Formal definition

Let ${\displaystyle X}$ be a connected differential manifold and ${\displaystyle G}$ be a subgroup of the group of diffeomorphisms of ${\displaystyle X}$ which act analytically in the following sense:

if ${\displaystyle g_{1},g_{2}\in G}$ and there is a nonempty open subset ${\displaystyle U\subset X}$ such that ${\displaystyle g_{1},g_{2}}$ are equal when restricted to ${\displaystyle U}$ then ${\displaystyle g_{1}=g_{2}}$

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

A ${\displaystyle (G,X)}$-structure on a topological space ${\displaystyle M}$ is a manifold structure on ${\displaystyle M}$ whose atlas' charts has values in ${\displaystyle X}$ and transition maps belong to ${\displaystyle G}$. This means that there exists:

• a covering of ${\displaystyle M}$ by open sets ${\displaystyle U_{i},i\in I}$ (i.e. ${\displaystyle M=\bigcup _{i\in I}U_{i}}$);
• open embeddings ${\displaystyle \varphi _{i}:U_{i}\to X}$ called charts;

such that every transition map ${\displaystyle \varphi _{i}\circ \varphi _{j}^{-1}:\varphi _{j}(U_{i}\cap U_{j})\to \varphi _{i}(U_{i}\cap U_{j})}$ is the restriction of a diffeomorphism in ${\displaystyle G}$.

Two such structures ${\displaystyle (U_{i},\varphi _{i}),(V_{j},\psi _{j})}$ are equivalent when they are contained in a maximal one, equivalently when their union is also a ${\displaystyle (G,X)}$ structure (i.e. the maps ${\displaystyle \varphi _{i}\circ \psi _{j}^{-1}}$ and ${\displaystyle \psi _{j}\circ \varphi _{i}^{-1}}$ are restrictions of diffeomorphisms in ${\displaystyle G}$).

Riemannian examples

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

Often the examples of ${\displaystyle X}$ are homogeneous under ${\displaystyle G}$, for example one can take ${\displaystyle X=G}$ with a left-invariant metric. A particularly simple example is ${\displaystyle X=\mathbb {R} ^{n}}$ and ${\displaystyle G}$ the group of euclidean isometries. Then a ${\displaystyle (G,X)}$ manifold is simply a flat manifold.

A particularly interesting example is when ${\displaystyle X}$ is a Riemannian symmetric space, for example hyperbolic space. The simplest such example is the hyperbolic plane, whose isometry group is isomorphic to ${\displaystyle G=\mathrm {PGL} _{2}(\mathbb {R} )}$.

Pseudo-Riemannian examples

When ${\displaystyle X}$ is Minkowski space and ${\displaystyle G}$ the Lorentz group the notion of a ${\displaystyle (G,X)}$-structure is the same as that of a flat Lorentzian manifold.

Other examples

When ${\displaystyle X}$ is the affine space and ${\displaystyle G}$ the group of affine transformations then one gets the notion of an affine manifold.

When ${\displaystyle X=\mathbb {P} ^{n}(\mathbb {R} )}$ is the n-dimensional real projective space and ${\displaystyle G=\mathrm {PGL} _{n+1}(\mathbb {R} )}$ one gets the notion of a projective structure.^[1]

Developing map

Let ${\displaystyle M}$ be a ${\displaystyle (G,X)}$-manifold which is connected (as a topological space). The developing map is a map from the universal cover ${\displaystyle {\tilde {M}}}$ to ${\displaystyle X}$ which is only well-defined up to composition by an element of ${\displaystyle G}$.

A developing map is defined as follows:^[2] fix ${\displaystyle p\in {\tilde {M}}}$ and let ${\displaystyle q\in {\tilde {M}}}$ be any other point, ${\displaystyle \gamma }$ a path from ${\displaystyle p}$ to ${\displaystyle q}$, and ${\displaystyle \varphi :U\to X}$ (where ${\displaystyle U}$ is a small enough neighbourhood of ${\displaystyle p}$) a map obtained by composing a chart of ${\displaystyle M}$ with the projection ${\displaystyle {\tilde {M}}\to M}$. We may use analytic continuation along ${\displaystyle \gamma }$ to extend ${\displaystyle \varphi }$ so that its domain includes ${\displaystyle q}$. Since ${\displaystyle {\tilde {M}}}$ is simply connected the value of ${\displaystyle \varphi (q)}$ thus obtained does not depend on the original choice of ${\displaystyle \gamma }$, and we call the (well-defined) map ${\displaystyle \varphi$ :{\tilde {M}}\to X} a developing map for the ${\displaystyle (G,X)}$-structure. It depends on the choice of base point and chart, but only up to composition by an element of ${\displaystyle G}$.

Monodromy

Given a developing map ${\displaystyle \varphi }$, the monodromy or holonomy^[3] of a ${\displaystyle (G,X)}$-structure is the unique morphism ${\displaystyle h:\pi _{1}(M)\to G}$ which satisfies

${\displaystyle \forall \gamma \in \pi _{1}(M),p\in {\tilde {M}}:\varphi (\gamma \cdot p)=h(\gamma )\cdot \varphi (p)}$.

It depends on the choice of a developing map but only up to an inner automorphism of ${\displaystyle G}$.

Complete (G,X)-structures

A ${\displaystyle (G,X)}$ 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 ${\displaystyle X}$ is simply connected the structure is complete if and only if the developing map is a diffeomorphism.

Examples