# Compact GX-manifolds

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?

• Your question does not make much sense. What are $G$ and $X$? What does the equation $M=(G,X)$ mean? Furthermore, you need to tell us more about what you already know regarding your question, and what you don't understand, otherwise potential answerers will not wish to waste their time writing things which you might already know. Feb 17 '16 at 22:17
• Sorry about this, but the so-called (G,X)-structures is already involved, and a intro can be found here. I was hoping to catch some expert. For someone who know this stuff the question is well-defined. Feb 17 '16 at 22:24
• Do you suppose that $G$ acts on $X$ by isometries? Feb 17 '16 at 22:30

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.

• Yes, this is all true in the Riemannian case. However in the Lorentzian case $\tilde{M}$ need not be compete as Hopf-Rinow is false. The spaceforms $X$ are Minkowskispace, de-Sitter and Anti-de-Sitter (or its universal cover). Feb 17 '16 at 22:46
• Usually, the sectional curvature is defined in the Riemannian case, Feb 17 '16 at 22:47
• I work in the Lorentzian setting. Feb 17 '16 at 22:48
• Then perhaps I could ask you a related question (not having looked at your link). It is stated that a flat compact lorentzian manifold will have affine structure in a paper by Bruno Klingler (building on the idea of Carriere, that you refed). Is this immediate? Feb 17 '16 at 23:07
• flat Lorentzian implies an affine structure by definition, flat means that the curvature of the associated connection vanishes, the torsion of a connection associated to a metric vanishes, and a manifold endowed with a connection whose torsion and curvature vanish is affine. Feb 17 '16 at 23:10

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.