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
A Cauchy surface can possess corners, and thereby need not be a differentiable submanifold of the spacetime; it is however always continuous (and even Lipschitz continuous). By using the flow of a vector field chosen to be complete, smooth, and timelike, it is elementary to prove that if a Cauchy surface $S$ is $C^{k}$-smooth then the spacetime is $C^{k}$-diffeomorphic to the product $S × R$, and that any two such Cauchy surfaces are $C^{k}$-diffeomorphic.

Robert Geroch proved in 1970 that every globally hyperbolic spacetime has a Cauchy surface $S$, and that the homeomorphism (as a $C^{0}$-diffeomorphism) to $S × R$ can be selected so that every surface of the form $S × \{a\}$ is a Cauchy surface and each curve of the form $\{s\} × R$ is a continuous timelike curve.

Various foundational textbooks, such as George Ellis and Stephen Hawking's The Large Scale Structure of Space-Time and Robert Wald's General Relativity, asserted that smoothing techniques allow Geroch's result to be strengthened from a topological to a smooth context. However, this was not satisfactorily proved until work of Antonio Bernal and Miguel Sánchez in 2003. As a result of their work, it is known that every globally hyperbolic spacetime has a Cauchy surface which is smoothly embedded and spacelike. As they proved in 2005, the diffeomorphism to $S × R$ can be selected so that each surface of the form $S × \{a\}$ is a spacelike smooth Cauchy surface and that each curve of the form $\{s\} × R$ is a smooth timelike curve orthogonal to each surface $S × \{a\}$.