Jet bundle (first jets of sections)

Let $\pi\colon E\to M$ be a smooth fiber bundle over a smooth manifold .

Definition (1-jet of a section)

Let $x\in M$. Two smooth local sections $s,t\colon U\to E$ (with $x\in U$) are 1-jet equivalent at $x$ if:

1. $s(x)=t(x)$, and
2. in any local trivialization of $E$ near $x$, their first derivatives at $x$ agree.

The equivalence class of $s$ is denoted $j_x^1 s$ and is called the 1-jet of $s$ at $x$.

Definition (1-jet bundle)

The 1-jet bundle $J^1E$ is the set of all 1-jets $j_x^1 s$ of local sections, with the smooth structure making the projection

a smooth fiber bundle over $E$, and the composite $\pi\circ \pi_{1,0}\colon J^1E\to M$ a smooth fiber bundle over $M$.

The fiber of $J^1E\to M$ at $x$ encodes “value + first derivative” data at $x$. More precisely, $J^1E\to E$ is an affine bundle modeled on $\mathrm{Hom}(T_xM, V_eE)$, where $T_xM$ is the tangent space and $V_eE$ is the vertical tangent space at $e\in E_x$.

A fundamental application is that for a principal bundle $P\to M$, the quotient $J^1P/G$ is the bundle of connections .

Examples

1. 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 $(f(x), df_x)$. Thus $J^1(M\times \mathbb{R})$ identifies with $M\times \mathbb{R}\times T_x^*M$.
2. 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 $(x, f(x), df_x)$.
3. 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.