Equivariant Moving Frames for Euclidean Surfaces

The purpose of this note is to explain how to use the equivariant method of moving frames, [1, 10], to derive the differential invariants, invariant differential operators and invariant differential forms for surfaces in three-dimensional Euclidean space. This is, of course, a very classical problem and the results are not new; for the classical moving frame derivation, see, for instance, [2, 3]. But there are several reasons for performing this calculation. First, in contrast to the classical treatment, the equivariant moving frame approach does not require any a priori insight into surface geometry, relying only on the explicit formulas for the Euclidean group transformations and their infinitesimal generators; the resulting curvature and higher order differential invariants, invariant differential operators, invariant differential forms, etc., all follow by direct and algorithmic calculations. Further systematic calculations will produce the invariant contact forms, and associated invariant variational bicomplex, [4], leading to the explicit formulas governing Euclidean signatures, used to solve the equivalence problem for surfaces under Euclidean motions, [6, 7], Euclidean-invariant variational problems, [4], and Euclidean-invariant geometric surface flows, [8]. Furthermore, some of these formulae — the invariant differential operators and invariant differential forms — do not, as far as I know, appear in the existing literature, making this a useful exercise for further developing such applications. We assume the reader is familiar with the basics of Lie transformation groups, jet bundles, [6], and the equivariant approach to moving frames, [1, 10]. In this computation, we will concentrate on the right-equivariant moving frame map. As we will see, the classical Darboux moving frame for Euclidean surfaces, [3], can be interpreted as a left-equivariant moving frame map, which is the group-theoretic inverse of our map: the orthonormal frame vectors at a point on the surface form its orthogonal components, while the point on the