In the context of cosmology, space is commonly described as potentially having a global curvature that can be positive, zero, or negative. A common way that textbooks describe positive curvature is by saying that one example of a 3D space with positive curvature would be the 'surface' of a 3-sphere. In the process of trying to understand this in more detail, I ran into the common question of whether a universe with global positive curvature, if we assumed the curvature was homogeneous on large scales, would need to imply the existence of a fourth spatial dimension for the universe to curve 'into' and from all the answers I have seen, it seems the answer is 'no' - that what we are measuring is the *intrinsic* curvature of space and that it does not imply that there would **have** to be *extrinsic* curvature into an additional dimension (but please correct me if I am wrong!).

What I don't quite understand is the following: if we consider a 2D universe in which inhabitants measured positive curvature that was consistent with what would be measured if their 2D universe lived on top of a 2-sphere, and also noticed that their universe was finite and had no boundaries (such that they could travel around their universe in a 'straight line' and come back to the starting point), could they not prove that there *had* to be a third dimension? In other words, are there 2D manifolds that would have these properties that do not require a third dimension and do not have *extrinsic* curvature? (all the examples of closed 2D manifolds I have found are defined on a 3D surface - sphere, torus, Klein bottle)

If the inhabitants of this 2D universe determined that there had to be a third dimension to make sense of the observed intrinsic curvature and boundary properties of their universe, why would it not also be the case that a 3D finite/closed universe with positive curvature require a fourth dimension? Could one explain a finite 3D universe with positive intrinsic curvature where you can travel in one direction and travel along a 'straight line' and come back to your starting point without a fourth dimension?