On a Riemannian manifold $M$, a geodesic is defined as a minimum of the energy functional. The Riemannian metric on $M$ makes $M$ into a metric space with metric $d$, say. We know that any geodesic $\gamma:I\rightarrow M$ is locally length minimising, i.e. if $t\in I$ there is $\epsilon >0$ such that $$|u-t|<\epsilon \Rightarrow d(\gamma(u),\gamma(t)) = |u-t|.$$ The Lusternik-Fet theorem says that any compact manifold has a non-trivial closed geodesic.
For a closed geodesic $\gamma:[0,T] \rightarrow M$, with $\gamma(0)=\gamma(T)$, I define it to be globally length minimising if for any $u,t\in[0,T]$ $$d(\gamma(u),\gamma(t)) = |u-t| \mod T.$$ Granted, this is not truly globally length minimising, as the distance might be greater travelling the geodesic one way than the other. So if you know of another name for this property, or can suggest a better one please let me know.
My question is: does every compact Riemannian manifold admit a non-trivial globally length minimising geodesic?
As I am writing this, I have realised another MSE question asks whether every compact Riemannian manifold admits a non-self-intersecting closed geodesic. If I am to trust the answer, it looks like this is an open problem. A self-intersecting closed geodesic cannot be globally length minimising, and therefore it seems like my question is also an open problem.
Unless there is a counterexample? I would be very interested to see one. Any other thoughts on this are appreciated!
