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In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms).

Overview

The k-th order jet group G_k consists of jets of smooth diffeomorphisms φ: R → R such that φ(0)=0.

The following is a more precise definition of the jet group.

Let k ≥ 2. The differential of a function f: R → R can be interpreted as a section of the cotangent bundle of R given by df: R → T*R. Similarly, derivatives of order up to m are sections of the jet bundle J(R) = R × W, where

${\displaystyle W=\mathbf {R} \times (\mathbf {R} ^{*})^{k}\times S^{2}((\mathbf {R} ^{*})^{k})\times \cdots \times S^{m}((\mathbf {R} ^{*})^{k}).}$

Here R* is the dual vector space to R, and S denotes the i-th symmetric power. A smooth function f: R → R has a prolongation jf: R → J(R) defined at each point p ∈ R by placing the i-th partials of f at p in the S((R*)) component of W.

Consider a point ${\displaystyle p=(x,x')\in J^{m}(\mathbf {R} ^{n})}$. There is a unique polynomial f_p in k variables and of order m such that p is in the image of jf_p. That is, ${\displaystyle j^{k}(f_{p})(x)=x'}$. The differential data x′ may be transferred to lie over another point y ∈ R as jf_p(y) , the partials of f_p over y.

Provide J(R) with a group structure by taking

${\displaystyle (x,x')*(y,y')=(x+y,j^{m}f_{p}(y)+y')}$

With this group structure, J(R) is a Carnot group of class m + 1.

Because of the properties of jets under function composition, G_k is a Lie group. The jet group is a semidirect product of the general linear group and a connected, simply connected nilpotent Lie group. It is also in fact an algebraic group, since the composition involves only polynomial operations.

Notes

1. Kolář, Michor & Slovák (1993, pp. 128–131)

References

• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operations in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2014-05-02
• Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, ISBN 80-210-0165-8
• Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN 0-521-36948-7

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