Introduction

Construction

For every open subset of $B$ , we denote by $\Gamma(U,X)$ the set of sections of $p$ over $U$ , i.e. the set of ${\cal C}^r$ functions $s\colon U\to p^{-1}U$ such that $p\vert_{p^{-1}U}\circ s=\mathop{\rm id}_U$ . Every point of $B$ has an open subset $U$ such that there exists at least one section of $p$ over $U$ , due to the assumption that $p$ is a surjective submersion. For all $x\in X$ , we define the fibre of $\J^k_BX$ above $x$ by $$ \J^k_BX(x)=\{(U,s)\colon U\subset B\hbox{ open}, x\in U,s\in\Gamma(U,X),s(p(x))=x\}/\mathord\sim, $$ where the equivalence relation $\sim$ is defined by $s\sim s'\Leftrightarrow s$ and $s'$ induce the same map between the fibres at $p(x)$ and $x$ of the $k$ -th iterated tangent bundles of $B$ and $X$ , respectively. (Note that the fibres in the $k$ -th iteration are the same if and only if the induced maps are already the same in the $(k-1)$ -st iteration). We will denote the equivalence class of a pair $(U,s)$ by $[U,s]$ . As a set, $\J^k_BX$ is defined as the disjoint union of the sets $\J^k_BX(x)$ with $x\in X$ . Write $\pi\colon \J^k_BX\to X$ for the `obvious' projection map, defined by $$ \pi([U,s])=x\hbox{ for }[U,s]\in\J^k_BX(x). $$ Notice that $\J^0_BX$ is just $X$ itself.

Suppose we have some section $s$ of $X$ over an open subset $U$ of $B$ . By sending every point $y\in U$ to the equivalence class $[U,s]\in\J^k_BX(s(y))$ we obtain a section of $\pi\circ p\colon \J^k_BX\to B$ over $U$ for each $k\ge 0$ , called the $k$ -th prolongation of $s$ . Composing this section on the left with $\pi$ gives back the original section $s$ of $X$ .

The bundle structure

We make $\J^1_BX$ into an affine bundle over $X$ , locally trivial of rank $\dim B(\dim X-\dim B)$ , in the following way. We cover $X$ with charts $(W,\phi,W')$ , where $\phi$ is a diffeomorphism between open subsets $W\subset X$ and $W'\subset\mathbb{R}^n$ . Without loss of generality, we assume that $p(W)$ is contained in the domain $V$ of a chart $(V,\psi,V')$ on $B$ , with $V'\subset\mathbb{R}^m$ . Here $n$ and $m$ are the local dimensions of $X$ and $B$ , respectively.

For all $x\in W$ and all $[U,s]\in\J^1_BX(x)$ , we have the tangent map $\T s(p(x))$ , which is a linear map from $\T V(p(x))$ to $\T W(x)$ . These tangent spaces are isomorphic to $\mathbb{R}^m$ and $\mathbb{R}^n$ via the chosen charts, so that $\T s(p(x))$ acts as a matrix $M_x([U,s])\in\mathbb{R}^{n\times m}$ :

We fix both the differentiable structure of $\pi^{-1}U$ and a local trivialisation of $\J^1_BX$ as a vector bundle by requiring that \begin{eqnarray*} \pi^{-1}W&\to&L \\ {}[U,s] &\mapsto&\left(\pi([U,s]),M_{\pi([U,s])}([U,s])\right) \end{eqnarray*}be a diffeomorphism and an $\mathbb{R}$ -linear map. Since the effect of a change of charts on $W$ or $V$ is multiplying each $M_x([U,s])$ by matrices depending differentiably on $x\in W$ (namely, the derivatives of the glueing maps), this gives a well-defined vector bundle structure on all of $\J^1_BX$ .

Iterating the above construction by defining $\J^{k+1}_BX$ as the first jet bundle of $\J^k_BX$ over $B$ , each jet bundle $\J^{k+1}_BX$ becomes a vector bundle over $\J^k_BX$ and a fibre bundle over $X$ . Normally, only $\J^1_BX$ is a vector bundle over $X$ .