We show that modeling a Grassmannian as symmetric orthogonal matrices $\operatorname{Gr}(k,\mathbb{R}^n) \cong\{Q \in \mathbb{R}^{n \times n} : Q^{\scriptscriptstyle\mathsf{T}} Q = I, \; Q^{\scriptscriptstyle\mathsf{T}} = Q,\; \operatorname{tr}(Q)=2k - n\}$ yields exceedingly simple matrix formulas for various curvatures and curvature-related quantities, both intrinsic and extrinsic. These include Riemann, Ricci, Jacobi, sectional, scalar, mean, principal, and Gaussian curvatures; Schouten, Weyl, Cotton, Bach, Pleba\'nski, cocurvature, nonmetricity, and torsion tensors; first, second, and third fundamental forms; Gauss and Weingarten maps; and upper and lower delta invariants. We will derive explicit, simple expressions for the aforementioned quantities in terms of standard matrix operations that are stably computable with numerical linear algebra. Many of these aforementioned quantities have never before been presented for the Grassmannian.

翻译：我们证明，将格拉斯曼流形建模为对称正交矩阵 $\operatorname{Gr}(k,\mathbb{R}^n) \cong\{Q \in \mathbb{R}^{n \times n} : Q^{\scriptscriptstyle\mathsf{T}} Q = I, \; Q^{\scriptscriptstyle\mathsf{T}} = Q,\; \operatorname{tr}(Q)=2k - n\}$ 能够为各种内蕴和外蕴曲率及曲率相关量导出极其简单的矩阵公式。这些量包括黎曼、里奇、雅可比、截面、标量、平均、主曲率和高斯曲率；舒滕、外尔、科顿、巴赫、普莱班斯基、余曲率、非度量性及挠率张量；第一、第二和第三基本形式；高斯映射和魏因加滕映射；以及上、下德尔塔不变量。我们将利用标准矩阵运算，推导出上述量的显式简单表达式，这些表达式可通过数值线性代数稳定计算。上述许多量此前从未在格拉斯曼流形中被给出过。