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

Forthe nonzero Christoffel symbols needed here areOn a line of constant longitude, $\dot{\varphi }=0$, and the geodesic equation reduces to $\ddot{\theta }=0$ after choosing an affine parameter. Thus every meridian is a geodesic.

For a nonconstant line of constant latitude, $\dot{\theta }=0$ and the $\theta$ equation requiresAway from the coordinate-degenerate poles, this holds only whenso only the equator is a constant-latitude geodesic.

Along the constant latitude, use $\varphi$ as parameter. The parallel transport equations areWith $W=\sin \theta V^{\varphi }$, these becomeThe initial components $(1,0)$ therefore evolve asAfter one circuit, the parallel transport around a latitude of the unit sphere gives