First of all, there exists a question on PSE which does seem to pertain to my question below, but not exactly. This is one of those questions which requires perhaps an intuitive rather than mathematical answer.
From a general perspective on curvature, it is known that converting a curved surface (embedded in some higher dimensional manifold) into a flat surface one meets a discontinuity. Explaining with the canonical example: Take a spherical surface, tear it apart and then flatten it up on a plane. We observe that there are discontinuities which on a first course of GR is pointed as an evidence of existence of curvature.
If I push this idea forward, it tells me that there is angular deficiency in converting one surface from negative curvature to zero and further to positive curvature. To ensure a continuous change, I'd need to supply my manifold (in this case the spherical surface we took) with more "material" to cover up the empty spaces generated.
Einstein's equation demand that curvature of spacetime is always evolving, dynamically responding to all forms of energy-momenta.
Here is the problem, then:
If my earlier remarks on discontinuity in variations of curvature are - in generally - true, then is the continuous dynamic evolution of spacetime curvature continuous? If yes, how?