Arguably, geodesics are the most important geometric objects on a differentiable manifold. They describe candidates for shortest paths and are guaranteed to be unique shortest paths when the starting velocity stays within the so-called injectivity radius of the manifold. In this work, we investigate the injectivity radius of the Stiefel manifold under the canonical metric. The Stiefel manifold $St(n,p)$ is the set of rectangular matrices of dimension $n$-by-$p$ with orthogonal columns, sometimes also called the space of orthogonal $p$-frames in $\mathbb{R}^n$. Using a standard curvature argument, Rentmeesters has shown in 2013 that the injectivity radius of the Stiefel manifold is bounded by $\sqrt{\frac{4}{5}}\pi$. It is an open question, whether this bound is sharp. With the definition of the injectivity radius via cut points of geodesics, we gain access to the information of the injectivity radius by investigating geodesics. More precisely, we consider the behavior of special variations of geodesics, called Jacobi fields. By doing so, we are able to present an explicit example of a cut point. In addition, since the theoretical analysis of geodesics for cut points and especially conjugate points as a type of cut points is difficult, we investigate the question of the sharpness of the bound by means of numerical experiments.

翻译：可以说，测地线是可微流形上最重要的几何对象。它们描述了最短路径的候选路径，并且当初始速度保持在流形所谓单射半径范围内时，测地线被保证是唯一的最短路径。本文研究了规范度量下Stiefel流形的单射半径。Stiefel流形$St(n,p)$是$n \times p$维正交列矩阵的集合，有时也称为$\mathbb{R}^n$中正交$p$-标架的空间。利用标准曲率论证，Rentmeesters于2013年证明Stiefel流形的单射半径存在上界$\sqrt{\frac{4}{5}}\pi$。该上界是否紧致仍是一个开放问题。通过基于测地线割点的单射半径定义，我们可以借助测地线研究获取单射半径的信息。具体而言，我们考察一类特殊测地线变分（称为Jacobi场）的行为。通过此方法，我们成功给出了一个割点的显式构造实例。此外，由于割点（特别是作为割点类型的共轭点）的理论分析具有较高难度，我们通过数值实验对该上界的紧致性问题进行了系统性探究。