Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 38D / a / ii / Solution

For $ds^2=d\theta^2+\sin^2\theta\,d\phi^2$, a vector initially having coordinate components $(V^\theta,V^\phi)=(1,0)$ returns after one constant-$\theta$ circuit with
$$
(V^\theta,V^\phi)
=\left(
\cos(2\pi\cos\theta),
-\frac{\sin(2\pi\cos\theta)}{\sin\theta}
\right).
$$


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 Incoming links: Parallel transport around a latitude of the unit sphere