[danbruc]

But in general relativity extrinsic curvature is the irrelevant thing, it's all about intrinsic curvature. We don't think of space-time being embedded in a higher dimensional space but we still talk about curvature of space-time, its intrinsic curvature.

These demonstrations with balls rolling on a stretchable rubber sheet to visualize gravity are really misleading in this regard because they use a good deal of extrinsic curvature to make things work but in reality mass doesn't deform space-time into a fifth dimension or at least it doesn't necessarily do so.