Flag manifolds, spaces of frames, and connectedness properties
Abstract
For integers 1≤ k < n and real numbers c1, …, ck>0 and d1, …, dn≥ 0, we investigate the space of all k× n matrices F such that the product FF* is equal to the diagonal matrix Diag(c1, …, ck) and the squared norms of the columns of F are equal to d1, …, dn respectively. Depending on the field where the coefficients of F are taken from, which can be of real or of complex numbers, we are mainly interested in determining whether the resulting space is path-connected or even simply connected relative to the subspace topology in the space of all k × n matrices. The criteria presented are closely related to results previously obtained by Cahill, Mixon, and Strawn (2017), Needham and Shonkwiler (2021), the last two authors together with Caine (2026), and the author of this work (2024 and 2026). Flag manifolds, namely orbits of the canonical conjugation actions of O(n) and U(n) on the spaces of symmetric real and Hermitian n× n matrices, respectively, play a central role in our development.
Related
- No Connectedness Lemma for Alexandrov spaces
- The intersection of two real flag manifolds in a complex flag manifold, and the canonical form of a compact symmetric triad
- Differential calculus on quantized irreducible flag manifolds
- Rational homotopy theory of flag manifolds
- CR Orbits on the complex flag manifolds of G2
BibTeX
@misc{mare2026flag,
title = {{Flag manifolds, spaces of frames, and connectedness properties}},
author = {Augustin-Liviu Mare},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17203},
eprint = {2609.17203},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17203},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Flag manifolds, spaces of frames, and connectedness properties
AU - Augustin-Liviu Mare
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17203
UR - https://arxiv.org/abs/2609.17203
AB - For integers 1≤ k < n and real numbers c_1, …, c_k>0 and d_1, …, d_n≥ 0, we investigate the space of all k× n matrices F such that the product FF^* is equal to the diagonal matrix Diag(c_1, …, c_k) and the squared norms of the columns of F are equal to d_1, …, d_n respectively. Depending on the field where the coefficients of F are taken from, which can be of real or of complex numbers, we are mainly interested in determining whether the resulting space is path-connected or even simply connected relative to the subspace topology in the space of all k × n matrices. The criteria presented are closely related to results previously obtained by Cahill, Mixon, and Strawn (2017), Needham and Shonkwiler (2021), the last two authors together with Caine (2026), and the author of this work (2024 and 2026). Flag manifolds, namely orbits of the canonical conjugation actions of O(n) and U(n) on the spaces of symmetric real and Hermitian n× n matrices, respectively, play a central role in our development.
ER -