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

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

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