inverse limit of the inverse system

\dots \to J^{k + 1} (π) \overset{π_{k + 1, k}}{⟶} J^{k} (π) \to \dots \to J^{1} (π) \overset{π_{1, 0}}{⟶} J^{0} (π) = E \overset{π}{\to} M

In this case, Cartan distribution is involutive (see wikipedia) and its dimension agrees with $dim M$ . We get a decomposition of $T J^{\infty} (E)$ in a vertical bundle (the subspace of $T_{p} J^{\infty} (E)$ tangent to the fibre) and an horizontal one (the subspace belonging to the Cartan distribution). This decomposition induces a similar decomposition in the cotangent space $T^{*} J^{\infty} (E)$ , and also in the ring of differential forms $Ω^{*} (J^{*} (E))$ giving rise to the variational bicomplex.

Second approach: Sketch of the formalization a la Olver for the extended jet bundle

Locally, any pair of submanifolds of a manifold $M$ of the same dimensions is given by the graphs of smooth functions $f$ and $\tilde{f}$ (with independent and dependent variables between them of $M$ ). These functions are said to be $k$ -equivalent at $x \in M$ if there is a coordinate chart such that the partial derivatives at $x$ coincide up to order $k$ :

\partial_{J} f^{α} (x_{0}) = \partial_{J} {\tilde{f}}^{α} (x_{0}), α = 1, \dots, q, 0 ⩽ # J ⩽ k

This idea is independent of the coordinate chart, and therefore we can define an equivalence relation between submanifolds being tangent up to order $k$ at $x$ . The equivalence class $[N]_{x}^{k}$ is called a jet. The union of all jets at $x$ is the jet space at $x$ , $J^{k} (M)_{x}$ and the jet bundle is

J^{k} (M) = ⋃_{x \in M} J^{k} (M)_{x}

See @olver86 page 218 for more details. There he doesn't treat the case of sections of bundles but the construction let define the extended jet bundle (something like a projective version).

Third approach: Sketch of the formalization of Wikipedia

Between functions $f, g : R \mapsto R$ we define an equivalence relation $E_{x}^{k}$ for $x \in R$ : $f E_{x}^{k} g$ iff

f^{j)} (x) = g^{j)} (x)

for $j = 0 \dots k$ . The equivalence class of $f$ is denoted by $[f]_{x}^{k}$ and is called the $k$ -jet of $f$ .

For $α, β : R \mapsto M$ , being $M$ a manifold and such that $α (0) = β (0) = x \in M$ , we say that $α E_{x}^{k} β$ iff for every $φ : M \mapsto R$

(φ \circ α) E_{x}^{k} (φ \circ β)

This is like saying that curves $α$ and $β$ have $k$ -order contact. The equivalence class of $α$ is denoted by $[α]_{x}^{k}$ and is called the $k$ -jet of $α$ . We will denote by

J_{0}^{k} (R, M)_{x} $ $ t h e s e t o f a l l $ k $ - j e t s a t $ x $ . A s $ x $ v a r i e s o v e r $ M $ t h e s e t $ J_{0}^{k} (R, M)_{x} $ p r o d u c e s a f i b r e b u n d l e k n o w n a s t h e $ k $ - t h o r d e r t a n g e n t b u n d l e : $ T^{k} M $ . - I f $ f, g : M \mapsto N $ a r e m a p s o v e r t w o m a n i f o l d s, w e s a y t h a t $ f E_{x}^{k} g $ i f f f o r e v e r y $ γ : R \to M $

(f\circ \gamma) E^k_x (g\circ \gamma)

- I n t h e p r e v i o u s c a s e, i f w e t a k e $ N $ a s a f i b e r b u n d l e o v e r $ M $, w e c a n c o n s i d e r $ f $ a n d $ g $ t o b e s e c t i o n s . T h e s e t o f a l l $ k $ - j e t s a t $ x $ i s d e n o t e d b y $ J_{x}^{k} (M, N) $, a n d i f w e c o n s i d e r $ x $ v a r y i n g a t $ M $ w e c a n c o n s t r u c t a f i b e r b u n d l e o v e r $ M $ d e n o t e d $ J^{k} (N) $ .