dg.differential geometry – Example on pseudo-groups Answer

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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

[1] 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.

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

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