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Geroch's splitting theorem
In the theory of causal structure on Lorentzian manifolds, Geroch's theorem or Geroch's splitting theorem (first proved by Robert Geroch) gives a topological characterization of globally hyperbolic spacetimes.
The theorem
Let \( (M, g_{ab}) \) be a globally hyperbolic spacetime. Then \( (M, g_{ab}) \) is strongly causal and there exists a global "time function" on the manifold, i.e. a continuous, surjective map \( f:M \rightarrow \mathbb{R} \) such that:
For all \( t \in \mathbb{R}, f^{-1}(t) \) is a Cauchy surface, and
f is strictly increasing on any causal curve.
Moreover, all Cauchy surfaces are homeomorphic, and M is homeomorphic to \( S \times \mathbb{R} \) where S is any Cauchy surface of M.
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