Injectivity radius of the canonical Stiefel manifold
For integers $n \ge 4$ and $2 \le p \le n - 2$, give the Stiefel manifold $\mathrm{St}(n,p) = \{U \in \mathbb{R}^{n \times p} : U^{\mathsf T}U = I_p\}$ the canonical metric $$g_U(\xi, \zeta) = \operatorname{tr}\bigl(\xi^{\mathsf T}(I_n - \tfrac12 UU^{\mathsf T})\,\zeta\bigr), \qquad \xi, \zeta \in T_U\,\mathrm{St}(n,p).$$ It is homogeneous, so its injectivity radius is the same at every point, and by Klingenberg's lemma it is the smaller of the conjugate radius and half the length of a shortest geodesic loop. Zimmermann and Stoye proved that the sectional curvature is at most $5/4$; Stoye, Mataigne, Absil and Zimmermann showed that the shortest geodesic loops have length $2\pi$; and Absil and Mataigne exhibited conjugate points at distance $\sqrt2\,\tau$, where $\tau \in (\pi/2, \pi)$ is the smallest positive solution of $\tan \tau = -\tau$. Together these give $$\sqrt{4/5}\,\pi \le \operatorname{inj}\mathrm{St}(n,p) \le \sqrt2\,\tau \approx 0.9133\,\pi.$$ For $p = 1$ and $p = n - 1$ the injectivity radius is $\pi$.
Problem. Is $\operatorname{inj}\mathrm{St}(n,p) = \sqrt2\,\tau$ for all $n \ge 4$ and $2 \le p \le n - 2$?
Source: Absil and Mataigne (2025), conjectured from numerical experiments that their upper bound is the injectivity radius; Stoye, Mataigne, Absil and Zimmermann (2026), Section 7.2, leave the canonical metric open. Stated by AIM — Open Applied Problems (entry 666) for 2 ≤ p ≤ n − 2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- NA, OC
- Keywords
- AIM Open Applied Problems, conjugate points, injectivity radius, Stiefel manifold
References
@article{absil2025,
author = {Absil, P.-A. and Mataigne, Simon},
title = {The ultimate upper bound on the injectivity radius of the {S}tiefel manifold},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {46},
number = {2},
year = {2025},
pages = {1145--1167},
doi = {10.1137/24M1644808},
eprint = {2403.02079},
archivePrefix = {arXiv},
}
@article{zimmermann2025curvature,
author = {Zimmermann, Ralf and Stoye, Jakob},
title = {High curvature means low rank: On the sectional curvature of {G}rassmann and {S}tiefel manifolds and the underlying matrix trace inequalities},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {46},
number = {1},
year = {2025},
pages = {748--779},
doi = {10.1137/24M1655755},
eprint = {2403.01879},
archivePrefix = {arXiv},
}
@article{stoye2026,
author = {Stoye, Jakob and Mataigne, Simon and Absil, P.-A. and Zimmermann, Ralf},
title = {Shortest geodesic loops, sectional curvature, and injectivity radius of the {S}tiefel manifold},
journal = {BIT Numer. Math.},
volume = {66},
year = {2026},
pages = {Paper No. 42},
doi = {10.1007/s10543-026-01141-9},
eprint = {2511.01563},
archivePrefix = {arXiv},
}
@article{zimmermann2025euclidean,
author = {Zimmermann, Ralf and Stoye, Jakob},
title = {The injectivity radius of the compact {S}tiefel manifold under the {E}uclidean metric},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {46},
number = {1},
year = {2025},
pages = {298--309},
doi = {10.1137/24M1663818},
eprint = {2405.02268},
archivePrefix = {arXiv},
}
@misc{aim-canonical-stiefel-manifold-injectivity-radius,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Injectivity radius of the canonical Stiefel manifold}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/8729eab98bafe409b66c5e39253bf56569a62327/problems/666-canonical-stiefel-manifold-injectivity-radius.md},
note = {Entry 666 at commit 8729eab, 2026-10-23},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00917, "Injectivity radius of the canonical Stiefel manifold", clopen.solutions.