Closed geodesic on Poincaré disk and Gauss-Bonnet Theorem

Question

Consider the Poincaré disk model for the Hyperbolic plane, that is, the set $\{(x,y)\in\mathbb{R}^{2}\}$ endowed with the metric $ds^{2}=\frac{4dx^{2}+4dy^{2}}{(1-x^{2}-y^{2})^2}$.

Can there be a simple and closed geodesic?

My attempt: The answer appears to be no, suppose the contrary, then such geodesic encloses a set $D$ that is homeomorphic to a closed ball from $\mathbb{R}^{2}$, but since the curvature is negative for the Hyperbolic plane we get a contradiction with $\int_DK=2\pi.$ Is it right the apply this reasoning? I'm not quite sure that such portion of the Hyperbolic plane is homeomorphic to the closed ball.

Answer 1

The reasoning is good. The fact that a simple closed curve bounds a topological disk is known as the Schoenflies theorem.

An alternative proof would be to pick a point on the geodesic and apply a hyperbolic isometry that moves it to the center of the disk. Then recall that:

1. A geodesic is determined by a point and its tangent vector at that point.
2. For every tangent vector at $0$, there is a non-closed geodesic in that direction, namely the diameter of the disk.