The injectivity radius of the compact Stiefel manifold under the Euclidean metric

Autor:	Zimmermann, Ralf, Stoye, Jakob
Rok vydání:	2024
Předmět:	Mathematics - Differential Geometry Mathematics - Numerical Analysis 15B10 15B57 65F99 53C30 53C80
Druh dokumentu:	Working Paper
Popis:	The injectivity radius of a manifold is an important quantity, both from a theoretical point of view and in terms of numerical applications. It is the largest possible radius within which all geodesics are unique and length-minimizing. In consequence, it is the largest possible radius within which calculations in Riemannian normal coordinates are well-defined. A matrix manifold that arises frequently in a wide range of practical applications is the compact Stiefel manifold of orthogonal $p$-frames in $\mathbb{R}^n$. We observe that geodesics on this manifold are space curves of constant Frenet curvatures. Using this fact, we prove that the injectivity radius on the Stiefel manifold under the Euclidean metric is $\pi$. Comment: 10 pages
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2405.02268 Zobrazit plný text záznamu View this record from Arxiv