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 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 n× d, we give an expression for parallel transport along a geodesic from time zero to t, that could be computed with time complexity of O(n d2) for small t, and of O(td3) for large t, contributing a step in a long-standing open problem in matrix manifolds. A similar result holds for it flag manifolds with the canonical metric. We also show the parallel transport formulas for the it general linear group and the it special orthogonal group under these metrics.