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When trying to do physics on a more general pseudo-Riemannian manifold we want to require that there is a foliation of this manifold into three-dimensional subspaces. By this I mean we would like to have a way of splitting our spacetime $M$ into a part that we call space and one we call time. This is necessary, for example, in order to write the two-form $F$ that represents the electromagnetic field as $F = E \wedge dt + B$. I know that this is possible if we assume $M$ to be compact, but this implies that there are closed time-like geodesics (I have been told I would not know a proof). What are the necessary and sufficient conditions for such a foliation to exist? I would also appreciate it if you could inform me about books/resources that go into these sorts of questions. I have looked into books about GR, but they don't seem to talk about these sorts of things.

Red Act
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Anonymous
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  • I tried to improve your title, but had to (subjectively) interpret your question in order to nicely formulate it: Please feel free to roll back my edit and replace the title by something better if you want to. – Danu Feb 03 '16 at 11:32
  • I think you can use the Frobenius theorem to show foliation of the manifold into spacelike hypersurfaces – Slereah Feb 03 '16 at 12:26
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    Related MO.SE posts: http://mathoverflow.net/q/198552/13917 , http://mathoverflow.net/q/214683/1391 and links therein. – Qmechanic Feb 03 '16 at 12:41
  • @Slereah as far as I know, Frobenius gives equivalent conditions for a foliation to be integrable. It does not tell you much (anything?) about which spaces would admit certain foliations. – Danu Feb 03 '16 at 13:23
  • Thanks guys! The articles that can be found in the links suggested by Qmechanic seem very interesting and give quite weak conditions for such a foliation to exist! I will certainly look into them! – Anonymous Feb 03 '16 at 18:14
  • Closely related: https://physics.stackexchange.com/q/696289/226902 https://physics.stackexchange.com/q/712737/226902 https://physics.stackexchange.com/q/328722/226902 https://physics.stackexchange.com/q/761289/226902 – Quillo Apr 26 '23 at 12:50
  • This was investigated by Geroch, see: "Topology in GR" J. Math. Phys. 8, 782–786 (1967) https://doi.org/10.1063/1.1705276 and "Domain of Dependence", J. Math. Phys. 11, 437–449 (1970) https://doi.org/10.1063/1.1665157 .. see also https://physics.stackexchange.com/a/92271/226902 – Quillo Apr 26 '23 at 13:10

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