Non-Euclidean Geometry: Objects on which every line is a closed curve, e.g. a sphere

Question

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.

Answer 1

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.