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.
Let
(
M
,
g
a
b
)
be a globally hyperbolic spacetime. Then
(
M
,
g
a
b
)
is strongly causal and there exists a global "time function" on the manifold, i.e. a continuous, surjective map
f
:
M
→
R
such that:
For all
t
∈
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
×
R
where
S
is any Cauchy surface of
M
.
