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.
Best Answer
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
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
and
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:
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.