Manifold over a pseudogroup

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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.