# Reference request for parallel transport

I am learning about parallel transport on a Riemannian manifold equipped with an affine connexion. It seems (if I understand it well) that, in general, we might not be able to compute the parallel transport of a tangent vector in closed form (For example, on the sphere $\mathbb{S}^{2} \subset \mathbb{R}^{3}$, the parallel transport of a tangent vector along a curve (not necessarily geodesic) can be obtain in closed form.) We can only obtain an approximation of this transport through a numerical scheme.

For example, the Schild's Ladder (see Schild's Ladder Parallel Transport Procedure for an arbitrary connexion) is a first-order numerical scheme which allows to approximate the parallel transport of a tangent vector along a curve.

My questions are the following (I apologize if these questions are naive) :

1. Is there a reference, a proof of how fast this approximation converges to the exact parallel transport ?
2. Do you know if there exists a second-order approximation of the parallel transport ?
• Parallel transport is the solution of an ODE. Doesn't this answer both questions? – user25199 Oct 7 '13 at 14:08
• And not even some complicated system of ODE's, but just a linear first order system of ODE's. Any standard numerical scheme for solving such a system should work. – Deane Yang Oct 7 '13 at 15:28
• Strictly speaking it is one ODE for every local coordinate system used to describe the curve. If the curve passes through many coordinate neighborhoods then you accumulate lots of error if you naively use the solution to one ODE as the initial condition for the next, and you can reduce this error by thinking more carefully. – Paul Siegel Oct 7 '13 at 19:17