Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 25I / a / Solution

For a local parametrization $X(u^1,u^2)$, the first fundamental form is the Euclidean inner product restricted to each tangent plane:Writing $u^1=u,u^2=v$, this is

After choosing an orientation, the Gauss map isThe second fundamental form is $II(v,w)=⟨S(v),w⟩$, where the shape operator is $S=-dN$. In coordinates,The Gaussian curvature and mean curvature are respectively the determinant and half the trace of $S$:Changing the chosen normal reverses $II$ and $H$ but leaves $K$ unchanged.