In V.I. Arnold's Mathematical Methods of Classical Mechanics the first appendix is on Riemannian curvature. He starts with parallel translation of a tangent vector to a surface:
- Along a geodesic the parallel translation of a given tangent vector is the vector of the same norm having the same inner product with the tangent vector to the geodesic.
- Along a piecewise geodesic curve you do the same thing along the different segments.
- A general piecewise smooth curve is approximated by piecewise geodesic curves, and in an appropriate limit gives you the parallel tranlated vector.
Assuming you know what the geodesics are, in this way you can in principle compute parallel transport. Note that much more than giving a recipe or a rigorous definition, the aim is to give an intuition for parallel transport on a surface.
As an example he shows how to compute the parallel translation of a vector pointing north on a sphere around the parallel at latitude $\lambda$, which corresponds to a rotation through $2\pi(1 -\sin\lambda)$. The way in which this is computed looks very interesting, but I don't manage to understand how it works. It looks like it should be possible to do this with a minimal amount of computation. The given explanation is:
It is sufficient to translate the vector along the same circle on the cone formed by the tangent lines to the meridian, going through all the points of the parallel. This cone then can be unrolled onto the plane, after which parallel translation on its surface becomes ordinary parallel translation on the plane.
The only thing I see clearly is that the tangents to geodesics starting at the parallel and ending at the apex of the cone are the straight lines in the rolled-out cone. I don't see what the reasoning is here, and I don't understand the left image where most vectors are not even tangent vectors.
I found that the question of parallel translation along parallels on a sphere has been asked a few times before, and this construction is mentioned, but without saying why this should work, with one exception: in this answer it is stated that
For the purposes of parallel transport along a particular circle of latitude, the sphere can be replaced by the cone which is tangent to the sphere along that circle, since a "flatlander" living on the surface and travelling along the circle would experience the same “twisting of the tangent plane in the ambient space” regardless of whether the surface is a sphere or a cone.
This is sufficient explanation given some definitions of parallel translation (and actually gives a nice intuitive interpretation of parallel transport). However, can this still easily (let's say without integration) be seen if the only definition you know is the one in terms of piecewise geodesic curves?