A Lorentzian manifold is called globally hyperbolic if it admits a well-defined time evolution from initial data of physical fields on it.
There are several equivalent definitions of global hyperbolicity. A simple one is:
In this form the characterization of global hyperbolicity appears for instance in the paragraph at the bottom of page 211 in (HE). The equivalence of this to more traditional definitions is (HE, prop. 6.6.3) together with (HE, prop. 6.6.8), due to (Geroch1970). The latter in fact implies the following stronger statement:
So in particular for a globally hyperbolic spacetime there is a homeomorphism
A standard textbook exposition is section 6.6 of
The fact that a single Cauchy surface implies a foliation by Cauchy surfaces is due to
The refinement of this statement to a smooth splitting is in