Suppose that $\langle M, g\rangle$ is a Lorentzian manifold, and that $\xi$ is a timelike vector in $T_pM$, at some point $p \in M$.
Let $S$ be a surface consisting of points that lie on some geodesic passing through $p$ orthogonal to $\xi$. In general, I believe, this surface will be space-like in at most a neighborhood of $p$.
I would like to know: if $M$ is globally hyperbolic (i.e. it contains at least one Cauchy surface), does it follow that $S$ is everywhere spacelike?
Also, I'd like to know if $S$ is everywhere spacelike, does it follow that $S$ is a Cauchy surface?
I'd be interested to know what counterexamples to these claims are like, if there are any. Any help would be greatly appreciated!