Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 24H / i / Solution

The Theorema Egregium states that Gaussian curvature is determined by the first fundamental form, and therefore preserved by local isometries. For a local surface parametrization $r$, write $g_{ij}=r_i\cdot r_j$, choose a unit normal $N$, and set $b_{ij}=r_{ij}\cdot N$. Decomposing the second derivatives into tangent and normal parts givesCommute the third partial derivatives of $r$ and take tangent components. The terms involving derivatives of $b$ are normal; the tangent terms give the Gauss equationwhere $R$ is computed from derivatives and products of the displayed Christoffel symbols, using $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$. For example the tangent term from differentiating $b_{ij}N$ is $-b_{ij}{b}_k{}^{\ell }{r}_{\ell }$, which produces exactly the difference of the two products on the right. ConsequentlyThe last expression depends only on $g$ and its derivatives. Pulling back $g$ under a local isometry therefore preserves $K$, proving the theorem.