Interpretation of the Poisson's summation formula on $S^1$ by the geodesic flow

Question

I am reading Marklof's notes on the Selberg's trace formula and I have a question concerning the Poisson summation formula on $S^1$:

$$\sum_n h(m)=\sum_n \int_{\mathbb R} h(\rho)e^{2\pi in\rho}d\rho,$$

where $h$ satisfies some good decay properties.

The author claims that the RHS in turn has a geometric interpretation as a sum over the periodic orbits of the geodesic flow on $S^1$, but I don't see why. Can anyone explain this in more details for me?

Answer 1

The geodesic flow on a Riemannian manifold $M$ (say geodesically complete) refers to a flow defined on the tangent bundle $\operatorname{T}(M)$ of a manifold in the following way:

1. Take a tangent vector $v \in \operatorname{T}(M)$. We want to flow this by time $t$.
2. There is a unique unit speed geodesic $\phi: \mathbb R \to M$ such that $\phi'(0) = v$.
3. Define $\phi'(t)$ to be the flow of $v$ by time $t$.

In other words, move $v$ forward along a "straight line" through $v$ by time $t$.

For example, consider the embedded unit circle $S^1 \subset \mathbb R^2$ with the induced metric. Consider the tangent vector $((1, 0), (0, 1))$ which is just a vector pointing up based at $(1, 0) \in S^1$. If we flow this by time $\frac{\pi}{2}$, we end up with the tangent vector $((0, 1), (-1, 0))$ which is the vector pointing left based at $(0, 1) \in S^1$.

Notice that the geodesic flow preserves the length of the tangent vectors. So it is hopeless to try and say anything about ergodicity and mixing on $\operatorname{T}(M)$ because tangent vector of some length cannot flow to a tangent vector of a different length. Hence in dynamics, it makes sense to restrict the flow to the unit tangent bundle $\operatorname{T}^1(M) \subset \operatorname{T}(M)$ consisting of unit tangent vectors.

Specifically in the 1-dimensional case like $S^1$, there are only two unit tangent vectors based at each point (one pointing in each direction). In this case, the orientation of the tangent vectors are preserved under the geodesic flow, so it makes sense to restrict to unit vectors in only one of the directions. With this further restriction, there is a unique tangent vector over each point in $S^1$. Thus we can simply identify the two. Now with this indentification, if you go back and unravel the definitions again, the flow by time $t$ on $S^1$ is just rotation by angle $t$. In complex notation, the rotation is $e^{it}$.