From my very limited experience and understanding, I have come to realize that some people study "closed" geodesics. I understand that to mean that for some a, b x(a)=x(b) for the geodesic curve x. In what situations would a manifold admit a geodesic that is not closed? Perhaps I am just not "seeing" it :-). Thank you for any help.
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There is an additional condition that belongs here. A geodesic is called closed when your $x(a) = x(b)$ AND $x'(a) = x'(b),$ meaning that it makes a closed smooth curve, and keeps going over the same same set of points forever.
It is easy to find manifolds with self-intersecting geodesics that are not closed geodesics.
The traditional example, the surface of revolution $z = x^2 + y^2,$ is an exercise in do Carmo. If a geodesic goes through $(0,0,0)$ it just goes on forever. Otherwise, it intersects itself infinitely often, but is not closed. Here we go, pages 258-260 in Differential Geometry of Curves and Surfaces
I met do Carmo at a conference once. He sent me Celso Costa's dissertation. Which is why there are bound copies in the libraries of U.C. Berkeley and U.C. San Diego.
In the very simple case of the euclidean spaces, the geodesics are the straight lines and they are not closed (null curvature).
In the case of the sphere, geodesics are closed (strictly positive curvature).
I strongly suspect that whenever the curvature is negative, all geodesics cannot be closed (see the Poincaré half plane and its geodesics).