Simple matrix expressions for the curvatures of Grassmannian

TL;DR

This paper presents remarkably simple matrix formulas for a wide range of curvature-related quantities of the Grassmannian, facilitating stable numerical computation and providing many new explicit expressions.

Contribution

It introduces straightforward matrix expressions for various geometric quantities of the Grassmannian using symmetric orthogonal matrices, many of which are novel.

Findings

Explicit formulas for Riemann, Ricci, and sectional curvatures

Simple expressions for fundamental forms and Weingarten maps

Many quantities derived are presented for the first time

Abstract

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.