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

TL;DR

This paper determines that the injectivity radius of the compact Stiefel manifold under the Euclidean metric is π, which is crucial for understanding geodesic uniqueness and numerical computations on this manifold.

Contribution

It provides a rigorous proof that the injectivity radius of the compact Stiefel manifold under the Euclidean metric is π, leveraging the properties of geodesics as space curves with constant Frenet curvatures.

Findings

Injectivity radius of the Stiefel manifold is π.

Geodesics are space curves with constant Frenet curvatures.

The result aids in numerical applications involving the manifold.

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 $p$-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 $\pi$.