Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 26G / c / Solution

For $v\in T_{p}S$, let ${\gamma }_v$ be the maximal geodesic satisfyingThe exponential map ison the domain of the exponential mapThis is an open, star-shaped neighbourhood of zero. In local coordinates the geodesic equation is a smooth ordinary differential equation, so smooth dependence on its initial position and velocity proves that $(p,v)\mapsto {\exp }_p(v)$ is smooth on its domain.

For both requested phenomena consider the embedded surfacewith $p=(1,0,0)$. Its geodesics are Euclidean straight lines for as long as they remain in $S$. The initial vector $v=(-2,0,0)$ would givewhich reaches the missing origin at $t=1/2$. Thus $v\notin {\mathcal{D}}_p$, and the domain is not all of $T_{p}S$.

The point $q=(-1,0,0)$ is not in the image of ${\exp }_p$. Any geodesic from $p$ to $q$ would have to be their unique Euclidean straight line, which passes through the omitted origin. Hence this exponential map is also not surjective, as summarized by exponential map of the punctured plane.

Solved by gpt-5.6-sol high.