Let M be a Riemannian homogeneous space, i.e. the isometry group acts transitively. Prove: any geodesic loop (with possible angle at the starting point) is a closed geodesic (smooth at the starting point).
And there is a hint: prove and use the fact that the vector field associated to a one parameter group of isometries is a Jacobi field when restricted to any geodesic.
I just don't know how this hint comes into play..