Jet bundle (first jets of sections)

Let $\pi:E\to M$ be a smooth fiber bundle. Two smooth local sections $s,t$ , defined near $x\in M$ , have the same first jet at $x$ if

s(x)=t(x)=e,\qquad ds_x=dt_x:T_xM\longrightarrow T_eE,

where $ds_x$ and $dt_x$ are their differentials. The equivalence class is denoted $j_x^1s$ .

The first jet bundle $J^1E$ consists of all such classes, with projections

\pi_{1,0}(j_x^1s)=s(x),\qquad \pi_1(j_x^1s)=x.

Its smooth structure is defined by the following jet charts. In local bundle coordinates $(x^i,y^\alpha)$ , assign to $j_x^1s$ the coordinates

\left(x^i,y^\alpha(s(x)),y_i^\alpha\right),\qquad y_i^\alpha=\frac{\partial(y^\alpha\circ s)}{\partial x^i}(x).

The derivative coordinates range freely over real matrices; changes of jet coordinates are the smooth transformations obtained by the chain rule. These charts define the smooth bundle structures $J^1E\to E$ and $J^1E\to M$ , not merely smooth projections of an unspecified structure.

Affine structure and connections

For fixed $e\in E_x$ , the fiber of $J^1E\to E$ identifies with linear maps $L:T_xM\to T_eE$ satisfying $d\pi_e\circ L=\operatorname{id}$ . Such maps are precisely the differentials of local sections with value $e$ . Their differences lie in $\operatorname{Hom}(T_xM,V_eE)$ , so this is an affine space modeled on that vector space. Here $V_eE=\ker d\pi_e$ is the vertical tangent space.

For a principal bundle $P\to M$ , the quotient $J^1P/G$ is the bundle of connections.

Examples

Jets of functions. For the trivial real line bundle $E=M\times \mathbb{R}$ , a section is a function $f\colon M\to \mathbb{R}$ , and $j_x^1 f$ is determined by $(x,f(x),df_x)$ . Thus $J^1(M\times \mathbb{R})$ identifies with $\mathbb{R}\times T^*M$ over $M$ .
Trivial bundle with fiber F. For $E=M\times F$ , a section is a map $f\colon M\to F$ , and $j_x^1 f$ records $(x, f(x), df_x)$ .
Local coordinate description. In coordinates $(x^i)$ on $M$ and fiber coordinates $(y^\alpha)$ on $E$ , a jet is described by $(x^i, y^\alpha, y^\alpha_i)$ , where $y^\alpha_i$ represent the first partial derivatives of the section components.

Differential relations

A first-order differential relation selects allowed jets. Formal and holonomic solutions distinguish assigned derivative data from actual derivatives.