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.
- 1 Definition and examples
- 2 Developing map and completeness
- 3 (G,X)-structures as connections
- 4 Notes
- 5 References
Definition and examples
- if and there is an 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 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 ).
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.
When is the affine space and the group of affine transformations then one gets the notion of an affine manifold.
Developing map and completeness
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 and a path from to a,d (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 include . 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 basepoint and chart, but only up to composition by an element of .
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 .
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.
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.
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 is 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
- Thurston, William (1997). Three-dimensional geometry and topology. Vol. 1. Princeton University Press.