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

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 $\R^n$. 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 $\pi$.