Minimal ideal

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 $(M,g_{ab})$ be a globally hyperbolic spacetime. Then $(M,g_{ab})$ is strongly causal and there exists a global "time function" on the manifold, i.e. a continuous, surjective map $f:M\to \mathbb{ℝ}$ such that:

• For all $t\in \mathbb{ℝ}$, $f^{-1}(t)$ is a Cauchy surface, and
• $f$ is strictly increasing on any causal curve.

Moreover, all Cauchy surfaces are homeomorphic, and $M$ is homeomorphic to $S\times \mathbb{ℝ}$ where $S$ is any Cauchy surface of $M$. Travel Agency Manager Alfonzo from Manotick, has numerous hobbies including comics, commercial property developers in singapore developers in singapore and creating a house. Recollects what an incredible area it was having traveled to Tubbataha Reefs Natural Park. Template:Differential-geometry-stub