The injectivity radius of the compact Stiefel manifold under the Euclidean metric
Ralf Zimmermann, Jakob Stoye
TL;DR
The paper determines the injectivity radius of the compact Stiefel manifold $St(n,p)$ under the Euclidean metric. By proving that Euclidean Stiefel geodesics are curves of constant Frenet curvatures and that the shortest closed geodesics have length $2\pi$, it combines this with known sectional curvature bounds ($-\tfrac{1}{2}\le K\le 1$) and Klingenberg's theorem to establish $\,\mathrm{inj}(St(n,p))=\pi$ for all $n\ge p$. The approach leverages an embedding-space Frenet-geometry view to connect geodesic behavior with global injectivity properties, yielding a result with direct implications for Riemannian normal coordinates and numerics on Stiefel manifolds. The work also points to future extensions to other metrics in the one-parameter family and to numerical verification of the exponential map along generic directions.
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 $π$.