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.
Let ( M , g a b ) be a globally hyperbolic spacetime. Then ( M , g a b ) is strongly causal and there exists a global "time function" on the manifold, i.e. a continuous, surjective map f : M → R such that:
For all t ∈ 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 × R where S is any Cauchy surface of M .
