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 [math]\displaystyle{ (M, g_{ab}) }[/math] be a globally hyperbolic spacetime. Then [math]\displaystyle{ (M, g_{ab}) }[/math] is strongly causal and there exists a global "time function" on the manifold, i.e. a continuous, surjective map [math]\displaystyle{ f:M \rightarrow \mathbb{R} }[/math] such that:

• For all [math]\displaystyle{ t \in \mathbb{R} }[/math], [math]\displaystyle{ f^{-1}(t) }[/math] is a Cauchy surface, and
• [math]\displaystyle{ f }[/math] is strictly increasing on any causal curve.

Moreover, all Cauchy surfaces are homeomorphic, and [math]\displaystyle{ M }[/math] is homeomorphic to [math]\displaystyle{ S \times \mathbb{R} }[/math] where [math]\displaystyle{ S }[/math] is any Cauchy surface of [math]\displaystyle{ M }[/math].