Spacetime interval and basic properties of light

[Ibix]

I think Carroll means the surface of a hyper-cylinder (or whatever the name for it is), where every $\mathbb{S}^2$ is identical and stacked one on top of another, not ever-smaller spheres nested one inside the other, which is what I think you mean. Every point is identical (diffeomorphic to any other), but not every direction.

 

[cianfa72]

I think Carroll means the surface of a hyper-cylinder (or whatever the name for it is), where every $\mathbb{S}^2$ is identical and stacked one on top of another, not ever-smaller spheres nested one inside the other, which is what I think you mean. Every point is identical (diffeomorphic to any other), but not every direction.

Yes, if we take in account (as required) the metric. However --from just a topological point of view-- I believe the 'infinite spherical onion with internal hole' might make sense though.

Ah ok, on the hypercylinder surface the homogeneity is 'encoded' by the diffeomorphism between every pair of points on it.