a (G,X)-structure

Imagine you want to give a doughnut surface a perfectly flat geometry — every patch should look like a piece of the ordinary plane, with no stretching when you move from patch to patch. You cannot lay one flat sheet over the whole doughnut, but you can cover it with many flat patches and demand that wherever two patches overlap, the change of coordinates is a rigid motion of the plane (a Euclidean symmetry). The result is a flat torus: locally indistinguishable from the plane, globally something new. A (G,X)-structure is the general machine behind this idea, where 'plane' is replaced by a chosen model space X and 'rigid motion' by a chosen symmetry group G.

Precisely, fix a homogeneous space X — a space on which a Lie group G acts transitively, so X looks the same from every point (think of X = R^n with G the Euclidean group, X = S^n with G = O(n+1), or X = H^n hyperbolic space with G its isometries). A (G,X)-structure on a manifold M is an atlas of charts mapping open pieces of M into X such that every transition map, on each overlap, is the restriction of an element of G. Because G usually acts by real-analytic maps, a transition map is determined on a whole overlap by its restriction to any small piece. This is exactly Klein's Erlangen viewpoint — a geometry IS a pair (G, X) — promoted from the model space to an arbitrary manifold modeled on it.

The payoff is that local model geometry and global topology become separable questions you can study together. The same surface can carry many inequivalent (G,X)-structures, and the space of all of them up to natural equivalence is a deformation space — the object Teichmüller theory studies for hyperbolic structures on surfaces. Two derived invariants organize everything: the developing map (a global unfolding of the charts into X) and the holonomy representation (a homomorphism pi_1(M) -> G recording how the unfolding fails to close up). A structure is called complete when its developing map is a covering of X.