Work

The Injectivity Radius of the Compact Stiefel Manifold under the Euclidean Metric

Year: 2025

Type: article

Abstract: . 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 [Formula: see text]-frames in [Formula: see text]. 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 [Formula: see text].

Source: SIAM Journal on Matrix Analysis and Applications

DOI: 10.1137/24m1663818

Authors: Ralf Zimmermann, Jakob Stoye

Institutions: University of Southern Denmark, Technische Universität Braunschweig

Language: English

Topic: Advanced Differential Geometry Research

Subfield: Astronomy and Astrophysics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W4407108970

FWCI: 1.827

Cites: 24

Cited by: 4

Related to: 10