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What is the shape of space in a spacetime? One way of addressing this issueis to consider edgeless spacelike submanifolds of the spacetime. An alternativeis to foliate the spacetime by timelike curves and consider the quotientobtained by identifying points on the same timelike curve. In this article weinvestigate each of these notions and obtain conditions such that it yields ameaningful shape of space. We also consider the relationship between these twonotions and find conditions for the quotient space to be diffeomorphic to anyedgeless spacelike hypersurface. In particular, we find conditions in whichmerely local behavior (being spacelike) combined with the correct behavior onthe homotopy level guarantees that a putative shape of space really isprecisely that.Comment: 19 pages, AMSTe

Causal monotonicity, omniscient foliations and the shape of space