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?
 A: This question has already been answered, but I just wanted to point out to others who may arrive at this question while reading do Carmo that this result in fact follows from the results in Chapter 12. 
Theorem 2.2 tells you that if you that every compact manifold has some closed geodesic in a given free homotopy class. 
Proposition 2.6 tells you that given a covering transformation, $\alpha$, of the universal cover $\tilde{M}$ of $M$, with $M$ compact, there is a geodesic which is fixed by $\alpha$. We can associate covering transformations to free homotopy classes in $M$ in the usual way. If you go through the proof of this proposition, you'll see that without changing anything, the proof tells you that in fact for every closed geodesic in this free homotopy class, the lift to $\tilde{M}$ is preserved by $\alpha$ (this is not what the proposition says, but it follows from the proof).
Lemma 3.3 tells you that if $K < 0$ on $M$, then given any covering transformation $\alpha$ of $\tilde{M}$, there is at most one geodesic fixed by $\alpha$.
Put this altogether and you get that if $K < 0$ and $M$ is compact, then there is exactly one closed geodesic in a given free homotopy class (since free homotopy classes correspond to covering transformations and invariant geodesics in $\tilde{M}$ correspond to closed geodesics in $M$ in the corresponding free homotopy class). 
A: 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
A: First chapter of Farb and Margalit : A primer on mapping class group. 
A: Thurston's Work on Surfaces, chapter three.
Here is a link:
https://wikis.uit.tufts.edu/confluence/download/attachments/21933250/flp.pdf
A: I think the following works:
Nonpositive curvature implies that the distance function is convex. So if you have two minimizing geodesics then you actually have a 1-parameter family of minimizing geodesics. Then look at the Jacobi field of such a 1-parameter family, say with one endpoint fixed. In the case of nonpositive curvature the Jacobi equation will yield a contradiction.
