In geometry, if X is a manifold with an action of a topological group G by analytical diffeomorphisms, the notion of a (G, X)-structure on a topological space is a way to formalise it being locally isomorphic to X with its G-invariant structure; spaces with a (G, X)-structure are always manifolds and are called (G, X)-manifolds.
This notion is often used with G being a Lie group and X a homogeneous space for G. Foundational examples are hyperbolic manifolds and affine manifolds.
be a connected differential manifold and
be a subgroup of the group of diffeomorphisms of
whose atlas' charts has values in
This means that there exists: such that every transition map
a Riemannian manifold with a faithful action of
by isometries then the action is analytic.
to be the full isometry group of
(i.e. every point has a neighbourhood isometric to an open subset of
the group of euclidean isometries.
The simplest such example is the hyperbolic plane, whose isometry group is isomorphic to
the Lorentz group the notion of a
-structure is the same as that of a flat Lorentzian manifold.
is the n-dimensional real projective space and
-manifold which is connected (as a topological space).
A developing map is defined as follows:[2] fix
) a map obtained by composing a chart of
thus obtained does not depend on the original choice of
It depends on the choice of base point and chart, but only up to composition by an element of
which satisfies It depends on the choice of a developing map but only up to an inner automorphism of
structure is said to be complete if it has a developing map which is also a covering map (this does not depend on the choice of developing map since they differ by a diffeomorphism).
is simply connected the structure is complete if and only if the developing map is a diffeomorphism.
its full group of isometry, then a
In particular, in this case if the underlying space of a
In general compactness of the space does not imply completeness of a
For example, an affine structure on the torus is complete if and only if the monodromy map has its image inside the translations.
But there are many affine tori which do not satisfy this condition, for example any quadrilateral with its opposite sides glued by an affine map yields an affine structure on the torus, which is complete if and only if the quadrilateral is a parallelogram.
Interesting examples of complete, noncompact affine manifolds are given by the Margulis spacetimes.
In the work of Charles Ehresmann