# How to think about elliptic (positive curvature) space? Is it really a sphere?

I have been interested in hyperbolic (negatively curved) space and I have been reading enough about it to feel that I understand it relatively well intuitively (e.g. the Poincare disc). But the problem is when I flip the curvature sign in my head to positive curvature and elliptic space nothing seems to make sense.

For instance:

• In elliptic space, locally "parallel" lines will converge and intersect given a great enough distance, but if there is really a positive curvature, then it seems like the lines will eventually converge after the intersection to intersect again. This leads to the result that the lines will intersect an infinite number of times. Is this correct? I cannot find the right search terms to find this answer.
• Is elliptic space really a sphere? Does the positive curvature mean that if you travel far enough, you will end up back at the starting point? If so, then it seems to imply that elliptic space is finite, where Euclidean and hyperbolic space are infinite, which seems wrong.

I likely have some misunderstanding of elliptic space, so If you can correct me where I may have gone wrong, please do. Thanks

The Riemannian manifolds of constant curvature fall into one of the following three cases:

1. elliptic geometry – constant positive sectional curvature

2. Euclidean geometry – constant vanishing sectional curvature

3. hyperbolic geometry – constant negative sectional curvature.

Theorem (Killing–Hopf): The universal cover of a manifold of constant sectional curvature is one of the model spaces:

1. Sphere (sectional curvature positive)

2. Euclidean space (sectional curvature zero)

3. Hyperbolic manifold (sectional curvature negative)

Reference: J.A. Wolf, "Spaces of constant curvature" , Publish or Perish (1977)

• does that then mean that the size of an elliptic space is determined by the amount of curvature? It seems that a "stronger" curvature would yield a smaller sphere and therefore a smaller space.
– Joff
Commented Aug 14, 2023 at 12:44
• Take the circle $S^1$. It has constant curvature $1/r$, where $r$ is the radius. So, if the circle is large, its curvature $1/r$ is small. If the circle is small, $1/r$ is then large, so the curve is more curved. Commented Aug 14, 2023 at 13:16
• I see. So as far as I can tell, my intuition was correct about this. Very rare! Thanks
– Joff
Commented Aug 15, 2023 at 3:22