Moving frames and differential invariants in centro-affine geometry

Abstract

Abstract. Explicit formulas for the generating differential invariants and invariant differential operators for curves in two-and three-dimensional centro-equi-affine and centroaffine geometry and surfaces in three-dimensional centro-equi-affine geometry are constructed using the equivariant method of moving frames. In particular, the algebra of centro-equi-affine surface differential invariants is shown to be generated by a single second order invariant.

Invariants of objects and their images under surjective maps

“…We now seek to express the projective curvature η of the projected curve Π 0 ( C) in terms of centro-affine differential invariants of C. We begin by summarizing the equivariant moving frame calculations in [42]. We choose the cross-section to the prolonged centro-affine action on J k (M, 1) defined by the normalization equations…”

“…Remarkably, η is the projective curvature and ζ is z times an equi-affine invariant of the image curve. In Section 3.2, we will express η and ζ in terms of the third and fourth order centro-equi-affine invariants derived in [42].…”

“…Warning: We have switched the designation of κ and τ from that used in [42], and also deleted a factor of 3 in τ to slightly simplify the formulas. Our choice of notation is motivated by the fact that the condition τ = 0 is equivalent to the curve being contained in a plane P ⊂ R 3 , thus mimicking the Euclidean torsion of a space curve.…”

“…More generally, we consider the action of the affine group A(n) = GL(n) ⋉ R n given by z → A z + b for A ∈ GL(n), b ∈ R n . This action forms the foundation of affine geometry, and, for this reason, the previous linear action of GL(n) is sometimes referred to as the centroaffine group, underlying centro-affine geometry, [15,42]. We also consider the action of the projective group PGL(n) = GL(n)/ { λ I | 0 = λ ∈ R } on the projective space RP n−1 along with its local, linear fractional action on the dense open subset R n−1 ⊂ RP n−1 obtained by omitting the points at infinity.…”

“…We concentrate on the effect of such projections on space curves, leaving the analysis of surfaces to subsequent investigations. We will, in particular, derive relatively simple formulas relating the centroaffine invariants of a space curve, as classified in [42], to the projective curvature invariant of its projections.…”