Moving frames

Although the ideas date back to the early nineteenth century (see [1; Chapter 5] for detailed historical remarks), the theory of moving frames (“repères mobiles”) is most closely associated with the name of Élie Cartan, [12], who molded it into a powerful and algorithmic tool for studying the geometric properties of submanifolds and their invariants under the action of a transformation group. In the 1970’s, several researchers, cf. [13, 16, 17, 24], began the attempt to place Cartan’s intuitive constructions on a firm theoretical foundation. A significant conceptual step was to disassociate the theory from reliance on frame bundles and connections, and define a moving frame as an equivariant map from the manifold or jet bundle back to the transformation group. More recently, [14, 15], Mark Fels and I formulated a new, constructive approach to the equivariant moving frame theory that can be systematically applied to general transformation groups. These notes provide a quick survey of the basic ideas underlying our constructions. New and significant applications of these results have been developed in a wide variety of directions. A promising inductive version of the method that uses the moving frame of a subgroup to induce that of a larger group appears in [27]. In [6, 40], the theory was applied to produce new algorithms for solving the basic symmetry and equivalence