Simple non-closed geodesic.

In torus there exists simple non-closed geodesic. One example is take the irrational slope curves in $\mathbb{R}^2$ and project it down to torus. Is this thing can happen in closed hyperbolic surface i.e.

Does there exists a simple non-closed geodesic in a closed (i.e. compact without boundary) hyperbolic surface?

Solution 1:

For closed hyperbolic surfaces (with constant negative curvature) the answer is yes. This is shown by exhibiting a transitive geodesic flow on the surface. In this paper by Hedlund, the sufficient property of the flow is called regional transitivity and implies the existence of a simple dense geodesic on the surface $M$. As the geodesic is dense in $M$, it can not be closed.

In the above paper, it is stated in Theorem 3.1 that there are a countable number of periodic geodesics and so in some sense 'most' geodesics on $M$ are non-closed. To actually construct such a geodesic (although not necessarily simple), the standard technique is to use 'symbolic trajectories' which Hedlund hints at in the very last section of the paper. Essentially, you can lift any geodesic on $M$ to its universal cover by the hyperbolic plane and create a bi-infinite sequence associated to the geodesic which tells you which edges of the fundamental regions the geodesic passes through as you 'walk along' the geodesic. Such a sequence is sometimes called a cutting sequence. Admissible cutting sequences can be totally classified by local configuration rules and so it is then just a matter of finding a non-periodic admissible cutting sequences, which will correspond to non-periodic (and hence non-closed) geodesics on the surface $M$.

For an introduction to cutting sequences, and their beautiful links with continued fractions (as well as a direct construction of a non-periodic geodesic on the modular surface) I would strongly recommend reading this wonderful and easy to read article by Caroline Series.

Solution 2:

Let's call $\Sigma$ a closed hyperbolic surface. Each free homotopy class of curves on a hyperbolic surface has exactly one geodesic representative. Conversely, a closed geodesic gives a free homotopy class. Thus (up to basepoint) there is a bijective correspondence $$\bigg\{ \mbox{ closed geodesics }\bigg\} \longleftrightarrow \pi_1(\Sigma).$$

Since $\pi_1(\Sigma)$ is finitely generated, it is a countable set. In particular, it lifts to a countable set of geodesics in $\mathbb{H}^2$. Any and every geodesic in $\Sigma$ can be obtained by pushing a geodesic of $\mathbb{H}^2$ forward under the covering map, so the set of all geodesics of $\Sigma$ lifts to the set of geodesics of $\mathbb{H}^2$, which is uncountable. So there are many more open geodesics than closed geodesics on $\Sigma$.

Here's a simple example on a non-closed hyperbolic cylinder that illustrates the phenomenon. Let $\Sigma = \mathbb{H}^2/\mathbb{Z}$, where the $\mathbb{Z}$ action is given by $z\mapsto \lambda z$, for some $\lambda > 1$. The single closed geodesic is the image (under quotient) of the $y$-axis. Take any other geodesic which limits to $0$ and push it down under the covering map; it will spiral in from one side of the cylinder and limit to the closed geodesic.