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## dg.differential geometry – Example on pseudo-groups

Definition:
‎A pseudo group is a collection $$\mathcal{G}$$ of (locally defined) invertible smooth diffeomorphisms of a manifold $$M$$. ‎The simplest example of a pseudo-group is the collection of all local diffeomorphism of a manifold $$M$$.

Example:
Consider the infinite-dimensional Lie pseudo-group‎

$$‎X=x,\qquad Y=y\,f(x)+\phi (x),\qquad Z=z(f(x))^x+\psi(x)‎,$$

‎where $$f,\phi,\psi\in C^\infty (\mathbb{R})$$ other $$f(x)>0$$‎.

Obviously, $$X,Y$$ other $$Z$$ are the target coordinates. Indeed, the pseudo-group acts on the coordinate $$(x,y,z)$$ and transforms it to the target coordinates $$(X,Y,Z)$$.

My questions:

1. what are the group parameters??

2. If we split the cotangent bundle into horizontal and vertical forms, what are the vertical forms?

Reference

 Olver, Peter J.; Pohjanpelto, Juha; Valiquette, Francis, On the structure of Lie pseudo-groups, SIGMA, Symmetry Integrability Geom. Methods Appl. 5, Paper 077, 14 p. (2009). ZBL1241.58008.

 https://math.stackexchange.com/questions/4449167/example-on-pseudo-groups

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