So I'm trying to find the components of a unit vector, initially on the sphere's equator parallel to the $\phi=0$ line, after parallel transport to $\phi=\phi_0$, then to the pole and then back to the equator at $\phi=0$ I've calculated the connection coefficients and get the following equations of motion for the components: (note $Y^\phi $ and $ Y^\theta $ are the phi and theta components of the vector).
$$ \frac{dY^{\phi}}{d\phi}=-\frac{cos\theta}{sin\theta}Y^{\theta} $$ $$ \frac{dY^{\phi}}{d\theta}=-\frac{cos\theta}{sin\theta}Y^{\phi} $$ $$ \frac{dY^\theta}{d\theta}=0 $$ $$ \frac{dY^\theta}{d\phi}=sin\theta cos\theta Y^\phi$$
Solving the motion around the equator is fine, I get just get the equator simple harmonic motion. The problem comes from moving the vector to the pole where the equation of motion blows up (integrating the 2nd equation gives):
$$ Y^\phi = \frac{A}{sin\theta} $$ which diverges as the pole. I've tried using the fact that the length of the vector is conserved along geodesics, however, I still fail to get a reasonable answer. Any suggestions as to what I'm doing wrong? Thanks!