I recently came across this exercise.
Let $S$ be a closed orientable surface of genus strictly greater than one and let $g$ be a Riemannian metric on S with constant negative curvature. Then, given any infinite family of simple closed geodesics on S, there are at least two geodesics in this family that intersect in a point (in other words, a family of closed simple geodesics that are disjoint two by two must be finite).
So far, I only managed to prove that, if $(\gamma_n)_{n\in\mathbb{N}}$ is a sequence of simple closed geodesics, $\gamma_n : S^1 \to S $, then $$ \inf_{n,m\in\mathbb{N}}\mathrm{d}(\gamma_n,\gamma_m)=0, $$ where $\mathrm{d}$ is the distance induced by $g$ on $S$. This infimum must be zero because, for any $t\in S^1$, the sequence $(\gamma_n(t))_{n\in\mathbb{N}}$ is contained in $S$, which is compact, so it admits a converging subsequence $(\gamma_{n_k}(t))_{k\in\mathbb{N}}$, which is a Cauchy sequence, and thus the equality above holds. Now, my (qualitative) idea is that the limit point is very near to all the geodesics in the sequence (for $k$ sufficiently big) and, since on a surface there is not much room, the geodesics are forced to meet in a point. Anyway, I do not how to make it rigorous nor if my idea is correct.
I am also aware of some basic results concerning hyperbolic manifolds, for example Cartan-Hadamard Theorem or the one to one corrispondence between closed geodesics and free homotopy loop classes, but I am not really familiar with them, so I do not know how to apply these theorems.
Any hint will be really appreciated.
Suppose that $S$ is a closed connected oriented surface of genus $g\ge 2$ and $L$ is a 1-dimensional submanifold in $S$ such that the complement $S-L$ contains no components which are annuli or disks. Prove (using the Euler characteristic) that the number of components of $L$ is $\le 3g-3$.
Now, if $S$ is equipped with a Riemannian metric of negative curvature, then a closed geodesic in $S$ cannot be null-homotopic and two such geodesics cannot be homotopic unless their images are the same. Use these observation to conclude that if $L\subset S$ is a submanifold each component of which is geodesic then the number of components is at most $3g-3$.