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.

Definition and examples

Formal definition 

Let  be a connected differential manifold and  be a subgroup of the group of diffeomorphisms of  which act analytically in the following sense:

if  and there is a nonempty open subset  such that  are equal when restricted to  then 

(this definition is inspired by the analytic continuation property of analytic diffeomorphisms on an analytic manifold).

A -structure on a topological space  is a manifold structure on  whose atlas' charts has values in  and transition maps belong to . This means that there exists:

a covering of  by open sets  (i.e. );
open embeddings  called charts;

such that every transition map  is the restriction of a diffeomorphism in .

Two such structures  are equivalent when they are contained in a maximal one, equivalently when their union is also a  structure (i.e. the maps  and  are restrictions of diffeomorphisms in ).

Riemannian examples 

If  is a Lie group and  a Riemannian manifold with a faithful action of  by isometries then the action is analytic. Usually one takes  to be the full isometry group of . Then the category of   manifolds is equivalent to the category of Riemannian manifolds which are locally isometric to  (i.e. every point has a neighbourhood isometric to an open subset of ).

Often the examples of  are homogeneous under , for example one can take  with a left-invariant metric. A particularly simple example is  and  the group of euclidean isometries. Then a  manifold is simply a flat manifold.

A particularly interesting example is when  is a Riemannian symmetric space, for example hyperbolic space. The simplest such example is the hyperbolic plane, whose isometry group is isomorphic to .

Pseudo-Riemannian examples 

When  is Minkowski space and  the Lorentz group the notion of a -structure is the same as that of a flat Lorentzian manifold.

Other examples 

When  is the affine space and  the group of affine transformations then one gets the notion of an affine manifold.

When  is the n-dimensional real projective space and  one gets the notion of a projective structure.

Developing map and completeness

Developing map 

Let  be a -manifold which is connected (as a topological space). The developing map is a map from the universal cover  to  which is only well-defined up to composition by an element of .

A developing map is defined as follows: fix  and let  be any other point,  a path from  to , and  (where  is a small enough neighbourhood of ) a map obtained by composing a chart of  with the projection . We may use analytic continuation along  to extend  so that its domain includes . Since  is simply connected the value of  thus obtained does not depend on the original choice of , and we call the (well-defined) map  a developing map for the -structure. It depends on the choice of base point and chart, but only up to composition by an element of .

Monodromy 

Given a developing map , the monodromy or holonomy of a -structure is the unique morphism  which satisfies

.

It depends on the choice of a developing map but only up to an inner automorphism of .

Complete (G,X)-structures 

A  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). For example, if  is simply connected the structure is complete if and only if the developing map is a diffeomorphism.

Examples

Riemannian (G,X)-structures 

If  is a Riemannian manifold and  its full group of isometry, then a -structure is complete if and only if the underlying Riemannian manifold is geodesically complete (equivalently metrically complete). In particular, in this case if the underlying space of a -manifold is compact then the latter is automatically complete.

In the case where  is the hyperbolic plane the developing map is the same map as given by the Uniformisation Theorem.

Other cases 

In general compactness of the space does not imply completeness of a -structure. 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.

(G,X)-structures as connections 

In the work of Charles Ehresmann -structures on a manifold  are viewed as flat Ehresmann connections on fiber bundles with fiber  over , whose monodromy maps lie in .

Notes

References 

Differential geometry
Structures on manifolds