Globally hyperbolic Lorentzian manifold I am currently trying to work through properties of globally hyperbolic Lorentzian manifolds and there are some things which aren't clear to me:
I have the following definition of globally hyperbolic:
Let $(M,g)$ be a connected time-oriented Lorentzian manifold. Then$(M,g)$ is called globally hyperbolic, if one of the following equivalent conditions hold:

*

*There exists a Cauchy hypersurface in $M$.

*There exists a smooth spacelike Cauchy hypersurface in $M$.

*$M$ is isometric to $\mathbb{R} \times S$ with metric
\begin{align*}
- Rdt^2 +\sigma_t
\end{align*}
where $R$ is a smooth positive function, $(S, \sigma_t)$ is a Riemannian manifold, $\sigma_t$ depending smoothly on $t$. Moreover, $\{t\} \times S$ is a Cauchy hypersurface in $M$ for each $t \in \mathbb{R}$.

*$M$ satisfies the strong causality condition and for all $p,q \in M$ s.t. $p<q$, it holds that $J^+(p) \cap J^-(q)$ is compact.

where a Cauchy hypersurface is defined as:
$S$ is called a Cauchy hypersurface, if it is met exactly one by exery inextendible timelike curve $\gamma$ in $M$.
First question: what does "smooth Cauchy hypersurface" mean? In the sources I use, this is never defined.
Second question: As far as I understand, a Cauchy hypersurface is not necessarily spacelike, right? And if we look at the Cauchy hypersurface from 3. this one is spacelike, because we have a Riemannian metric, right?
 A: The definition of a Cauchy hypersurface you cited only implies that it is a continuously ($C^0$) embedded submanifold. This leads to $M$ being only homeomorphic to $\mathbb{R}\times S$. That the definition of global hyperbolicity (your item 4. is the actual original definition) is equivalent to the existence of a ($C^0$) Cauchy hypersurface and the topological splitting $M=\mathbb{R}\times S$ is the classical result proven by Geroch in the 1970s.
On your questions: The Cauchy hypersurface is smooth if it is a smoothly embedded submanifold (usually $C^\infty$). This is an additional condition for the Cauchy hypersurface. The Cauchy hypersurface is spacelike if additionally the restriction of the metric is Riemannian or, equivalently, all inextendible causal curves hit it at most once. This is another additional condition put on the Cauchy hypersurface.
For a long time it was an open question if a globally hyperbolic spacetime actually always admits a smooth spacelike hypersurface and hence if it is actually diffeomorphic to $\mathbb{R}\times S$. This was only proven at the begin of this century by Bernal and Sánchez. You might want to consult their papers for details:

*

*https://arxiv.org/pdf/gr-qc/0306108.pdf

*https://arxiv.org/pdf/gr-qc/0401112.pdf

*https://arxiv.org/pdf/gr-qc/0512095.pdf
