# Can the universe be round but still infinite?

1. Can the universe still be infinite in space if its curvature is > 1?

2. Is a manifold of positive curvature necessarily compact?

3. Does the Tarski paradox have any bearing on the finite or infinite character of a compact manifold?

• Would Mathematics be a better home for this question? Replace the word universe with the word manifold, and it is a pure mathematical question. – Qmechanic Apr 12 '15 at 13:33
• But here part of the question is: are these the right mathematics to answer this question or is there a more appropriate framework to discuss a possible answer. – John Shedletsky Apr 13 '15 at 21:34
• In particular, I believe Tarski's may have no bearing. – John Shedletsky Apr 13 '15 at 21:34
• Subquestion 2 (v2) has trivial counterexamples. Subquestion 3 (v2) is a possible duplicate of physics.stackexchange.com/q/20370/2451 – Qmechanic Apr 13 '15 at 21:46
• @Qmechanic Would you care to give some details ?- I'd certainly upvote such an answer - In your counterexamples, do you mean curvature bounded from below by a positive $\epsilon>0$ or can the curvature be locally nought in your examples: I have read that Bonnet-Myers theorem has some kind of analogue in a Lorentzian manifold but would like to understand this further. Also, I think this question, although mathematical, has direct physical interest, especially for the cosmologically naive like me. – Selene Routley Apr 16 '15 at 6:07