TITLE: Looking for theorem of form: If [curvature hypotheses, etc] then every class in $\pi_1(M)$ is uniquely represented by a closed geodesic
QUESTION [9 upvotes]: I recall a conversation in which somebody described to me a result that sounded something this:

Let $M$ be a closed Riemann manifold satisfying [some hypotheses, most significantly some kind of negative curvature hypothesis]. Then, every class in the fundamental group $\pi_1(M)$ is represented by exactly one unit speed geodesic loop.

There are some technical issues with the above statement (e.g. what about basepoints?), but I just wanted to convey the essence of the statement. Now at the time of the conversation, I was very impressed by this result and since then I have had it in the back of my mind to look up this result and at least understand where something like this could come from. I finally got around to this, but ran immediately ran into the issue that I couldn't remember the exact statement of the theorem, nor could I find a result like it by searching for relevant terms.

Question: Does somebody know of a result resembling what is stated above? If so, I would very much appreciate a precise statement, and a reference. 

Possibly there is no result like this, and I misunderstood what was being said in this conversation.

REPLY [3 votes]: I did some background reading and understand what's going on a little now. Let's consider the based version of the result as in Lee Mosher's answer. I follow his lead by only considering geodesics which are constant speed and parametrized by the unit inverval. 

Theorem: Let $M$ be a complete Riemann manifold with non-positive sectional curvature and choose a basepoint $p \in M$. Then, every class in $\pi_1(M,p)$ is represented by exactly one geodesic.

This result sounds very surprising at first, but becomes less so once one knows about the Cartan-Hadamard theorem which essentially says that if a complete Riemann manifold of non-positive sectional curvature is simply-connected, then it is diffeomorphic to $\mathbb{R}^n$ via any of its exponential maps. This at least sounds plausible, since the non-positive curvature hypothesis should make the geodesics "spread out", but I was still rather amazed to learn this result. In particular, this result gives us:

Corollary of C-H: Each pair of points in a simply-connected, complete Riemann manifold of non-positive sectional curvature is connected by precisely one geodesic.

Using the universal cover, it is not difficult to see this corollary proves the theorem we're after. First, lifting the Riemann structure of $M$ to the universal cover $\widetilde M$ gives us a simply-connected, complete Riemann manifold of non-positive sectional curvature. By lifting, every homotopy class of loops based at $p$ in $M$ is in bijective correspondence with the collection of all paths from $\widetilde p$ to $\widetilde p'$ where $\widetilde p, \widetilde p' \in \widetilde M$ are two lifts of $p$. This correspondence preserves geodesics, so the above corollary gives us the theorem directly.