  We explore some fundational issues regarding the splitting of D-dimensional
EFE's w.r.t timelike and spacelike (D-1)-dimensional hypersurfaces, first
without and then with thin matter sheets such as branes. We begin to implement
methodology, that is well-established for the GR CP and IVP, in the new field
of GR-based braneworlds, identifying and comparing many different choices of
procedure. We abridge fragmentary parts of the literature of embeddings,
putting the Campbell--Magaard theorem into context. We recollect and refine
arguments why York and not elimination methods are used for the GR IVP. We
compile a list of numerous mathematical and physical impasses to using timelike
splits, whereas spacelike splits are known to be well-behaved. We however
pursue both options to make contact with the current braneworld literature
which is almost entirely based on timelike splits. We look at the
Shiromizu-Maeda -Sasaki braneworld by means of reformulations which emphasize
different aspects from the original formulation. We show that what remains of
the York method in the timelike case generalizes heuristic bulk construction
schemes. We formulate timelike (brane) versions of the thin sandwich
conjecture. We discuss whether it is plausible to remove singularities by
timelike embeddings. We point out how the braneworld geodesic postulates lead
to further difficulties with the notion of singularities than in GR where these
postulates are simpler. Having argued for the use of the spacelike split, we
study how to progress to the construction of more general data sets for spaces
partially bounded by branes. Boundary conditions are found and algorithms
provided. Working with (finitely) thick branes would appear to facilitate such
a study.
