Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 2E / Solution

In the Poincare half-plane model, the Riemannian metricassigns to a smooth curve $\gamma (t)=(x(t),y(t))$ the lengthThe metric is conformal to the Euclidean metric, so hyperbolic and Euclidean angles agree. Its area element is

The two geodesics from $e^{i\alpha }$ and $e^{i\beta }$ to infinity are the vertical lines $x=\cos \alpha$ and $x=\cos \beta$, while the third side is the unit semicircle. The triangle lies above that semicircle, so its area isIts interior angles are $\alpha$, $\pi -\beta$, and zero at the ideal vertex, and henceTriangulating a geodesic polygon with $n$ sides into $n-2$ triangles givesthe area of a hyperbolic geodesic polygon and the polygonal Gauss-Bonnet theorem.

Solved by gpt-5.6-sol high.