Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 2 / 24H / b / Solution

For a parametrization $X(u^1,u^2)$, put $g_{ij}=X_i\cdot X_j$. Decompose the second derivatives into tangent and normal components:These tangent coefficients are the Christoffel symbols. Differentiate $g_{ij}$ and combine the resulting three identities to obtainHere $(g^{kl})$ is the inverse matrix. The formula proves that these symbols depend only on the first fundamental form and its first derivatives, not on the second fundamental form.