The
k-th order
jet group ''G'
n'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: 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 = \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')\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'. 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^mf_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, ''G'
n'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==