(G,X)-manifold

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")-structures 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 formula_1 be a connected differential manifold and formula_2 be a subgroup of the group of diffeomorphisms of formula_1 which act analytically in the following sense:
(this definition is inspired by the analytic continuation property of analytic diffeomorphisms on an analytic manifold).
A formula_9-structure on a topological space formula_10 is a manifold structure on formula_10 whose atlas' charts has values in formula_1 and transition maps belong to formula_2. This means that there exists:
such that every transition map formula_18 is the restriction of a diffeomorphism in formula_2.
Two such structures formula_20 are equivalent when they are contained in a maximal one, equivalently when their union is also a formula_9 structure (i.e. the maps formula_22 and formula_23 are restrictions of diffeomorphisms in formula_2).
Riemannian examples.
If formula_2 is a Lie group and formula_1 a Riemannian manifold with a faithful action of formula_2 by isometries then the action is analytic. Usually one takes formula_2 to be the full isometry group of formula_1. Then the category of formula_9 manifolds is equivalent to the category of Riemannian manifolds which are locally isometric to formula_1 (i.e. every point has a neighbourhood isometric to an open subset of formula_1).
Often the examples of formula_1 are homogeneous under formula_2, for example one can take formula_35 with a left-invariant metric. A particularly simple example is formula_36 and formula_2 the group of euclidean isometries. Then a formula_38 manifold is simply a flat manifold.
A particularly interesting example is when formula_1 is a Riemannian symmetric space, for example hyperbolic space. The simplest such example is the hyperbolic plane, whose isometry group is isomorphic to formula_40.
Pseudo-Riemannian examples.
When formula_1 is Minkowski space and formula_2 the Lorentz group the notion of a formula_9-structure is the same as that of a flat Lorentzian manifold.
Other examples.
When formula_1 is the affine space and formula_2 the group of affine transformations then one gets the notion of an affine manifold.
When formula_46 is the n-dimensional real projective space and formula_47 one gets the notion of a projective structure.
Developing map and completeness.
Developing map.
Let formula_10 be a formula_38-manifold which is connected (as a topological space). The developing map is a map from the universal cover formula_50 to formula_1 which is only well-defined up to composition by an element of formula_2.
A developing map is defined as follows: fix formula_53 and let formula_54 be any other point, formula_55 a path from formula_56 to formula_57, and formula_58 (where formula_7 is a small enough neighbourhood of formula_56) a map obtained by composing a chart of formula_10 with the projection formula_62. We may use analytic continuation along formula_55 to extend formula_64 so that its domain includes formula_57. Since formula_50 is simply connected the value of formula_67 thus obtained does not depend on the original choice of formula_55, and we call the (well-defined) map formula_69 a "developing map" for the formula_38-structure. It depends on the choice of base point and chart, but only up to composition by an element of formula_2.
Monodromy.
Given a developing map formula_64, the "monodromy" or "holonomy" of a formula_38-structure is the unique morphism formula_74 which satisfies
It depends on the choice of a developing map but only up to an inner automorphism of formula_2.
Complete ("G","X")-structures.
A formula_38 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 formula_1 is simply connected the structure is complete if and only if the developing map is a diffeomorphism.
Examples.
Riemannian ("G","X")-structures.
If formula_1 is a Riemannian manifold and formula_2 its full group of isometry, then a formula_38-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 formula_38-manifold is compact then the latter is automatically complete.
In the case where formula_1 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 formula_38-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 formula_9-structures on a manifold formula_10 are viewed as flat Ehresmann connections on fiber bundles with fiber formula_1 over formula_10, whose monodromy maps lie in formula_2.