The Lyusternik-Fet theorem states that every compact Riemannian manifold has at least one closed geodesic.
Are there any easy-to-construct1 examples of compact Riemannian manifolds for which it is easy to see they only have one closed geodesic?2
If there aren't any such examples, are there any easy-to-construct examples that only have one closed geodesic but where proving this might be difficult?
And if there aren't any examples of this, are there any examples at all of compact manifolds with only one closed geodesic?
1 Of course, the $1$-sphere $S^1$ contains just one closed geodesic, but I'm interested in examples besides this one.
2 By the theorem of the three geodesics, this example cannot be a topological sphere.