Past exam of the geometry course of the University of Oxford / 2008 / A / 4 / iii / Solution

If two distinct geodesic arcs enclosed a disk, regard its boundary as a curvilinear two-sided polygon, with interior angles $\alpha ,\beta$. These are positive; zero angle would make the arcs tangent and local geodesic uniqueness would make them coincide. The geodesic-curvature terms vanish, so Gauss-Bonnet givesThis contradicts $K\le 0$. Thus no such disk-bounding geodesic bigon exists. The disk hypothesis matters because it fixes the Euler characteristic on the right side.