Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 25I (Differential Geometry)

An isometry is a smooth diffeomorphism with $I_x(v,w)=I_{\varphi (x)}(d{\varphi }_{x}v,d{\varphi }_{x}w)$.

If $f\circ \varphi =f$, then $x$ and $\varphi (x)$ have the same value, so every level set is preserved.

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.

An isometry preserves both foliations. Independence makes their level curves transverse near $x$, so their pair of values gives local coordinates. The isometry fixes the intersection corresponding to every pair of nearby values, hence fixes every point in a sufficiently small neighbourhood.

The one-dimensional kernel of $d{K}_x$ is the tangent to its level curve; the metric gives two unit normals and $d{K}_x(e_x)>0$ selects one. Isometries preserve the metric, $K$, and this orientation choice, so they carry $e_x$ to $e_{\varphi (x)}$. Therefore $g(x)=d{K}_x(e_x)=|\nabla K(x)|$ is isometry invariant.