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 be a globally hyperbolic spacetime. Then is strongly causal and there exists a global "time function" on the manifold, i.e. a continuous, surjective map such that:

• For all , is a Cauchy surface, and
• is strictly increasing on any causal curve.

Moreover, all Cauchy surfaces are homeomorphic, and is homeomorphic to where is any Cauchy surface of .