Given a global hyperbolic space-time $M$, Geroch's work implies existence of a three dimensional Cauchy surface $\Sigma$ (the 1+3 splitting of space-time). Then,
Given a metric $g_{\mu\nu}$ in $M$, are there any known ways to construct $\Sigma$ using mathematical objects derived entirely from the metric?
If yes, then are such constructions generally covariant (tensorial)?
I think they are completely independent concepts. In particular, I think you can study foliations by talking about sets and maps between sets (think about how you would foliate $\mathbb{R}^2$, it is just a bunch of straight lines with a random slope).
This does not mean, however, it is not useful to use the metric to define a foliation. For example, one could compute the Killing vectors, and for each of them define a leave as the surface with the Killing vector as its normal vector.
I have not studied this in detail, but I would say that following this procedure your foliation may have nicer properties than an arbitrary foliation.
