Affine Curvature - Derivation of The Curvature By Affine Invariance

Derivation of The Curvature By Affine Invariance

The special affine curvature can be derived explicitly by techniques of invariant theory. For simplicity, suppose that an affine plane curve is given in the form of a graph y = y(x). The special affine group acts on the Cartesian plane via transformations of the form

with ad − bc = 1. The following vector fields span the Lie algebra of infinitesimal generators of the special affine group:

An affine transformation not only acts on points, but also on the tangent lines to graphs of the form y = y(x). That is, there is an action of the special affine group on triples of coordinates

The group action is generated by vector fields

defined on the space of three variables (x,y,y′). These vector fields can be determined by the following two requirements:

• Under the projection onto the xy-plane, they must to project to the corresponding original generators of the action, respectively.
• The vectors must preserve up to scale the contact structure of the jet space

Concretely, this means that the generators X(1) must satisfy

where L is the Lie derivative.

Similarly, the action of the group can be extended to the space of any number of derivatives

The prolonged vector fields generating the action of the special affine group must then inductively satisfy, for each generator X ∈ {T₁,T₂,X₁,X₂,H}:

• The projection of X(k) onto the space of variables (x,y,y′,…,y(k−1)) is X(k−1).
• X(k) preserves the contact ideal:

where

Carrying out the inductive construction up to order 4 gives

The special affine curvature

does not depend explicitly on x, y, or y′, and so satisfies

The vector field H acts diagonally as a modified homogeneity operator, and it is readily verified that H(4)k = 0. Finally,

The five vector fields

form an involutive distribution on (an open subset of) R6 so that, by the Frobenius integration theorem, they integrate locally to give a foliation of R6 by five-dimensional leaves. Concretely, each leaf is a local orbit of the special affine group. The function k parameterizes these leaves.