Parallel Transport on Matrix Manifolds and Exponential Action
Du Nguyễn, Stefan Sommer · SIAM Journal on Matrix Analysis and Applications · 2025
Abstract. We express parallel transport for several common matrix Lie groups with a family of pseudo-Riemannian metrics in terms of the matrix exponential and exponential actions. The metrics are constructed from a deformation of a bi-invariant metric and are naturally reductive. There is a similar picture for homogeneous spaces when taking quotients satisfying a general condition. In particular, for a Stiefel manifold of orthogonal matrices of size [Formula: see text], we give an expression for parallel transport along a geodesic from time zero to [Formula: see text] that could be computed with time complexity of [Formula: see text] for small [Formula: see text] and of [Formula: see text] for large [Formula: see text], contributing a step in a long-standing open problem in matrix manifolds. A similar result holds for flag manifolds with the canonical metric. We also show the parallel transport formulas for the general linear group and the special orthogonal group under these metrics.