Jet of a smooth map - Imported Knowlpedia Content

Definition

Let $M,N$ be smooth manifolds, $x\in M$ , and $r\geq 0$ an integer. Two smooth maps $f,g$ , defined near $x$ , have the same $r$ -jet at $x$ if $f(x)=g(x)$ and, in local coordinates about $x$ and their common value, all partial derivatives of their coordinate representatives of total order at most $r$ agree at $x$ . This relation is independent of the chosen coordinates. The equivalence class is denoted $j_x^r f$ . Varying $x$ , the value $f(x)$ , and the class produces the jet bundle $J^r(M,N)$ .

Source, target, and prolongation

The maps

\alpha(j_x^r f)=x,\qquad \beta(j_x^r f)=f(x)

are the source and target projections $J^r(M,N)\to M$ and $J^r(M,N)\to N$ . Every smooth map $f:M\to N$ has an $r$ -jet prolongation $j^r f:M\to J^r(M,N)$ , $x\mapsto j_x^r f$ . For $s\leq r$ , forgetting derivatives of order greater than $s$ defines a truncation $J^r(M,N)\to J^s(M,N)$ .

Special orders and composition

A $0$ -jet remembers only $(x,f(x))$ , so $J^0(M,N)\cong M\times N$ . A $1$ -jet additionally remembers the linear map $df_x:T_xM\to T_{f(x)}N$ . Jets compose: $j_{f(x)}^r g$ and $j_x^r f$ determine $j_x^r(g\circ f)$ by the chain rule through order $r$ .

Relation to jets of sections

When $E\to M$ is a smooth fiber bundle, jets of local sections are precisely those map jets $j_x^r s$ for which the source is $x$ and the composite $M\xrightarrow{s}E\to M$ is the identity. The section jet bundle $J^rE$ is therefore a constrained version of the general map-jet construction. For $r>1$ , these bundles are generally affine in their highest-order data rather than canonically vector bundles.

References

M. W. Hirsch, Differential Topology, Springer, 1976. DOI record. Relevant: Chapter 2.
M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer, 1973. DOI record. Relevant: Chapter II.