Moving coframes: II. Regularization and theoretical foundations

The primary goal of this paper is to provide a rigorous theoretical justifi cation of Cartan's method of moving frames for arbitrary finite-dimensional Lie group actions on manifolds. The general theorems are based a new regul arized version of the moving frame algorithm, which is of both theoretical and practical use. Applications include a new approach to the construction and classification of differential invariants and invariant differential op erators on jet bundles, as well as equivalence, symmetry, and rigidity theo rems for submanifolds under general transformation groups. The method also leads to complete classifications of generating systems of differential inv ariants, explicit commutation formulae for the associated invariant differe ntial operators, and a general classification theorem for syzygies of the h igher order differentiated differential invariants. A variety of illustrati ve examples demonstrate how the method can be directly applied to practical problems arising in geometry, invariant theory, and differential equations . Mathematics Subject Classifications (1991): 53A55, 58D19, 58H05, 68U10.