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 Gnk 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 differential of a function f: Rk → R can be interpreted as a section of the cotangent bundle of RK given by df: Rk → T*Rk. Similarly, derivatives of order up to m are sections of the jet bundle Jm(Rk) = Rk × W, where

W = R × ( R ∗ ) k × S 2 ( ( R ∗ ) k ) × ⋯ × S m ( ( R ∗ ) k ) . {\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 smooth function f: Rk → R has a prolongation jmf: Rk → Jm(Rk) defined at each point p ∈ Rk by placing the i-th partials of f at p in the Si((R*)k) component of W.

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

Provide Jm(Rn) with a group structure by taking

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

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

Because of the properties of jets under function composition, Gnk 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

  • 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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