Jet group

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 Gⁿ_k consists of jets of smooth diffeomorphisms φ: Rⁿ → Rⁿ 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^k → R can be interpreted as a section of the cotangent bundle of R^K given by df: R^k → T*R^k. Similarly, derivatives of order up to m are sections of the jet bundle J^m(R^k) = R^k × W, where

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

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

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

Provide J^m(Rⁿ) with a group structure by taking

$(x,x^{\prime })*(y,y^{\prime })=(x+y,j^m{f}_p(y)+y^{\prime })$

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

Because of the properties of jets under function composition, Gⁿ_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.

References

• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993) (PDF), Natural operations in differential geometry, Springer-Verlag, http://www.emis.de/monographs/KSM/kmsbookh.pdf, 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, https://archive.org/details/geometryofjetbun0000saun