I'm learning about meromorphic (!) quadratic differentials on Riemann surfaces, and would like to determine the closed trajectories [EDIT: I mean closed geodesics, not just closed trajectories; trajectories seems to usually mean horizontal geodesics] in some simple cases. I'll give an example that I think I understand, to illustrate what I'm trying to do, and another example that I don't quite understand and for which I'd like advice. Both examples are on the Riemann sphere $\mathbf{CP}^{1}=\mathbf{C} \cup \infty$, where $z$ is a coordinate on $\mathbf{C}$.
Example 1: Consider the meromorphic quadratic differential $q=dz^{2}/z^{2}$, which evidently has a pole of order $2$ at $0 \in \mathbf{C}$ and also at $\infty$. It's easy to check that the horizontal distribution associated to $q$ is just radial lines coming out of $0$, and that the vertical distribution is concentric circles centred at $0$. In particular, those concentric circles give closed geodesics (in fact, trajectories).
Moreover, I claim there are no other closed geodesics. Now, a piece of a closed trajectory through a point $z_{0}$ is the image of a straight line under the inverse of a coordinate defined in a neighbourhood of $z_{0}$. That's well-defined, because the coordinates are well-defined up to translation and multiplication by $-1$, and that doesn't change which lines are straight. Now to compute that coordinate, we take a square root of the quadratic differential and integrate. In a small neighbourhood of $z_{0}$, that will give $\log(z)$ up to a constant, so the inverse will be $\exp(z)$, and the only lines that map to periodic geodesics under $z \mapsto \exp(z)$ are vertical.
Example 2: Consider the quadratic differential $q=dz^{2}/(1-z^2)$, which has poles of order $-1$ at $1$ and $-1$ and a pole of order $2$ at $\infty$. Playing the same game with coordinates, we find that (up to a shift of variable) the inverse of the local coordinate is given by $z \mapsto \sin(z)$ (inverse of contour integral of $dz/\sqrt{1-z^{2}}$). Evidently, horizontal lines map to periodic geodesics (in fact, trajectories) under $z \mapsto \sin(z)$, by the usual periodicity of the $sin$ function, and in fact, those seem to be the only lines that map to closed geodesics .
Are there any other periodic geodesics that I'm missing? Can I not see them by picking a single coordinate like this, but have to patch them together somehow? How to do so, if necessary?
