I want to show that there exists a closed geodesic on an arbitrarily chosen surface where the Gaussian curvature is negative (on the entire surface).
One example would be the "inside" of a ring Torus, where the Gaussian curvature is:
$K = \frac{\cos(v)}{a(c+a\cos(v))} <0$ for $v \in(\frac{\pi}{2},\frac{3\pi}{2}) $
I'm confident that there has to be a closed geodesic because of its symmetry but I'm not sure how to prove it.