A Fanning Scheme for the Parallel Transport Along Geodesics on Riemannian Manifolds

Maxime Louis (1, 2) , Benjamin Charlier (2, 3) , Paul Jusselin (1, 2) , Susovan Pal (1, 2) , Stanley Durrleman (1, 2)

1 ARAMIS - Algorithms, models and methods for images and signals of the human brain = Algorithmes, modèles et méthodes pour les images et les signaux du cerveau humain [ICM Paris]
2 ICM - Institut du Cerveau = Paris Brain Institute
3 IMAG - Institut Montpelliérain Alexander Grothendieck

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

Mots clés: Jacobi field; Numerical scheme; Parallel Transport; Riemannian manifold

Domaines: Géométrie différentielle [math.DG]; Analyse numérique [math.NA]

Maxime Louis, Benjamin Charlier, Paul Jusselin, Susovan Pal, Stanley Durrleman. A Fanning Scheme for the Parallel Transport Along Geodesics on Riemannian Manifolds. SIAM Journal on Numerical Analysis, 2018, 56 (4), pp.2563-2584. ⟨10.1137/17M1130617⟩. ⟨hal-01560787v2⟩