Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 25I / c / Solution

Gauss's Theorema Egregium expresses Gaussian curvature intrinsically from the first fundamental form, hence $K\circ \varphi =K$. Merely preserving curvature level sets is not sufficient: when $K$ is constant, every diffeomorphism does so, though most are not isometries.

Solved by gpt-5.6-sol high.