Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 029, 43 pages      arXiv:1304.1616      https://doi.org/10.3842/SIGMA.2013.029
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Solving Local Equivalence Problems with the Equivariant Moving Frame Method

Francis Valiquette
Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada

Received July 21, 2012, in final form March 31, 2013; Published online April 05, 2013

Abstract
Given a Lie pseudo-group action, an equivariant moving frame exists in the neighborhood of a submanifold jet provided the action is free and regular. For local equivalence problems the freeness requirement cannot always be satisfied and in this paper we show that, with the appropriate modifications and assumptions, the equivariant moving frame constructions extend to submanifold jets where the pseudo-group does not act freely at any order. Once this is done, we review the solution to the local equivalence problem of submanifolds within the equivariant moving frame framework. This offers an alternative approach to Cartan's equivalence method based on the theory of G-structures.

Key words: differential invariant; equivalence problem; Maurer-Cartan form; moving frame.

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