Let G be a groupacting on a manifold X via diffeomorphisms—i.e. for each , the map from X to itself is a diffeomorphism. A manifold M which satisfies the following conditions is called a (G, X)-manifold:
There exists an open cover of M and a family of diffeomorphisms taking each Uα to an open subset Vα of X.
For each Uα, Uβ with nonempty intersection, there exists a such that for all . In other words, viewing the elements of G as diffeomorphisms, each transition map is the restriction of an element of G to .