Let $S^1$ be the unit sphere in $\mathbb R^3$, and let $$C=\{(r\cos t, r\sin t, h)\colon t\in \mathbb R\}$$ with $r^2+h^2=1$ be a circle in $S^2$.
I want to compute the holonomy around this circle.
I know that if $(\theta, \phi)$ are spherical coordinates, where $\theta$ measures the angle from the $x$-axis (as in polar coordinates) and $\phi$ measures the angle from the $z$-axis, we can pull back the Euclidean metric to find $$ds^2=d\phi^2 + \sin^2 \phi \ d\theta^2.$$
Then traveling around the circle means fixing $\phi$ and moving theta from $0$ to $2\pi$.
However, actually computing parallel transport along this curve in coordinates seems really messy. Is there a way to do it in the ambient coordinates that would make the calculation easier?
I tried to do the computation in coordinates and it didn't go so well.
We are looking to parallel transport vectors along $\gamma(t)=(t,\phi_0)$, with $t\in[0,2\pi]$. We have $\gamma'(t)=(1,0)$. Any vector field $V(\theta)$ along the curve extends in the obvious way: $\tilde V(\theta, \phi)=V(\theta)$, so it suffices to look at extensions that are just a function of $\theta$. Note $D_t=\nabla_\theta$. If $$V=a(\theta)\partial_\theta + b(\theta)\partial_\phi,$$ we need $$D_tV=0,$$ or $$\nabla_\theta(a(\theta)\partial_\theta + b(\theta)\partial_\phi)=0.$$ Writing everything out and noting that a few of the Christoffel symbols vanish, I get the following system.
$$a'(\theta)+b(\theta)\Gamma^\theta_{\theta \phi}=0.$$ $$b'(\theta) + a(\theta)\Gamma^\phi_{\theta\phi}=0.$$
I get $$\Gamma^\theta_{\theta \phi} = \frac{\cos \phi}{\sin \phi},$$ $$\Gamma^\phi_{\theta\phi}=-\cos \phi \sin \phi.$$
We get solutions $$a(\theta)=\cos(\theta \cos \phi),$$ $$b(\theta)=\sin (\phi) \sin(\theta \cos \phi).$$
At $\theta=0$, we have $X=(1,0)$. At $\theta=2\pi$, we have $$Y=(\cos(2\pi \cos \phi), \sin\phi \cdot \sin(2\pi \cos \phi).$$ These are both norm $\sin^2 \phi$ vectors. We can compute the angle $$\cos \alpha = \frac{\langle X, Y \rangle_g}{|X||Y|}.$$ Then $\alpha = 2\pi \cos\phi$, or $2\phi h$.
Hint : Parallel transport along $\gamma (t)=(t,\phi_0)$ : Consider a cone $C$ tangent to a sphere $S$ s.t. $$ C\cap S = {\rm Im}\ \gamma $$
So we consider a parallel transport on a cone.