(G,X)-manifold

In mathematics, the concept of a (G, X)-manifold generalizes several different kinds of manifolds, including 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]

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 Uα to an open subset Vα of X.
2. For each Uα, Uβ 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 $V_\alpha\cap V_\beta$.

Citations

1. ^ Thurston 2002, p. 27