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. Essentially a jet group describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms).

The k-th order jet group Gⁿ_k consists of jets of smooth diffeomorphisms φ: Rⁿ → Rⁿ such that φ(0)=0.^[1]

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

Let k ≥ 2. The gradient 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

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

Here R* is the dual vector space to R, and Sⁱ denotes the i-th symmetric power. A 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 ${\displaystyle p=(x,x')\in J^{m}(\mathbf {R} ^{n})}$. There is a unique polynomial f_p in k variables and of order m such that p is in the image of j^mf_p. That is, ${\displaystyle j^{k}(f_{p})(x)=x'}$. 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

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

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.

Notes

1. Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operations in differential geometry (PDF), Springer-Verlag, pp. 128–131.

References

• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operations in differential geometry (PDF), Springer-Verlag
• 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