# Are there positively-curved spaces of infinite extent?

Unbounded flat Euclidean spaces can be either infinite (e.g., an infinite plane) or finite--e.g., a flat torus, constructed by starting with a square and identifying opposite edges.

Meanwhile, the most obvious example of a space with constant positive curvature is the sphere, which is finite. And there doesn't seem to be an obvious way to cut a piece out and tile it like you could to go from a closed torus to an open plane. So, are constant-positive-curvature spaces inherently an unavoidably finite, or is it possible to construct an infinite space with constant positive curvature?

• What do you mean by "infinite" and "finite"? Do you mean max distance is finite? Volume is finite? Something else? Oct 8, 2021 at 18:32
• @JasonDeVito It did not occur to me that those could be different things. I suppose I mean area or volume. Can you take an open plane and give it constant positive curvature? Oct 8, 2021 at 18:38
• If you view the plane as, say, the northern hemisphere of a sphere, does that answer your question? For the distance vs volume issue, if your graph is something like the graph of $e^{-x}$ for $x\geq 1$ rotated around the $x$-axis, then volume (which means surface area in this context) is finite, while max distance is infinite. I don't have an example where surface area is infinite but max distance is finite. Maybe that can't happen? Oct 8, 2021 at 18:44