Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 25I / a / Solution

For $v$ in a sufficiently small neighbourhood of $0\in T_{p}S$, let ${\gamma }_v$ be the geodesic satisfying ${\gamma }_v(0)=p$ and ${\dot{\gamma }}_v(0)=v$. The exponential map at $p$ isChoose an oriented orthonormal basis of $T_{p}S$, write $e(\theta )=(\cos \theta ,\sin \theta )$, and define geodesic polar coordinates by

The Gauss lemma says that radial and angular coordinate curves are orthogonal. Since $r\mapsto \varphi (r,\theta )$ is a unit-speed geodesic,Moreover, torsion-freeness of the surface connection gives $D_r{\varphi }_{\theta }=D_{\theta }{\varphi }_r$, and thereforeAt $r=0$, ${\varphi }_{\theta }=0$, so $⟨{\varphi }_r,{\varphi }_{\theta }⟩=0$ for every sufficiently small $r$. Thus the first fundamental form isand $\sqrt{G}/r\to 1$ as $r\downarrow 0$ because $d({\exp }_p{)}_0$ is the identity.