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.{{harvtxt|Kolář|Michor|Slovák|1993|pp=128-131}}

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 $p=(x,x')\in J^m(\mathbf R^n)$ . 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, $j^k(f_p)(x)=x'$ . 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

: $(x,x') * (y, y') = (x+y, j^mf_p(y) + y')$

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

References

{{citation|last1=Kolář|first1=Ivan|last2=Michor|first2=Peter|last3=Slovák|first3=Jan|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf|format=PDF|title=Natural operations in differential geometry|year=1993|publisher=Springer-Verlag|access-date=2014-05-02|archive-url=https://web.archive.org/web/20170330154524/http://www.emis.de/monographs/KSM/kmsbookh.pdf|archive-date=2017-03-30|url-status=dead}}
{{citation|last1 = Krupka|first1=Demeter|last2=Janyška|first2=Josef|title=Lectures on differential invariants|year = 1990|publisher = Univerzita J. E. Purkyně V Brně|isbn=80-210-0165-8}}
{{citation|last1 = Saunders|first1 = D.J.|title = The geometry of jet bundles|year = 1989|publisher = Cambridge University Press|isbn = 0-521-36948-7|url-access = registration|url = https://archive.org/details/geometryofjetbun0000saun}}

Category:Lie groups