 Solution

ID: 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

α, β

. 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 gives

\int_{D} K d A + (π - α) + (π - β) = 2 π, \int_{D} K d A = α + β > 0.

This contradicts

K \leq 0

. Thus no such disk-bounding geodesic bigon exists. The disk hypothesis matters because it fixes the Euler characteristic on the right side.

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