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I was dabbling in hyperbolic/spherical geometry when I had the thought, "Why does an ant walking on a spherical plane have to come back to the same point it started on?" I knew that the answer is because it lives on a sphere(-ical plane), but, hypothetically, what if it wasn't a sphere?

I abstractly thought of a plane that is infinite like the Euclidian and hyperbolic plane but has constant positive curvature like the spherical plane. I am unsure if this 2-space can be embedded in Euclidian 3-space.

This space is different from the spherical plane because an ant walking on the plane will not cross its own path or walk where it has walked before. In relation to Euclid's 5th postulate, given two lines that are both perpendicular to a third line, if they are extended far enough in one direction (as in not backwards), then they will intersect only once (as opposed to twice in the spherical plane).

I feel these differences are big enough that it should have a name to it, but small enough that it should've already been conceived, so that makes me question if it could even be possible.

Question 1: Would such a thing exist (mathematically of course), or is there some formula/logic barrier that directly implies that any space with constant positive curvature must always have a line run into itself?

Question 2: If the previous question allows for this space to exist, has this already been thought of and given a name to?

ThePretzelMan
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    Since you have a constant sectional curvature $\kappa>0$ complete (sub)manifold, you know its universal cover must be the model space $\mathbb{M}^n_\kappa=\mathbb{S}^n(\kappa^{-1/2})$ so the claim follows. – user10354138 Jul 19 '21 at 08:46
  • IIRC, we can remove a pair of antipodal points from a round sphere and form the "universal cover"; the resulting surface has constant curvature, a single geodesic of infinite length, and (most of it) can be immersed as a surface with helicoidal symmetry. The hitch is, the surface is not complete.... – Andrew D. Hwang Aug 03 '21 at 14:33

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There is no such thing as an "infinite spherical plane", and in fact this non-existence holds under rather broad hypotheses.

Myer's Theorem says that every complete Riemannian manifold $M$ whose Ricci curvature is bounded below by a positive constant has finite diameter. In particular if $M$ is a Riemannian $n$-manifold of constant positive sectional curvature, then its Ricci curvature is also constant and positive, and hence $M$ has finite diameter.

Lee Mosher
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