arXiv 1705.040932017-05-11math.DGcs.NAmath.NA

矩阵流形的主丛结构

Principal bundle structure of matrix manifolds

Ecole Centrale de Nantes（南特中央理工学院）；Universidad CEU Cardenal Herrera（CEU圣心红衣主教大学）

Marie Billaud-Friess, Antonio Falco, Anthony Nouy

发表信息：: Mathematics 2021, 9(14), 1669
DOI：: 10.3390/math9141669

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AI总结：

本文提出了一种新的几何描述方法，用于描述固定秩矩阵流形。通过定义一个流形的Atlas，证明其具有分析主丛结构，并进一步证明非满秩矩阵流形的结构，同时展示了矩阵空间作为流形的拓扑结构。

摘要：

在本文中，我们介绍了一种新的几何描述方法，用于描述固定秩矩阵流形。起点是线性子空间的Grassmann流形$\mathbb{G}_r(\mathbb{R}^k)$的几何描述，该描述避免了等价类的使用。集合$\mathbb{G}_r(\mathbb{R}^k)$配备了提供其为分析流形的Atlas，其模型为$\mathbb{R}^{(k-r) imes r}$。然后我们定义了全秩矩阵集合$\mathcal{M}_r(\mathbb{R}^{k imes r})$的Atlas，并证明所得到的流形是一个分析主丛，其底空间为$\mathbb{G}_r(\mathbb{R}^k)$，典型纤维为$\mathrm{GL}_r$，即$\mathbb{R}^{k imes k}$中的可逆矩阵一般线性群。最后，我们定义了非满秩矩阵集合$\mathcal{M}_r(\mathbb{R}^{n imes m})$的Atlas，并证明所得到的流形是一个分析主丛，其底空间为$\mathbb{G}_r(\mathbb{R}^n) imes \mathbb{G}_r(\mathbb{R}^m)$，典型纤维为$\mathrm{GL}_r$。$\mathcal{M}_r(\mathbb{R}^{n imes m})$的Atlas以流形本身为索引，允许对给定矩阵自然定义一个邻域，该邻域被证明具有李群的结构。此外，$\mathcal{M}_r(\mathbb{R}^{n imes m})$配备由Atlas诱导的拓扑，被证明是矩阵空间$\mathbb{R}^{n imes m}$的嵌入子流形，配备子空间拓扑。所提出的几何描述因此导致矩阵空间$\mathbb{R}^{n imes m}$，作为流形$\mathcal{M}_r(\mathbb{R}^{n imes m})$的并集，被视作一个分析流形，其拓扑结构中矩阵秩是一个连续映射。

英文摘要：

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold $\mathbb{G}_r(\mathbb{R}^k)$ of linear subspaces of dimension $r<k$ in $\mathbb{R}^k$ which avoids the use of equivalence classes. The set $\mathbb{G}_r(\mathbb{R}^k)$ is equipped with an atlas which provides it with the structure of an analytic manifold modelled on $\mathbb{R}^{(k-r)\times r}$. Then we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{k \times r})$ of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^k)$ and typical fibre $\mathrm{GL}_r$, the general linear group of invertible matrices in $\mathbb{R}^{k\times k}$. Finally, we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^n) \times \mathbb{G}_r(\mathbb{R}^m)$ and typical fibre $\mathrm{GL}_r$. The atlas of $\mathcal{M}_r(\mathbb{R}^{n \times m})$ is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space $\mathbb{R}^{n \times m}$ equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space $\mathbb{R}^{n \times m}$, seen as the union of manifolds $\mathcal{M}_r(\mathbb{R}^{n \times m})$, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

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