# 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

Further information: topological manifold

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

### Differential manifold

Further information: differential manifold

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

### Real-analytic manifold

Further information: real-analytic manifold

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

### Measured manifold

Further information: measured manifold

A measured manifold is a manifold over the pseudogroup of all volume-preserving diffeomorphisms between open sets, with the model space being $\R^n$.