Differential Invariants of Equi–Affine Surfaces

We show that the algebra of equi-affine differential invariants of a suitably generic surface S ⊂ R is entirely generated by the third order Pick invariant via invariant differentiation. The proof is based on the new, equivariant approach to the method of moving frames. The goal of this paper is to prove that, in three-dimensional equi-affine geometry, all higher order differential invariants of suitably nondegenerate surfaces S ⊂ R are generated by the well-known Pick invariant, [1, 5, 8, 14, 15], through repeated invariant differentiation. Thus, in surprising contrast to Euclidean surface geometry, which requires two generating differential invariants — the Gauss and mean curvatures, [3, 9, 15] — equiaffine surface geometry is, in a sense, simpler, in that the local geometry, equivalence and symmetry properties of generic surfaces are entirely prescribed by the single Pick differential invariant. Our proof is based on the equivariant approach to Cartan’s method of moving frames that has been developed over the last decade by the author and various collaborators, [2, 11, 12]. One immediate advantage of the equivariant method is that it is not tied to geometrically-based actions, but can, in fact, be directly applied to any transformation group. In geometrical contexts, the equivariant approach mimics the classical moving frame construction, [3, 5], but goes significantly further, in that it supplies us with the † Supported in part by NSF Grant DMS 05-05293.