Under what conditions splitting (e.g. $m + p$ foliation of a $m + p$ dimensional manifold) of a general pseudo-Riemannian manifold (with any arbitrary signature) possible? If it is too general then I am happy knowing the answer for manifolds with Euclidean and Lorentzian signatures.
If a manifold doesn't admit this type of foliation, hence we cannot specify field data on "spacelike" surfaces, can we still do field theory on that manifold?
There is a proof that we can always 3+1 foliate on a finite patch of a (smooth) spacetime. How large that patch is depends on the spacetime, of course, but if a caustic shows up, the expectation (I don't know if this expectation is just conjecture or if there is a formal proof), is that one could take the finite-volume slice from the initial evolution, re-foliate it in a way that deals with the caustic in the foliated coordinates, evolve the spacetime further, and eventually get N patches over the whole spacetime for some finite (or at least countable) N.
