Jet space or jet bundle

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

First approach

(@saunders1989geometry)

Let $\pi: E\mapsto M$ be a smooth vector bundle, with $dim(E_x)=q$. We can interpret it as $M$ being a manifold of independent variables and sections of $\pi$ 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 $[s]^k_x$ 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

$$ J^k(E)=\bigcup_{x\in M} J^k (E)_x. $$

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

* $\tilde{U}=\{[s]_x^k: x\in M, s(x)\in U\}$

* $\tilde{u}=(x^i, u^{\alpha},u_I^{\alpha})$ where

$x^i([s]_x^k)=x^i(s(x))$

$u^{\alpha}([s]_x^k)=u^{\alpha}(s(x))$

$u_I^{\alpha}([s]_x^k)=\dfrac{\partial f}{\partial x_I}$ 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

$$ p+q {p+k \choose k}. $$

If $P\in J^k(E)_x$, with $P=[s]_x^k$, we define $\pi_0(P)=s(x) \in E$. See example 2.24 in @olver86.

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

$$ x\mapsto [s]_x^k, $$

being $s:M\rightarrow E$ a section, are called holonomic sections or $k$-graphs (see @bryant2013exterior page 23 Theorem 3.2.) 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

$$ \cdots \rightarrow J^{k+1}(\pi) \stackrel{\pi_{k+1, k}}{\longrightarrow} J^{k}(\pi) \rightarrow \cdots \rightarrow J^{1}(\pi) \stackrel{\pi_{1,0}}{\longrightarrow} J^{0}(\pi)=E \stackrel{\pi}{\rightarrow} M $$

In this case, Cartan distribution is involutive (see wikipedia and its dimension agrees with $\mbox{dim}M$. We get a decomposition of $TJ^{\infty}(E)$ 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 $\Omega^*(J^*(E))$ 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 $k$:

$$ \partial_{J} f^{\alpha}\left(x_{0}\right)=\partial_{J} \tilde{f}^{\alpha}\left(x_{0}\right), \quad \alpha=1, \ldots, q, \quad 0 \leqslant \# J \leqslant k $$

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

$$ J^k(M)=\bigcup_{x\in M} J^k(M)_x $$

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 $f,g:\mathbb{R} \mapsto \mathbb{R}$ we define an equivalence relation $E^k_x$ for $x\in \mathbb{R}$: $f E^k_x g$ iff

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

for $j=0\ldots k$. The equivalence class of $f$ is denoted by $[f]^k_x$ and is called the $k$-jet of $f$.

For $\alpha, \beta: \mathbb{R} \mapsto M$, being $M$ a manifold and such that $\alpha(0)=\beta(0)=x\in M$, we say that $\alpha E^k_x \beta$ iff for every $\varphi:M\mapsto \mathbb{R}$

$$ (\varphi \circ \alpha) E^k_x (\varphi \circ \beta) $$

This is like saying that curves $\alpha$ and $\beta$ have $k$-order contact. The equivalence class of $\alpha$ is denoted by $[\alpha]^k_x$ and is called the $k$-jet of $\alpha$. We will denote by

$$ J_{0}^{k}(\mathbb{R}, M)_{x} $$the set of all $k$-jets at $x$. As $x$ varies over $M$ the set $J_{0}^{k}(\mathbb{R}, M)_{x}$ produces a fibre bundle known as the $k$-th order tangent bundle: $T^k M$.

If $f,g: M\mapsto N$ are maps over two manifolds, we say that $f E^k_x g$ iff for every $\gamma : \mathbb{R} \rightarrow M$

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

In the previous case, if we take $N$ as a fiber bundle over $M$, we can consider $f$ and $g$ to be sections. The set of all $k$-jets at $x$ is denoted by $J_x^k(M,N)$, and if we consider $x$ varying at $M$ we can construct a fiber bundle over $M$ denoted $J^k(N)$.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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