Can a Lorentzian manifold be compact (without time loops)? Consider a manifold with metric signature (+++-). Which means that it is a curved 4 dimensional surface but also locally is Minkowski and can be assigned light-cones. 
Now, in the 3 space directions there is no problem with it being closed. e.g. it could be $\mathbb{R}\times S_3$ for example.
But can we also close the surface off in the `time' direction? Because there will be at least one point (possibly more) where we can't assign a light cone. e.g. at the south and north poles. (Assuming no time loops).
Is there a mathematical theory of closed Lorenzian manifolds? 
(Hawking suggested bolting on a Euclidean space to close off the manifold using "imaginary time" but that doesn't really make sense to me.)
I would like to know what the mathematicians think of this.
 A: It appears that you are asking the following question:
Does there exist a compact Lorentzian manifold which contains no time-like (casual) loops? 
The answer to this question is negative: Compactness implies existence of such loops. A proof is not hard, see Lemma 10 on page 407 of 
O'Neill's book "Semi-Riemannian geometry", which is the standard source for mathematical treatment of Lorentzian manifolds. 
One can even prove that a compact Lorentzian manifold contains a casual loop which is "almost geodesic". 
As for mathematical literature on compact Lorentzian manifolds, it is quite extensive. Take a look at a relatively recent paper


*

*S. Suhr, Closed geodesics in Lorentzian surfaces.  Trans. Amer. Math. Soc. 365 (2013), no. 3, 1469–1486. 


and references therein. 
Here is just a couple of results concerning geodesic completeness of compact Lorentzian manifolds. Recall that, according to the Hopf-Ronow theorem, every compact Riemannian manifold is geodesically complete, i.e. every geodesic extends indefinitely. In the Lorentzian case 
every compact Lorentzian manifold of constant sectional curvature is geodesically complete, see  


*Y.Carrière, Autour de la conjecture de L. Markus sur les variétés affines. Invent. Math. 95 (1989), no. 3, 615–628. (zero curvature case) 


and 


*B. Klingler, Complétude des variétés lorentziennes à courbure constante. Math. Ann. 306 (1996), no. 2, 353–370. (general case) 


Thus, every compact (locally) flat Lorentzian $(n+1)$-dimensional manifold is isometric to one of the form
$$
{\mathbb R}^{n,1}/\Gamma
$$
where ${\mathbb R}^{n,1}$ is the standard Lorentzian space-time and $\Gamma$ is a properly discontinuous (torsion-free) group of isometries of ${\mathbb R}^{n,1}$. A great deal is known about the structure of such groups, for instance, it is known (W.Goldman) that $\Gamma$ contains a polycyclic subgroup of finite index, i.e. is "close to" being commutative. An easy example is a flat Lorentzian metric on the $n+1$-dimensional torus (see Tsemo's answer). But there are more complex examples such that $\Gamma$ contains no commutative subgroups of finite index.  
In contrast, there are incomplete Lorentzian metrics on 2-dimensional tori, see this paper for a survey: 


*M. Sánchez, An introduction to the completeness of compact semi-Riemannian manifolds. Séminaire de Théorie Spectrale et Géométrie, No. 13, Année 1994–1995, 37–53, Sémin. Théor. Spectr. Géom., 13, Univ. Grenoble I, Saint-Martin-d'Hères, 1995.


And here is an open problem: 
Question. Is it true that every compact Lorentzian manifold contains a closed geodesic? 
Lastly, because of the existence of casual loops, physicists tend to regard compact Lorentzian manifolds as non-physical. This I never understood: For all what we know, casual loops exist in "our" space-time, it's just to traverse them takes more than the life-time of our universe. 
A: The tranlations preserve the the flat $b$ metric of signature $(+,+,+,-)$ of $\mathbb{R}^4$. $T^4$, the $4$-dimensional torus is the quotient of $\mathbb{R}^4$ by four translations whose directions are independent, thus $b$ induces a Lorentzian metric on $T^4$.
