Manifold over a pseudogroup

Definition

Given data

A pseudogroup $G$ acting on a topological space $X$.

Definition part

A topological space $M$ is termed a $G$-manifold if there is an open cover $U_{\alpha }$ of $M$ such that $U_{\alpha }$ are isomorphic to open sets $V_{\alpha }$ in $X$, and such that the transition functions are elements within $G$.

The underlying space $X$ here is termed the model space for the $G$-manifold structure.

Examples

Topological manifold

A topological manifold is a manifold over the pseudogroup of all homeomorphisms between open sets, with the model space being $\mathbb {R} ^{n}$.

Differential manifold

A differential manifold is a manifold over the pseudogroup of all diffeomorphisms between open sets, with the model space being $\mathbb {R} ^{n}$.

Real-analytic manifold

A real-analytic manifold is a manifold over the pseudogroup of all real-analytic maps between open sets in $\mathbb {R} ^{n}$, with the model space being $\mathbb {R} ^{n}$.