Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 25I (Differential Geometry)

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.

Because the profile is parametrized by arc length,The tangent vectors areso the first fundamental form of an arc-length surface of revolution isWith the orientation determined by ${\varphi }_u\times {\varphi }_v$, the Gauss map isThe second derivatives giveand hence

Differentiating $Y^{\prime 2}+Z^{\prime 2}=1$ yields $Y^{\prime }{Y}^{\prime \prime }+Z^{\prime }{Z}^{\prime \prime }=0$, from whichThereforewhilefor the chosen orientation.

A unit-speed surface curve $\alpha$ is a geodesic when its acceleration is normal to the surface, equivalently when its covariant acceleration vanishes. In coordinates $u^1,u^2$, the Christoffel symbols of the Levi-Civita connection areThe coordinate functions of a geodesic satisfy the geodesic equation

For the metricthe Christoffel symbols needed in the $v$ equation areThus a geodesic obeysMultiplication by $Y^2$ turns this intoConsequently the Clairaut first integral for a surface of revolution isIt is also the conserved momentum associated with the cyclic coordinate $v$.