If I understand correctly, a geodesic between two points $a$ and $b$ is the "most direct" path from $a$ to $b$. Geodesics on a plane are straight lines, geodesics on a sphere are great circle arcs. ...
There's a lot of work about the existence, number and other properties of closed geodesics on a Riemannian manifold (belonging to some specific class of manifolds). In the case of geodesics ...