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For any point $P$ on a sphere $S$, every line (geodesic?) containing $P$ is closed, i.e. wraps around $S$ and passes through $P$ "again."

1) Are there other objects besides spheres for which this property holds? Maybe a torus?

2) What tools would one use to do this type of analysis? For example, how would one prove for spheres that every line is a closed curve?

Thank you! Please feel free to correct my vocabulary or suggest some terminology, as I am new to the subject of non-Euclidean geometry.

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Wouldn't any polyhedron posess that property? – cirpis Jul 4 '14 at 19:34
A starting point: – Qiaochu Yuan Jul 4 '14 at 19:49
@cirpis Good question! The spirit of my question was about smooth surfaces, but now I'm curious about polyhedra too. This should be pretty easy to solve by "unfolding" the poly. Thanks! – ajnelson.alpha Jul 4 '14 at 20:06
A counterexample for the torus is as follows. Think of the torus as $\mathbb{R}^2 / \mathbb{Z}^2$. The geodesics on this thing (with the flat metric) are straight lines. A straight line passing through the origin is a closed geodesic iff it meets an element of $\mathbb{Z}^2$ iff its slope is rational. So any straight line with irrational slope is a counterexample; this in fact shows that "most" geodesics on a torus aren't closed. (They do, however, get arbitrarily close to their starting points; I believe this property is called being recurrent but I could be wrong.) – Qiaochu Yuan Jul 4 '14 at 20:06
@QiaochuYuan thanks for the link and counterexample! I had to wrap my head around the meaning of $\mathbb{R}^2 / \mathbb{Z}^2$, group theory was a few years ago haha – ajnelson.alpha Jul 4 '14 at 20:19
up vote 2 down vote accepted

Question 1: Yes, there are other examples, but a torus isn't one of them. The simplest non-sphere example is the real projective plane with a constant-curvature metric. Understanding all such spaces is a very interesting problem, and as far as I know it's unsolved.

Question 2: This problem requires quite a lot of sophisticated tools of Riemannian geometry, symplectic geometry, differential equations, algebraic topology, and probably other fields as well. There's a whole book about the topic: Manifolds All of Whose Geodesics Are Closed by Arthur Besse (a pseudonym for a group of prominent French differential geometers).

EDIT: fixed link.

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Your link seems to require University of Washington login information. – Mike Miller Jul 4 '14 at 19:50
Sorry about that. Fixed now. – Jack Lee Jul 4 '14 at 19:54
Thank you!${}{}$ – Mike Miller Jul 4 '14 at 19:55
Thanks professor! – ajnelson.alpha Jul 4 '14 at 20:22

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