I am struggling with this exercise that my professor of Riemannian Geometry gave to me.
Let S be the compact orientable surface without boundary with genus two. I know that there exists a metric on S with constant negative curvature on S. I have to find an infinite familiy of simple closed geodesics that intersect each other's (i.e., if $(\gamma_n)_{n\in\mathbb{N}}$ is the family of geodesics, for every $n_1\in\mathbb{N}$ there must be an $n_2\in\mathbb{N}$ such that $\gamma_{n_1}$ and $\gamma_{n_2}$ have a point in common).
My idea is the following: define the Poincaré half-plane by
\begin{equation}
\mathbb{H}^2 = \{z\in\mathbb{C} \; | \; \mathrm{Im}(z)>0 \}
\end{equation}
equipped with the usual metric
$$
g
=
\frac{1}{\mathrm{Im}(z)^2}\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}.
$$
I want to find a subgroup $\Gamma$ of the isometry group of $(\mathbb{H}^2,g)$ (which is isomorphic to $\mathrm{PSL}(2,\mathbb{R})$) such that $S=\mathbb{H}^2/\Gamma$ (this should be possible, as far as I know). If the resulting projection $\pi: \mathbb{H}^2 \to S $ is a local isometry, I should be able to study the geodesics of S passing through the geodesics of $\mathbb{H}^2$, but actually I'm quite stuck.
All suggestions, also for similar problems, are welcome.
Edit: Following the suggestion by Moishe Kohan, I think that the following procedure could work. First, consider this two lemmas:
Lemma 1: Let $S$ be a compact and complete Riemannian manifold with strictly negative sectional curvature. Then every free homotopy class of simple closed curves contains exacly one geodesic.
Lemma 2: Take $S$ as in the preceding Lemma. If $c_1$ and $c_2$ are two simple loops non freely homotopic one to the other and if they intersect in at least one point, then, if $\gamma_1$ and $\gamma_2$ are the geodesics freely homotopic to $c_1$ and $c_2$ respectively, also $\gamma_1$ and $\gamma_2$ intersect in at least one point.
The first one is a quite classical result about hyperbolic manifolds, while I am not 100% sure that also the second is correct (I tried to sketch a proof, but I have not found it on books, and this is suspicious).
If also the second Lemma holds, my problem reduces to find an infinite family of simple loops each in a different class of free homotopy which intersect each other's. Take two intersecting loops $a$ and $b$ on $S$, for example consider the single loop around one handle of $S$ and $b$ the maximum loop around the to holes of $S$. Then, applying the Dehn twist $T_a$ to $b$, the resulting curve $b_1$ is in the homotopy class of $b \ast a$. Iterating this step produces the desired family of curves.
Any feedback will be really appreciated.