# What are the allowed topologies for a FRW metric?

Given a spacetime that has the maximal amount of spacelike translations and rotations, what are the possible topologies it may take? I am mostly wondering about the "time" topology since the spatial one is fairly well documented.

I'm fairly sure that any FRW spacetime will be a foliation by spacelike hypersurfaces, so given a spacelike hypersurface $\Sigma$, either $\mathbb R \times \Sigma$ or $S \times \Sigma$ (or possibly some $\Sigma$ bundle over $\mathbb R$ or $S$), but looking through the literature (Hawking Ellis, Straumann, Ringström), everyone seems to assume global hyperbolicity outright.

Just given the Killing vector fields, can one show that the manifold has to be foliated by spacelike hypersurfaces?

• I could be wrong, but if you consider cases that lack time-translation symmetry, then it seems to me that if homogeneity and isotropy hold, then any scalar observable, such as temperature, is guaranteed to have level surfaces.
– user4552
Apr 2, 2018 at 19:35

As you know, 3-dimensional space need not be infinite even if the spatial curvature has sign $$k = 0$$ or $$k = -1$$. As George Ellis writes in "Issues in the philosophy of cosmology", in Butterfield & Earman eds. (2006):
Misconception 5: The space sections are necessarily infinite if $$k = 0$$ or −1. This is only true if they have their ‘natural’ simply connected topology. If their topology is more complex (e.g. a 3-torus) they can be spatially finite [Ellis, 1971a; Lachieze et al., 1995]. There are many ways this can happen; indeed if $$k = −1$$ there is an infinite number of possibilities.
Ellis, Maartens & MacCallum's excellent textbook Relativistic Cosmology (2012) contains a section "Topology", $$\S9.1.5$$. For $$k = 0$$ they mention generalisations of the Mobius band. For $$k > 0$$ they mention identifying antipodal points. Any $$k > 0$$ case is closed. They cite Wolf (1972), Ellis (1971), and Thurston (1997), and clearly there are many more references.