digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

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 $f:M \rightarrow \mathbb{R}$ 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 $M$.

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Geroch's_splitting_theorem — Please support Wikipedia.
A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.

Youtube says it doesn't have anything for Geroch's splitting theorem.

We're sorry, but there's no news about "Geroch's splitting theorem" right now.

 Limit to books that you can completely read online Include partial books (book previews) .gsc-branding { display:block; }

Oops, we seem to be having trouble contacting Twitter

# Talk About Geroch's splitting theorem

You can talk about Geroch's splitting theorem with people all over the world in our discussions.