Abstract. This article presents the equivariant method of moving frames for finitedimensional Lie group actions, surveying a variety of applications, including geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant flows. 1. Introduction. According to Akivis, [1], the method of moving frames originates in work of the Estonian mathematician Martin Bartels (1769–1836), a teacher of both Gauss and Lobachevsky. The field is most closely associated with Élie Cartan, [21], who forged earlier contributions by Darboux, Frenet, Serret, and Cotton into a powerful tool for analyzing the geometric

Abstract. The goal of this paper is to describe, in as much detail as possible, the structure of the algebra of differential invariants of a Lie pseudo-group. Under the assumption of local freeness of the prolonged pseudo-group action, we develop algorithms for locating a finite generating set of differential invariants, establishing the recurrence relations for the differentiated invariants, and fixing a finite system of generating differential syzygies. In particular, if the pseudo-group acts transitively on the base manifold, then the algebra of differential invariants forms a rational, non-commutative differential algebra. We show that the essential features of the differential invariant algebra are prescribed by a pair of commutative algebraic modules: the symbol module associated with the infinitesimal determining system of the pseudo-group, and the new “prolonged symbol module ” constructed from the symbols of the prolonged pseudo-group generators. Modulo low order complications, the generating differential invariants and differential syzygies are in one-to-one correspondence with the algebraic generators and syzygies of the prolonged symbol module. Our algorithms and proofs are all constructive, and rely on combining the moving frame approach developed in earlier papers with Gröbner basis algorithms from commutative algebra.

Abstract. The equivariant method of moving frames provides an algorithmic procedure for determining and analyzing the structure of algebras of differential invariants for both finite-dimensional Lie groups and infinite-dimensional Lie pseudo-groups. This paper surveys recent developments, including a few surprises and several open questions. 1. Introduction. Differential invariants are the fundamental building blocks for constructing invariant differential equations and invariant variational problems, as well as determining their explicit solutions and conservation laws. The equivalence, symmetry and rigidity properties of submanifolds are all governed by their differential invariants. Additional applications

Abstract. This paper surveys several new developments in the analysis of Lie pseudogroups and their actions on submanifolds. The main themes are direct construction of Maurer–Cartan forms and structure equations, and the use of equivariant moving frames to analyze the algebra of differential invariants and invariant differential forms, including generators, commutation relations, and syzygies. 1. Introduction. Inspired by Galois ’ introduction of group theory to solve polynomial equations, Lie founded his remarkable theory of transformation groups for the purpose of analyzing and solving differential equations. In Lie’s time, abstract groups were as yet unknown, and hence he made no significant distinction between finite-dimensional and infinitedimensional

Abstract. The action of a Lie pseudogroup G on a smooth manifold M induces a prolonged pseudogroup action on the jet spaces J n of submanifolds of M. We prove in this paper that both the local and global freeness of the action of G on J n persist under prolongation in the jet order n. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the pseudogroup action on J ∞. 1.