[vzweazef] See the back of your head

If the universe were "closed", they say if you looked through a telescope powerful enough, you would see the back of your head. Is this really true?

Geodesics on a sphere certainly loop back on themselves, but is this true for all geodesics on all closed manifolds in all dimensions? I'm guessing not; in fact I would not be surprised if "most" geodesics on "most" manifolds do not loop back on themselves: the sphere is special.

Consider a two-dimensional manifold of a unit square with corresponding opposite edges glued together into a torus. A beam of light fired with an irrational slope ("most" slopes) will not loop back on itself.

For any given point on a closed manifold, does there exist some geodesic which loops back to the point?

Update: There is at least one closed geodesic (Lyusternik and Fet).  It's probably not through your given point.