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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