AI 中文总结
本文研究研究较少的齐性空间 $\u039F_{m,n}(\u211D)/(\u039F_{p,q}(\u211D) \times \u039F_{m-p,n-q}(\u211D))$，即不定格拉斯曼流形，推导其性质并确定其几何与拓扑特征。
AI 中文摘要
齐性空间 $\u039F_{m,n}(\u211D)/(\u039F_{p,q}(\u211D) \times \u039F_{m-p,n-q}(\u211D))$ 是一个研究较少的对象，现有文献中仅两次简短提及，其中一次将其命名为不定格拉斯曼流形。本文从零开始推导其若干基本性质。我们会发现，除了齐性空间的描述外，不定格拉斯曼流形还可通过多种方式刻画：集合论层面，它是 $(m+n)$ 维空间中 $(p+q)$ 维不定子空间构成的流形；它是李群的伴随轨道；是矩阵的半代数光滑流形；是主丛的基空间，其全空间为对应的不定斯蒂费尔流形。它还可自然配备多种结构，成为伪黎曼流形、爱因斯坦流形、辛流形（仅在复数域上）和伪凯勒流形。作为本文核心，我们建立了不定格拉斯曼流形与标准格拉斯曼流形相关的两个性质：(i) 任何格拉斯曼流形都具有惠特尼分层，其最高维层为不定格拉斯曼流形；(ii) 不定格拉斯曼流形是两个格拉斯曼流形乘积的强形变收缩核，由此可完全确定前者的拓扑。我们还确定了不定格拉斯曼流形的若干可及特征，包括几何特征（黎曼曲率、里奇曲率、截面曲率、纯量曲率及第二基本形式）和拓扑特征（上同调环、同伦群、示性类）。
英文摘要
The homogeneous space $\operatorname{O}_{m,n}(\mathbb{R})/(\operatorname{O}_{p,q}(\mathbb{R}) \times \operatorname{O}_{m-p,n-q}(\mathbb{R}))$ is an object that has received scant attention, with just two brief mentions in existing literature, and christened the indefinite Grassmannian in one of them. In this article, we develop some of its basic properties, building it from ground up. We will see that, aside from its homogeneous space description, the indefinite Grassmannian may be characterized in several other ways: set-theoretically, it is the manifold of indefinite $(p+q)$-dimensional subspaces in $(m +n)$-dimensional space; it is an adjoint orbit of a Lie group; a semialgebraic smooth manifold of matrices; a base space of a principal bundle whose total space is the indefinite Stiefel manifold, a natural corresponding notion. It may also be naturally equipped with various structures, turning the indefinite Grassmannian into a pseudo-Riemannian manifold; a Einstein manifold; a symplectic manifold; and a pseudo-Kähler manifold (last two only over $\mathbb{C}$). As a centerpiece of this article, we establish two attributes of the indefinite Grassmannian in relation to the standard Grassmannian: (i) any Grassmannian has a Whitney stratification whose highest dimensional strata are indefinite Grassmannians; (ii) the indefinite Grassmannian is a strong deformation retract of a product of two Grassmannians, thereby allowing us to completely ascertain the topology of the former. We will also determine some of the indefinite Grassmannian's features that are within reach --- both geometric (Riemann, Ricci, sectional, and scalar curvatures; second fundamental form) and topological (cohomology ring, homotopy groups, characteristic classes).
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