TITLE: When can we factor out the time dimension?
QUESTION [5 upvotes]: First of all, my knowledge of both GR and differential geometry is quite weak, so forgive me if the physics here doesn't make much sense.
Let $(M, g)$ be a smooth, connected Lorentzian manifold of dimension $n$.  Let $f: \mathbb{R}\to M$ be a smooth curve such that the pullback of $g$ through $f$ is everywhere negative (where we've chosen an orientation on $\mathbb{R}$); we say that $f$ is time-like.  Say that we can "factor" $f$ out of $M$ if there exists a manifold $S$ of dimension $n-1$ and an isomorphism $M\simeq S\times \mathbb{R}$ so that the map $\mathbb{R}\to M\simeq S\times \mathbb{R}\to S$ is constant and the map $\mathbb{R}\to M\simeq S\times \mathbb{R}\to \mathbb{R}$ is the identity.  Intuitively, this factorization exhibits $f$ as "time" in some reference frame, and $S$ as space.  My question is:

For which $(M, g)$ can every time-like path be factored out?

Minkowski space seems like an obvious example unless I'm missing something; it seems one can take a tangent vector to $f$ at any point and consider a perpendicular subspace to that vector as $S$.  I'd accept as an answer a characterization of all such $(M, g)$ in dimension $4$, or some nice sufficient condition on $M$ for factorization to always work.
If the motivation isn't obvious already, this is supposed to codify the intuition that in my reference frame, I seem to be standing still -- and that the same is true for everyone else, even if they seem to me to be moving.  My apologies if I've overloaded terms, or used them incorrectly.

Added:  Note that this condition is much stronger than stable causality; indeed, it certainly implies stable causality, as choosing any timelike path $f$ and then considering the given projection to $\mathbb{R}$ gives a global time function.  However, I am asking for (1) a product structure on $M$ for each path $f$ and (2) in order to formalize the notion that I seem to be standing still (to myself), the projection of $f$ to $S$ must be constant.  

Added:  I don't think global hyperbolicity suffices either.  The theorem of Geroch (it and other splitting theorems are discussed here, for example) does indeed give a decomposition of $M$ as $\mathbb{R}\times S$.  But I don't think this is enough.  In particular, I am asking for the following---for every timelike path $f: \mathbb{R}\to M$, there is a product structure $M\simeq \mathbb{R}\times S$ such that the projection to $\mathbb{R}$ is a section of $f$, and that $f$ is constant upon projection to $S$.  This is much stronger than Geroch's splitting theorem, as far as I can tell.

Added:  As the accepted answerer rightly points out in the comments to his question, I was wrong to claim that my condition is stronger than global hyperbolicity.  They are in fact equivalent.

REPLY [5 votes]: This sounds to me like you're asking that your spacetime admit a family of Cauchy surfaces (modulo annoyances like having $f$ be closed and acausal).   There's a theorem of Geroch which guarantees that this is equivalent to global hyperbolicity.  
I don't think that globally hyperbolic and stably causal are equivalent conditions, but I'm not an expert on causality conditions in GR, so take this claim with a grain of salt.