Shortest Geodesic Loops, Sectional Curvature, and Injectivity Radius of the Stiefel Manifold

TL;DR

This paper calculates the shortest nontrivial geodesic loops and the injectivity radius of the Stiefel manifold for a family of Riemannian metrics, advancing understanding of its geometric properties.

Contribution

It provides exact values for the injectivity radius of the Stiefel manifold under various metrics, extending previous bounds with new curvature estimates.

Findings

Exact length of shortest geodesic loops determined

Injectivity radius computed for multiple metrics

Enhanced bounds on sectional curvature obtained

Abstract

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 H\"uper 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.