I'm having trouble understanding exactly what this manifold is. Let me draw an analogy: Say I have a flat map of the world. The map is a two-dimensional surface with a coordinate chart on it. However, its embedded in a higher three-dimensional space. So by analogy, is the four dimensional spacetime manifold of Einstein equivalent to our three spatial dimensions ("the map") that is embedded in a higher fourth time dimension, or are all four dimensions and related coordinates (x,y,z,t) part of the surface of the "map"? If that were the case, and the "map" is curved, then does that mean there must be at least five dimensions?