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Equivariant corner straightening is usually mentioned in the literature without further explanation. What would be a reference where this is done (more or less) carefully for compact Lie group actions ...

Suppose we have a compact manifold $M$ with the action of a compact group $G$. Consider the space of $C^l$ $G$-equivariant diffeomorphisms $\text{Diff}_G^{l}(M)$ with the $C^l$ topology and the space ...