Abstract. Parallel transport on Riemannian manifolds allows one to connect tangent spaces at 12 different points in an isometric way and is therefore of importance in many contexts, such as statistics 13 on manifolds. The existing methods to compute parallel transport require either the computation 14 of Riemannian logarithms, such as the Schild's ladder, or the Christoffel symbols. The Logarithm is 15 rarely given in closed form, and therefore costly to compute whereas the number of Christoffel symbols 16 explodes with the dimension of the manifold, making both these methods intractable. From an 17 identity between parallel transport and Jacobi fields, we propose a numerical scheme to approximate 18 the parallel transport along a geodesic. We find and prove an optimal convergence rate for the 19 scheme, which is equivalent to Schild's ladder's. We investigate potential variations of the scheme 20 and give experimental results on the Euclidean two-sphere and on the manifold of symmetric positive 21 definite matrices. 22 This manuscript is for review purposes only.
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