What is a non-complete simple geodesic on closed hyperbolic surface?

Question

I saw the words "complete simple geodesic" in many places. But I can't not figure out an example of simple geodesic which is not complete on closed hyperbolic surface. I think a gedesic ray can be non-complete but in case of gedesic, I don't know. I hope someone can help me. Thanks in advance

Answer 1

Let $\alpha:\mathbb{R} \to S$ be a complete closed geodesic that its image in $S$ intersects itself transversely in a single point, e.g. a geodesic representative (with respect to a chosen hyperbolic structure) of the homotopy class of a "figure eight" encircling the two holes of a closed genus two surface as you might draw this surface in the usual way in 3-space. Choose a maximal interval $I \subset \mathbb{R}$ such that $\alpha:I \to S$ is one-to-one. Then $\alpha$ is a simple geodesic, but it cannot be extended and still remain a simple geodesic.