Parallel transport on matrix manifolds and Exponential Action

TL;DR

This paper derives explicit formulas for parallel transport on matrix Lie groups and homogeneous spaces using matrix exponential actions, with efficient computation methods for Stiefel and flag manifolds, advancing understanding in matrix manifold geometry.

Contribution

It provides new explicit formulas for parallel transport on matrix Lie groups and homogeneous spaces using exponential actions, with computational complexity improvements for key manifolds.

Findings

Explicit parallel transport formulas for matrix Lie groups and homogeneous spaces.

Efficient computation methods for Stiefel and flag manifolds.

Contributes to solving a long-standing open problem in matrix manifold computations.

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 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 $n\times d$, we give an expression for parallel transport along a geodesic from time zero to $t$, that could be computed with time complexity of $O(n d^2)$ 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 {\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.