Differential Invariants of Conformal and Projective Surfaces

Abstract

Abstract. We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in three-dimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.

Moving frames and differential invariants in centro-affine geometry

“…Moreover, we prove that the algebra of surface differential invariants can, in fact, be generated by this single differential invariant, in the sense that all the higher order differential invariants can be obtained by repeatedly applying the two invariant differential operators associated with the moving frame to the generating invariant and taking functional combinations thereof. This is reminiscent of similar surprising recent results in Euclidean and equi-affine surface geometry, [12], as well as conformal and projective geometry, [7].…”

“…We will eventually prove, using moving frame methods, the following result, which is very much in the spirit of those established in [12] and [7]. Theorem 4.1.…”

Algebraic and Differential Invariants

“…For surfaces in 3-space, under the Euclidean and affine group, two curvatures with a syzygy describe the algebra of differential invariants. Using the tools presented here and their implementation, it was shown that the same is true for the projective and conformal group acting on surfaces [33]. Some cases of 3-dimensional submanifolds in 4-space were also given a computational treatment in [29,Section 7].…”

“…Many simple examples, as well as computationally challenging ones, are treated in the papers [14,29,30] from which the material of this section is drawn as well as in, for instance, [33,49,39,42,40]. Here we wish to illustrate how the presented material is put into action on a well known example.…”

“…Differential elimination algorithms can be applied to further reduce the number of generating differential invariants, as was done in [33,29]. Indeed, if a differential invariant can be written in terms of the others (and their invariant derivatives) it means that there is such a relationship in the differential ideal generated by the complete set of syzygies.…”