Connectivity properties of the Schur-Horn map for real Grassmannians
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916127434604544 |
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| author | Mare, Augustin-Liviu |
| author_facet | Mare, Augustin-Liviu |
| contents | To any $V$ in the Grassmannian ${\rm Gr}_k({\mathbb R}^n)$ of $k$-dimensional vector subspaces in ${\mathbb R}^n$ one can associate the diagonal entries of the ($n\times n$) matrix corresponding to the orthogonal projection of ${\mathbb R}^n$ to $V$. One obtains a map ${\rm Gr}_k({\mathbb R}^n)\to {\mathbb R}^n$ (the Schur-Horn map). The main result of this paper is a criterion for pre-images of vectors in ${\mathbb R}^n$ to be connected. This will allow us to deduce connectivity criteria for a certain class of subspaces of the real Stiefel manifold which arise naturally in frame theory. We extend in this way results of Cahill, Mixon, and Strawn. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_11596 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Connectivity properties of the Schur-Horn map for real Grassmannians Mare, Augustin-Liviu Differential Geometry 14M15, 53C42, 53D20 To any $V$ in the Grassmannian ${\rm Gr}_k({\mathbb R}^n)$ of $k$-dimensional vector subspaces in ${\mathbb R}^n$ one can associate the diagonal entries of the ($n\times n$) matrix corresponding to the orthogonal projection of ${\mathbb R}^n$ to $V$. One obtains a map ${\rm Gr}_k({\mathbb R}^n)\to {\mathbb R}^n$ (the Schur-Horn map). The main result of this paper is a criterion for pre-images of vectors in ${\mathbb R}^n$ to be connected. This will allow us to deduce connectivity criteria for a certain class of subspaces of the real Stiefel manifold which arise naturally in frame theory. We extend in this way results of Cahill, Mixon, and Strawn. |
| title | Connectivity properties of the Schur-Horn map for real Grassmannians |
| topic | Differential Geometry 14M15, 53C42, 53D20 |
| url | https://arxiv.org/abs/2309.11596 |