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A result in math says that $S^n$ carries a Lorentzian metric iff $n$ is odd.
Using it we can observe that a 2-sphere spacetime is impossible, a 3-sphere spacetime is geometrically possible, but again a 4-sphere spacetime is impossible, and so on.
Are there resources in the literature that deal with geometrically impossible spacetime or offering a general result to be used and checked?
$\begingroup$Looks like there is a result which says on non-compact maniforld Lorentzian metric is possible, and on compact iff the Euler characteristic is zero, are there any more relatd results? mathoverflow.net/questions/44861/…$\endgroup$
