Parallel transport on matrix manifolds and Exponential Action

Code

• github.com/dnguyend/par-trans
  OfficialIn paperjax★ 3

Abstract

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 expression for parallel transport is preserved by taking the quotient under certain scenarios. 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(nd^2) for small t, and of O(td^3) for large t, 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 generalized linear group, and the special orthogonal group under these metrics.

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