Simple matrix expressions for the curvatures of Grassmannian
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arXiv
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| Format: | Preprint |
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2024
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| author | Lai, Zehua Lim, Lek-Heng Ye, Ke |
| author_facet | Lai, Zehua Lim, Lek-Heng Ye, Ke |
| contents | 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, Plebański, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_11821 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simple matrix expressions for the curvatures of Grassmannian Lai, Zehua Lim, Lek-Heng Ye, Ke Differential Geometry Numerical Analysis Optimization and Control 14M15, 15B10, 53C21, 53C17 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, Plebański, 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. |
| title | Simple matrix expressions for the curvatures of Grassmannian |
| topic | Differential Geometry Numerical Analysis Optimization and Control 14M15, 15B10, 53C21, 53C17 |
| url | https://arxiv.org/abs/2406.11821 |