Lie pseudogroup

Remarks

The condition $j^k_x f \in R$ for all $x$ is a system of partial differential equations of order $k$ for $f$, and $\Gamma$ is its set of local invertible solutions. Definitions in the literature differ in details, for example in requiring the equations to be analytic or involutive.

Example: let $g$ be a Riemannian metric on $M$, and let $R \subseteq J^1_\times(M)$ be the set of $1$-jets whose derivative $T_x M \to T_y M$ is a linear isometry from $g_x$ to $g_y$. It is a closed subgroupoid and a submanifold, and a local diffeomorphism has all its $1$-jets in $R$ exactly when it is a local isometry. So local isometries form a Lie pseudogroup of order $1$. In the same way, area-preserving maps of $\R^2$ are the local diffeomorphisms whose $1$-jets satisfy $\det Df = 1$.

Non-example: the pseudogroup of all local homeomorphisms of $\R^n$ is not a Lie pseudogroup. It contains maps that are not smooth, so it is not even a pseudogroup of local diffeomorphisms.