Parallel transport around a latitude of the unit sphere
= Parallel transport around a latitude of the unit sphere
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
$$
(V^\theta,V^\phi)
=\left(
\cos(2\pi\cos\theta),
-\frac{\sin(2\pi\cos\theta)}{\sin\theta}
\right).
$$