Relative Invariants from Moving Frames on an Extended Manifold

TL;DR

The paper addresses the constructive extraction of relative invariants for smooth Lie group actions by extending the base manifold to an extended space $\widehat{\mathcal M}=\mathcal M\times\mathbb R^\times$ and encoding the multiplier as a 1-cocycle. Using a moving frame on the extended space, it shows that relative invariants on $\mathcal M$ correspond to absolute invariants on $\widehat{\mathcal M}$ and that invariantization of the multiplier yields a canonical relative invariant of weight $-1$. It also provides a cohomological interpretation showing all multipliers are coboundaries $\mu(g,x)=\frac{f(g\cdot x)}{f(x)}$, hence realizable within this framework. The approach is then specialized to the diagonal projective action of $PGL(3,\mathbb R)$, where an explicit moving frame, invariantized Jacobian, and joint absolute and relative invariants (including integral projective invariants) are constructed for image data, illustrating practical applications in projective geometry and computer vision.

Abstract

A constructive modification of the moving frame method is developed in this paper for the construction of relative invariants of regular Lie group actions. Let a relative invariant $I$ of weight $ω$ transform according to the rule $$ I(g \cdot \boldsymbol x) = μ(g, \boldsymbol x)^ω I(\boldsymbol x), $$ where $μ: G \times \mathcal{M} \to \mathbb{R}^\times$ is a scalar multiplier (1-cocycle). It is shown that the cocycle property of $μ$ is equivalent to the well-definedness of the twisted group action on the extended manifold $\widehat{\mathcal{M}} = \mathcal{M} \times \mathbb{R}^\times$, and that relative invariants on $\mathcal{M}$ are in one-to-one correspondence with absolute invariants of this action on $\widehat{\mathcal{M}}$. The main result is that, given a moving frame, the invariantization of the multiplier is a canonical relative invariant of weight $-1$. This enables the constructive realization of any weight and yields an explicit formula for an arbitrary relative invariant in terms of the fundamental absolute invariants and the invariantized multiplier. Examples are provided to demonstrate the application of the proposed approach for the projective group $PGL(3, \mathbb{R})$.

Theorems & Definitions (11)

• Theorem 1: Olver1999-1
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof
• Theorem 6