User:Foxjwill/(G,X)-Manifold

The concept of a $(G, X)$-manifold generalizes all the various different types of manifolds, e.g. Riemannian manifolds, affine manifolds, piecewise linear manifolds, etc.

Definition
Let $G$ be a group acting on a manifold $X$ via diffeomorphisms—i.e. for each $$g\in G$$, the map $$x\mapsto gx$$ from $X$ to itself is a diffeomorphism. A manifold $M$ which satisfies the following conditions is called a $(G, X)$-manifold :
 * 1) There exists an open cover $$\{U_\alpha\}$$ of $M$ and a family $$\{\varphi_\alpha\colon U_\alpha \to V_\alpha\}$$ of diffeomorphisms taking each $U_{&alpha;}$ to an open subset $V_{&alpha;}$ of $X$.
 * 2) For each $U_{&alpha;}, U_{&beta;}$ with nonempty intersection, there exists a $$g\in G$$ such that $$gx=\phi_\alpha\circ\phi_\beta^{-1}(x)$$ for all $$x\in V_\alpha\cap V_\beta$$. In other words, viewing the elements of $G$ as diffeomorphisms, each transition map $$\phi_\alpha\circ\phi_\beta^{-1}\colon V_\alpha\cap V_\beta\to V_\alpha\cap V_\beta$$ is the restriction of an element of $G$ to $$V_\alpha\cap V_\beta$$.