Parallel Transport on Matrix Manifolds and Exponential Action

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 ×, we give an expression for parallel transport along a geodesic from time zero to that could be computed with time complexity of ⁡(⁢2) for small and of ⁡(⁢3) for large , 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.