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  is -smooth then the spacetime is -diffeomorphic to the product , and that any two such Cauchy surfaces are -diffeomorphic.

Robert Geroch proved in 1970 that every globally hyperbolic spacetime has a Cauchy surface , and that the homeomorphism (as a -diffeomorphism) to  can be selected so that every surface of the form  is a Cauchy surface and each curve of the form  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  can be selected so that each surface of the form  is a spacelike smooth Cauchy surface and that each curve of the form  is a smooth timelike curve orthogonal to each surface .

References

Sources

 
 
 
 
 
 

Theorems in general relativity
Lorentzian manifolds
Theorems in mathematical physics