Reference: In every free homotopy class is a unique minimizing closed geodesic Does anyone know a reference for the following result:
Let $M$ be a compact hyperbolic manifold/manifold with strict negative curvature . Then in every non-trivial free homotopy class of $M$ there exists a unique minimizing closed geodesic.
The existence is proofed for example in Do Carmo's Riemannian Geometry, for which the negative curvature is not needed.
But for the uniquness I've not yet found a reference.
Can anyone help?

I have found a reference, if someone is interested:
in The Geometry and Topology of Three-Manifolds, William P. Thurston, S.88.
Here is an online version of chapter 5:
http://library.msri.org/books/gt3m/PDF/5.pdf
