Work

Parallel Transport on Matrix Manifolds and Exponential Action

Year: 2025

Type: article

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.

Source: SIAM Journal on Matrix Analysis and Applications

DOI: 10.1137/24m1684189

Authors: Du Nguyen, Stefan Sommer

Institution: University of Copenhagen

Language: English

Topic: Advanced Differential Geometry Research

Subfield: Astronomy and Astrophysics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: green

Funders: Villum Fonden, Strong, Novo Nordisk, Novo Nordisk Fonden

Awards: G1619525767, G6044743522, G7291770527, G7991563841