Moving frames and singularities of prolonged group actions

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Moving Frames for Lie Pseudo–Groups (2008, result): “M, as in [ 49 ], then, for n ≫ 0, the bundle H(n) can be locally identified with the principal bundle Jn × G introduced in [21].”
Invariant Euler–Lagrange Equations and the Invariant Variational Bicomplex (2003, background): “Even with these particular examples in hand, the general formula connecting (1.2) to (1.3) is not at all obvious.”
Generating differential invariants (2007, background): “…a (locally defined) moving frame, which, according to [3], is an equivariant map ρ: Jn → G. Equivariance can be with respect to either the right or left multiplication † A theorem of Ovsiannikov, [24], slightly corrected in [18], guarantees local freeness of the prolonged action at…”
Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces (2005, methodology): “A theorem by Ovsiannikov [Ov], corrected by Olver in [02], states that if a group acts locally effectively on subsets, then there exists an integer ko such that the prolonged action of the given group is locally free on an open and dense subset of the ko th order jet space.”