Reference: In every free homotopy class is a unique minimizing closed geodesic

Question

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?

Answer 1

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.

Answer 2

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

Answer 3

First chapter of Farb and Margalit : A primer on mapping class group.