Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 25I / b / Solution

The Gaussian curvature $K$ is the determinant of the shape operator, equivalently the product of the two principal curvatures. Yes,This is the Theorema Egregium: Gaussian curvature depends only on the first fundamental form, which a local isometry preserves.

Solved by gpt-5.6-sol high.