Compact GX-manifolds

Question

Let $M=(G,X)$ be a compact smooth Lorentzian manifold with constant sectional curvature, where $X$ is any of the well-known spaceforms $\mathcal{M}^n$, de-Sitter och Anti-de-Sitter and G is their associated isometry groups. There exists a local diffeomorphism $D:\tilde{M}\rightarrow X$ called the developing map. When is $D$ a covering map?

Answer 1

Suppose that $G$ is a subgroup of isometries of $X$, I assume that you take the model $(X,G)$ to be the 1-connected space $X$ endowed with a metric whose sectional curvature is constant. In this case there are three models for the euclidean, hyperbolic and elliptic geometry.

then $M$ is also a compact manifold endowed with a metric $<,>$_M whose sectional curvature is constant. Hopf Rinow implies that $<,>_M$ is complete, so the universal cover of $M$ is one of the model https://en.wikipedia.org/wiki/Sectional_curvature#Manifolds_with_constant_sectional_curvature for the hyperbolic, euclidean or elliptic geometry and coincide with the model $(X,G)$.

In general, the developing map is not always a covering. If $X=R^n$ and $G=Aff(R^n)$ is the group of affine transformations, Thurston and Sullivan have constructed a 3-dimensional compact manifold whose developing map is not a covering map.

Sullivan, D. and Thurston, W., Manifolds with canonical coordinates: Some examples, L’ens. Math.29 (1983), 15–25.

If you suppose that $X$ is a Lorentzian manifold,here is a partial answer to your question; Carriere has shown the Markus conjecture for Lorentzian manifolds equivalently he has shown that a compact flat Lorentzian manifold is complete this implies that its universal cover is $R^n$ and the developing map is a diffeomorphism.

Carri`ere, Y., Autour de la conjecture de L. Markus sur les vari ́et ́es affines Inv. Math.95 (1989), no. 3, 615–628

For a related question on similarity Lorentzian manifold, you can see this.

Aristide, Tsemo. "Closed similarity Lorentzian affine manifolds." Proceedings of the American Mathematical Society 132.12 (2004): 3697-3702.

Answer 2

Following Bruno Klingler's paper "Complétude des Variétés Lorentziennes à Courbure Constante", a $(G, X)$ manifold is defined as follows:

Definition A manifold $M$ is called a $(G, X)$ manifold if

• $X$ is a manifold, $G$ a Lie group acting transitively (from the left) on $X$
• $M$ has an atlas $\{U_i, \phi_i\}$ s.t. the coordinate changes $\phi_{i,j}$ are locally elements of G.

Klingler then proceeds with an equivalent definition:

Definition A manifod $M$ is a $(G, X)$ manifold if there exists a local diffeomorphism $D: \tilde{M} \rightarrow X$ and a group morphism $h: \pi_1M \rightarrow G$ s.t. for all $\gamma \in \pi_1M$, $D \circ \gamma = h(\gamma) \circ D$.

And a final definition:

Definition A $(G, X)$ manifold $M$ is called complete (as a $(G, X)$ manifold), if $D$ defines a covering map.

So you are interested in figuring out: Is a compact, Lorentzian manifold with constant sectional curvature complete as a $(G, X)$ manifold?

Klingler proves that the above statement/question is true in all three possibles cases (negative, positive or flat sectional curvature). The proof however is quite advanced and/or technical (and in French). It is still a reference you might find useful. Besides, chances are he is quoting further interesting sources.