# Geroch's splitting theorem

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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 . Template:Asbox Template:Differential-geometry-stub