Arguably, this is pure mathematical question, but most discussions of semi-riemannian manifolds are in the context of physics, so I post here. Can anyone state or point me to references discussing best known answers to the following: Given an arbitrary Semi-Riemannian 4-manifold, and an arbitrary open 4-d subset of it, under what conditions on the subset (e.g. orientable, torsion free, not closed, metric condition ....???) is it possible to achieve the following (on the subset): 1) The subset can be covered with some family of non-intersecting 3-surfaces on which the induced metric is Euclidean flat. 2) The subset can be covered with some family of non-intersecting 2x1 Minkowski flat 3-manifolds. ---- This is sort of a converse question the following example that the geometry of embedded surfaces can be very different from the geometry of the space they are embedded in: Flat Euclidean 3-space can be covered, except for one point, with family of non-intersecting 2-spheres.