Moving frames and singularities of prolonged group actions

Abstract

Abstract. The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie's theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have necessitated a detailed understanding of the geometry of prolonged transformation groups. This paper begins with a basic review of moving frames, and then focuses on the study of both regular and singular prolonged group orbits. Highlights include a corrected version of the basic stabilization theorem, a discussion of "totally singular points," and geometric and algebraic characterizations of totally singular submanifolds, which are those that admit no moving frame. In addition to applications to the method of moving frames, the paper includes a generalized Wronskian lemma for vector-valued functions, and methods for the solution to Lie determinant equations.

Differential invariant algebras

“…This implies, [35], that the prolonged action is locally free on a dense open subset V n ⊂ J n for n sufficiently large. In fact, in all known examples, the prolonged action is, in fact, free on an open subset of J n for n ≫ 0, although there is, frustratingly, no general proof or counterexample known as yet.…”

Joint Invariant Signatures

“…At present, the most general result in this direction is based on a theorem of Ovsiannikov, cf. [35,32]; additional details, including a correct extension of this result to smooth actions, can be found in [34]. Theorem 4.2.…”

Differential Invariants of Conformal and Projective Surfaces

“…Clearly, r 0 ≤ r 1 ≤ r 2 ≤ · · · ≤ r = dim G, and r n = r if and only if the action is locally free on an open subset of J n . Assuming G acts locally effectively on subsets, [31], this holds for n sufficiently large. We define the stabilization order s to be the minimal n such that r n = r. Locally, the number of functionally independent differential invariants of order ≤ n equals dim J n − r n .…”