Suppose I want to parallel transport a vector $v$ living in the tangent space of my surface at a point $p$ along a closed geodesic polygonal path. On each regular component I keep the angle between $v$ and the tangent fixed. How do I calculate the angle between my original vector and the one obtained after travelling ones around the curve?
I guess it should be the sum of the exterior angles?
For a concrete example, suppose we are parallel transporting a vector, along a Geodesic triangle in the sphere, and the area of the triangle is $\alpha$. Suppose, the initial angle between $v$ and the tangent is $\theta$. Is then the angle between the original vector and the parallel transported vector then $2\pi -\alpha$?