旗流形、标架空间与连通性性质
Flag manifolds, spaces of frames, and connectedness properties
AI总结：
本文研究满足特定范数和对角乘积条件的矩阵空间，在实或复系数下判定其道路连通性和单连通性，并利用旗流形给出判据，与已有结果紧密相关。
AI中文摘要：
对于整数 $1\le k < n$ 以及实数 $c_1, \ldots, c_k>0$ 和 $d_1, \ldots, d_n\ge 0$，我们研究所有 $k\times n$ 矩阵 $F$ 的空间，使得乘积 $FF^*$ 等于对角矩阵 ${\rm Diag}(c_1, \ldots, c_k)$，且 $F$ 的各列平方范数分别等于 $d_1, \ldots, d_n$。根据 $F$ 的系数取自实数域或复数域，我们主要关注确定所得空间在全体 $k \times n$ 矩阵空间的子空间拓扑下是否是道路连通的，甚至是否是单连通的。所提出的判据与先前由 Cahill、Mixon 和 Strawn（2017）、Needham 和 Shonkwiler（2021）、后两位作者与 Caine（2026）以及本工作作者（2024 和 2026）获得的结果密切相关。旗流形，即 ${\rm O}(n)$ 和 ${\rm U}(n)$ 分别对对称实矩阵和 Hermitian $n\times n$ 矩阵空间的典范共轭作用的轨道，在我们的发展中起着核心作用。
英文摘要：
For integers $1\le k < n$ and real numbers $c_1, \ldots, c_k>0$ and $d_1, \ldots, d_n\ge 0$, we investigate the space of all $k\times n$ matrices $F$ such that the product $FF^*$ is equal to the diagonal matrix ${\rm Diag}(c_1, \ldots, c_k)$ and the squared norms of the columns of $F$ are equal to $d_1, \ldots, d_n$ 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 $k \times n$ 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 ${\rm O}(n)$ and ${\rm U}(n)$ on the spaces of symmetric real and Hermitian $n\times n$ matrices, respectively, play a central role in our development.
