Parallel transport on matrix manifolds and Exponential Action
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 $n\times d$, we give an expression for parallel transport along a geodesic from time zero to $t$, that could be computed with time complexity of $O(nd^2)$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.06054
- arXiv:
- arXiv:2408.06054
- Bibcode:
- 2024arXiv240806054N
- Keywords:
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- Mathematics - Numerical Analysis;
- Computer Science - Computer Vision and Pattern Recognition;
- 15A16;
- 15A18;
- 15B10;
- 22E70;
- 51F25;
- 53C80;
- 53Z99