The ultimate upper bound on the injectivity radius of the Stiefel manifold

We exhibit conjugate points on the Stiefel manifold endowed with any member of the family of Riemannian metrics introduced by H"uper et al. (2021). This family contains the well-known canonical and Euclidean metrics. An upper bound on the injectivity radius of the Stiefel manifold in the considered metric is then obtained as the minimum between the length of the geodesic along which the points are conjugate and the length of certain geodesic loops. Numerical experiments support the conjecture that the obtained upper bound is in fact equal to the injectivity radius.