# Geroch's splitting theorem Geroch's splitting theorem This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Geroch's splitting theorem" – news · newspapers · books · scholar · JSTOR (February 2020) (Learn how and when to remove this template message) 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:Mrightarrow mathbb {R} } such that: For all {displaystyle tin 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 Stimes mathbb {R} } where {displaystyle S} is any Cauchy surface of {displaystyle M} .

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