Geroch's splitting theorem

From HandWiki

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

  • For all t, f1(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× where S is any Cauchy surface of M.