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Are there any known topological restrictions on what kinds of manifolds can form the Cauchy hypersurface of a Lorentzian manifold? I'm particularly interested about restrictions on Euler characteristics.

If the above question is too general, I am specifically interested in the two subcases:

1. What if we require the Cauchy hypersurface to be compact?
2. And/or what if we require the ambient manifold to be flat?

Motivation

(Copied from comments below)

I'm trying to find a certain kind of tiling system of Minkowski space, where there are finitely many polyhedral tile shapes (up to Poincare equivalence). There are local rules for assembling them into a Cauchy surface, and rules for evolving the surface. In 3d I haven't found an interesting example and am wondering if there is something I should know about Euler characteristic that would restrict my search.

I think in general with tiles and local rules for assembling them you can get arbitrary flat Lorentz manifolds including ones with compact Cauchy hypersurfaces (definitely seems true in 2d) so I'm looking for any results, compact or not, that might suggest something about my situation. I have a feeling that the finiteness of the tiling system might make compact results relevant, but who knows.

Are there any known topological restrictions on what kinds of manifolds can form the Cauchy hypersurface of a Lorentzian manifold? I'm particularly interested about restrictions on Euler characteristics.

If the above question is too general, I am specifically interested in the two subcases:

1. What if we require the Cauchy hypersurface to be compact?
2. And/or what if we require the ambient manifold to be flat?

Motivation

(Copied from comments below)

I'm trying to find a certain kind of tiling system of Minkowski space, where there are finitely many polyhedral tile shapes (up to Poincare equivalence). There are local rules for assembling them into a Cauchy surface, and rules for evolving the surface. In 3d (2 space and 1 time) I haven't found an interesting example and am wondering if there is something I should know about Euler characteristic that would restrict my search.

I think in general with tiles and local rules for assembling them you can get arbitrary flat Lorentz manifolds including ones with compact Cauchy hypersurfaces (definitely seems true in 2d) so I'm looking for any results, compact or not, that might suggest something about my situation. I have a feeling that the finiteness of the tiling system might make compact results relevant, but who knows.

Are there any known topological restrictions on what kinds of manifolds can form the Cauchy hypersurface of a Lorentzian manifold? I'm particularly interested about restrictions on Euler characteristics.

If the above question is too general, I am specifically interested in the two subcases:

1. What if we require the Cauchy hypersurface to be compact?
2. And/or what if we require the ambient manifold to be flat?

Motivation

(Copied from comments below)

I'm trying to find a certain kind of tiling system of Minkowski space, where there are finitely many polyhedral tile shapes (up to Poincare equivalence). There are local rules for assembling them into a Cauchy surface, and rules for evolving the surface. In 3d I haven't found an interesting example and am wondering if there is something I should know about Euler characteristic that would restrict my search.

I think in general with tiles and local rules for assembling them you can get arbitrary flat Lorentz manifolds including compact ones (definitely seems true in 2d) so I'm looking for any results, compact or not, that might suggest something about my situation. I have a feeling that the finiteness of the tiling system might make compact results relevant, but who knows.

Are there any known topological restrictions on what kinds of manifolds can form the Cauchy hypersurface of a Lorentzian manifold? I'm particularly interested about restrictions on Euler characteristics.

If the above question is too general, I am specifically interested in the two subcases:

1. What if we require the Cauchy hypersurface to be compact?
2. And/or what if we require the ambient manifold to be flat?

Motivation

(Copied from comments below)

I'm trying to find a certain kind of tiling system of Minkowski space, where there are finitely many polyhedral tile shapes (up to Poincare equivalence). There are local rules for assembling them into a Cauchy surface, and rules for evolving the surface. In 3d I haven't found an interesting example and am wondering if there is something I should know about Euler characteristic that would restrict my search.

I think in general with tiles and local rules for assembling them you can get arbitrary flat Lorentz manifolds including ones with compact Cauchy hypersurfaces (definitely seems true in 2d) so I'm looking for any results, compact or not, that might suggest something about my situation. I have a feeling that the finiteness of the tiling system might make compact results relevant, but who knows.

I know there is some theorem about the Euler characteristic of a compact $n$-dimensional Lorentzian spacetime being zero. Does something similar apply to Cauchy surfaces in the spacetime? What if the spacetime is flat?

Are there any known topological restrictions on what kinds of manifolds can form the Cauchy hypersurface of a Lorentzian manifold? I'm particularly interested about restrictions on Euler characteristics.

If the above question is too general, I am specifically interested in the two subcases:

1. What if we require the Cauchy hypersurface to be compact?
2. And/or what if we require the ambient manifold to be flat?

Motivation

(Copied from comments below)

I'm trying to find a certain kind of tiling system of Minkowski space, where there are finitely many polyhedral tile shapes (up to Poincare equivalence). There are local rules for assembling them into a Cauchy surface, and rules for evolving the surface. In 3d I haven't found an interesting example and am wondering if there is something I should know about Euler characteristic that would restrict my search.

I think in general with tiles and local rules for assembling them you can get arbitrary flat Lorentz manifolds including compact ones (definitely seems true in 2d) so I'm looking for any results, compact or not, that might suggest something about my situation. I have a feeling that the finiteness of the tiling system might make compact results relevant, but who knows.