jet bundle

0.1 Introduction

Let $p:X\to B$ be a surjective submersion of ${𝒞}^r$differential manifolds, where$r\in \{0,1,\mathrm{\dots }\}\cup \{\mathrm{\infty },\omega \}$ (${𝒞}^{\omega }$ meansreal analytic). For all integers $k\ge 0$ with $k\le r$, we willdefine a fibre bundle ${\mathrm{J}}_B^{k}X$ over $X$, called the $k$-th jetbundle of $X$ over $B$. The fibre of this bundle above a point $x\in X$ can be interpreted as the set of equivalence classes of localsections of $p$ $x$, where twosections are consideredequivalent if their first $k$ derivatives at $x$ are equal. Theequivalence class of a section is then the jet of that section;it indicates the direction of the section locally at $x$. Thisconcept has much in common with that of the germ of a smooth functionon a manifold: it not only the value of afunction at a point, but also some about the behaviour of the function near that point.

0.2 Construction

We will now define each jet bundle of $X$ over $B$ as a set with aprojection map to $X$, and we describe the concept ofprolongation of sections. After that, we give a slightlydifferent construction allowing us to put a manifold structure on eachof the jet bundles.

For every open subset of $B$, we denote by $\mathrm{\Gamma }(U,X)$ the set ofsections of $p$ over $U$, i.e. the set of ${𝒞}^r$ functions$s:U\to p^{-1}U$ such that ${p|}_{p^{-1}U}\circ s={id}_U$. Every point of $B$ has an open subset $U$ such that thereexists at least one section of $p$ over $U$, due to the assumptionthat $p$ is a surjective submersion. For all $x\in X$, we define thefibre of ${\mathrm{J}}_B^{k}X$ above $x$ by

$${\mathrm{J}}_B^{k}X(x)=\{(U,s):U\subset B\text{ open},x\in U,s\in \mathrm{\Gamma }(U,X),s(p(x))=x\}/\mathrm{\sim },$$

where the equivalence relation $\sim$ is defined by $s\sim s^{\prime }\iff s$ and $s^{\prime }$ induce the same map between the fibresat $p(x)$ and $x$ of the $k$-th iterated tangent bundles of $B$ and$X$, respectively. (Note that the fibres in the $k$-th iteration arethe 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, ${\mathrm{J}}_B^{k}X$ is defined as the disjointunion of the sets ${\mathrm{J}}_B^{k}X(x)$ with $x\in X$. Write $\pi :{\mathrm{J}}_B^{k}X\to X$ for the ‘obvious’ projection map, defined by

$$\pi ([U,s])=x\text{ for }[U,s]\in {\mathrm{J}}_B^{k}X(x).$$

Notice that ${\mathrm{J}}_B^{0}X$ 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$[U,s]\in {\mathrm{J}}_B^{k}X(s(y))$ we obtain a section of $\pi \circ p:{\mathrm{J}}_B^{k}X\to B$ over $U$ for each $k\ge 0$, called the $k$-thprolongation of $s$. Composing this section on the left with$\pi$ gives back the original section $s$ of $X$.

0.3 The bundle structure

Instead of defining all the jet bundles at once, we may choose todefine only the first jet bundle in the way described above. Afterequipping the first jet bundle with the structure of a differentialmanifold, which we will do below, we can then inductively define${\mathrm{J}}_B^{k+1}X$ as the first jet bundle of ${\mathrm{J}}_B^{k}X$ over $B$ for $k\ge 1$. This is useful because the manifold structure only needs to bedefined for ${\mathrm{J}}_B^{1}X$.

We make ${\mathrm{J}}_B^{1}X$ into an affine bundle over $X$, locally trivial ofrank $dimB(dimX-dimB)$, in the following way. We cover $X$ withcharts $(W,\varphi ,W^{\prime })$, where $\varphi$ is a diffeomorphism between opensubsets $W\subset X$ and $W^{\prime }\subset {ℝ}^n$. Without loss ofgenerality, we assume that $p(W)$ is contained in the domain $V$ of achart $(V,\psi ,V^{\prime })$ on $B$, with $V^{\prime }\subset {ℝ}^m$. Here $n$and $m$ are the local dimensions of $X$ and $B$, respectively.

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

The definition of the equivalence relation $\sim$ on ${\mathrm{J}}_B^{1}X(x)$ meansthat the association $[U,s]\mapsto M_x([U,s])$ is well-defined andinjective for each $x\in W$. The image of $M_x$ consists of thematrices with the property that multiplying them on the left with the$m\times n$ matrix corresponding to the tangent map $\mathrm{T}p(x)$ givesthe $m\times m$ identity matrix. These matrices form a$m(n-m)$-dimensional linear subspace $L_x\subset {ℝ}^{n\times m}$,and $L={\bigcup }_{x\in W}\{x\}\times L_x$ is a submanifold of$W\times {ℝ}^{n\times m}$.

We both the differentiable structure of${\pi }^{-1}U$ and a local trivialisation of ${\mathrm{J}}_B^{1}X$ as a vector bundleby requiring that

${\pi }^{-1}W$$\to$$L$

$[U,s]$$\mapsto$$(\pi ([U,s]),M_{\pi ([U,s])}([U,s]))$

be a diffeomorphism and an $ℝ$-linear map. Since the effectof a change of charts on $W$ or $V$ is multiplying each $M_x([U,s])$ bymatrices depending differentiably on $x\in W$ (namely, the derivativesof the glueing maps), this gives a well-defined vector bundlestructure on all of ${\mathrm{J}}_B^{1}X$.

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