Shortest Geodesic Loops, Sectional Curvature, and Injectivity Radius of the Stiefel Manifold
Provides exact geometric invariants for Stiefel manifolds, which are important in optimization and machine learning, but the results are incremental extensions of prior work.
The paper computes the length of the shortest nontrivial geodesic loops on the Stiefel manifold with a family of Riemannian metrics, and determines the exact injectivity radius for many metrics by combining curvature bounds.
We determine the length of the shortest nontrivial geodesic loops on the Stiefel manifold endowed with any member of the one-parameter family of Riemannian metrics introduced by Hüper et al. (2021). This family includes, in particular, the canonical and Euclidean metrics. By combining existing and new bounds on the sectional curvature, we determine the exact value of the injectivity radius of the Stiefel manifold under a wide range of members of the metric family.