Let $(M,g)$ be a connected time-oriented Lorentzian manifold.
$M$ is called globally hyperbolic, if $M$ 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 smooth spacelike Cauchy hypersurface in $M$ for each $t \in \mathbb{R}$.
I know that this is only one of a few equivalent definitions for global hyperbolicity, but I think this is the one I need for my question.
Then, in "Global existence theorems for hyperbolic harmonic maps", regularly hyperbolic manifolds are defined as follows:
A manifold $(M,g)$, where $M= \mathbb{R} \times S$, is called regularly hyperbolic, if it holds
- The induced metrics $g_t$ on $S_t := \{t\} \times S$ are Riemannian and uniformly equivalent to $g_0$, which is complete.
- Let $X$ be the tangent vector to the lines $\mathbb{R} \times \{x\}$ and $n$ the unit time-like normal to $S_t$ and define $N:= g(X,n)$. Then $X$ is time-like and there exist numbers $a,b>0$, s.t. \begin{align*} \inf\limits_M g(X,X) \geq a \text{ and } \sup\limits_M N \leq b \end{align*}
- Define the metric \begin{align*} \gamma:= (dx^t)^2- g_0 \end{align*} where $t$ denotes the coordinate in $\mathbb{R}$. Then the Riemann curvature of $g$ and its derivatives are bounded in the $\gamma$-norm.
My problem is: I am not quite sure if in the second definition, one already assumes that $(M,g)$ is globally hyperbolic, since in the introduction in the source paper, it says something like
"In this article we proof a local existence theorem for harmonic maps from a globally hyperbolic manifolg $(M,g)$ into...."
or if the definition given for regular hyperbolicity already implies, that $(M,g)$ is globally hyperbolic. As far as I see, this is not true, since $S_t$ is not necessarily a Cauchy hypersurface. Is that correct?