Consider a manifold, which is locally isomorphic to 3-sphere (three-dimensional sphere). Does it have to have the topology of 3-sphere? In particular, can it have a geodesic, which doesn't loop on itself?
By "isomorphism" I mean a function that preserves distances, so that our manifold has a curvature of a sphere at any given point.
Clearly that isn't true for a 1-sphere (a circle) -- a circle is locally isomorphic to $\mathbb{R}$, and yet $\mathbb{R}$ has different topology, and has an unbounded geodesic.
That is true for 2-sphere, although I have no proof. I seem to remember that 3-sphere behaves more like 1-sphere though.
Motivation of the question: I saw a claim that "If the Universe has positive curvature, it loops on itself". But, does it?
The Killing-Hopf theorem says that your manifold most be a quotient of the 3-spehere, and for that reason all geodesics have to loop on itself.