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 for this to make sense? 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?

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?