Let \((M,g)\) be a complete Riemannian manifold and \(\sigma:[0,1]\to M\) a curve between \(p,q\). Then there exists a minimizing geodesic between \(p,q\) that is homotopic to \(\sigma\).
Pass to universal cover. ∎
Closed geodesic.
Assume that \(M\) is compact. Let \(\sigma:S^{1}\to M\) be a continuous map. Then there exists a closed geodesic that is freely homotopic to \(\sigma\).
Example if \(M\) is not compact.
If \(\pi_{1}(M)\neq 1\), then there are closed geodesics.
Every compact Riemannian manifold has at least one closed geodesic.
It is a conjecture whether there are even infinitely many closed geodesics.