AIUI, strictly speaking, the universe could be "stretched on something". The manifold could be embedded in a higher dimensional space, like the 2D surface of a

AIUI, strictly speaking, the universe could be "stretched on something". The manifold could be embedded in a higher dimensional space, like the 2D surface of a balloon being embedded in 3D space. However, the maths is the same whether it's embedded like that (in any of an infinity of possible ways) or the manifold is all there is. My understanding is that physicists assume the latter simply because there's no practical benefit to doing otherwise, rather than because they have any evidence the universe isn't embedded in a higher space. (I believe this is a different thing to the various higher dimensional cosmological theories that exist - branes and the like - but such things are beyond me.)