Geroch's splitting theorem

From formulasearchengine
Jump to navigation Jump to search

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:

Moreover, all Cauchy surfaces are homeomorphic, and is homeomorphic to where is any Cauchy surface of . Template:Asbox Template:Differential-geometry-stub