Considering closed geodesics on a compact manifold M of even dimension, what does it mean to say that a curve (any closed geodesic) is locally energy minimizing but not globally ? For simplicity, say $M=S^2$. Intuitive answers would help rather than equations/semantics.
EDIT: What I am looking for is an example of a closed curve on a manifold, which is a local geodesic(locally energy minimizing) but not globally. I understand that $S^2$ might be a bad example because intuitively I think all geodesics are local and global minimizers of energy (please correct me If I am wrong) but in any case, I think a torus might be a great example to find one, especially in light of this article:
Shall give an example. Take a cone, start at point near vertex and wind a string around tautly, returning to the same point after it nears a minimum distance at vertex and gets back.
Repeat the same with starting angle 90 degrees to the generator, the string will just go away from vertex along a divergent path of no-return.
Both are geodesics. The first one is closed, minimizes associated energy.The energy of the diverging geodesic is more than the first. Accordingly it is necessary to distinguish the local energy minimizing path from the other global possibilities. That is to say, local minimum is minimum among all energy/length minimizing paths.