PreprintCC BY 4.0arXiv
No closed-form formula for Stiefel geodesics
Abstract
We use differential Galois theory to prove that given two arbitrary points X and Y on the Stiefel manifold, there is no closed-form analytic expression for the Riemannian logarithm logX Y, the geodesic between X and Y, or the geodesic distance between X and Y. In other words, these cannot be expressed in terms of elementary functions of the coordinates of X and Y; in fact, our result goes further --- even if we allow for repeated integrals of elementary functions, there is still no such expression.
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BibTeX
@misc{wang2026closedform,
title = {{No closed-form formula for Stiefel geodesics}},
author = {Rongbiao Thomas Wang and Lek-Heng Lim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37568},
eprint = {2610.37568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37568},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - No closed-form formula for Stiefel geodesics
AU - Rongbiao Thomas Wang
AU - Lek-Heng Lim
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37568
UR - https://arxiv.org/abs/2610.37568
AB - We use differential Galois theory to prove that given two arbitrary points X and Y on the Stiefel manifold, there is no closed-form analytic expression for the Riemannian logarithm log_X Y, the geodesic between X and Y, or the geodesic distance between X and Y. In other words, these cannot be expressed in terms of elementary functions of the coordinates of X and Y; in fact, our result goes further --- even if we allow for repeated integrals of elementary functions, there is still no such expression.
ER -