Parallel transport on a sphere

What you are seeing: a "Beltrami" spherical triangle on the unit sphere with one vertex at the north pole and the other two on a colatitude $\alpha$ circle, separated by longitude $\beta$. Parallel transport of any vector around the triangle returns it rotated by the enclosed solid angle $\Omega = (1 - \cos\alpha) \beta$. This is Gauss-Bonnet on the sphere of constant curvature $K = 1$: the angle excess $A + B + C - \pi$ equals the integrated curvature, which is just the area on the unit sphere.

For the full hemisphere ($\alpha = \pi/2$, $\beta = 2\pi$) the holonomy is $2\pi$: a full rotation. For a quarter octant ($\alpha = \beta = \pi/2$) the holonomy is $\pi/2$. For tiny triangles the holonomy is just the area, vanishing as $\alpha \to 0$.