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：测地线无疑是可微流形上最重要的几何对象，它描述了最短路径的候选者，且当初始速度保持在流形的所谓单射半径内时，测地线一定是唯一的最短路径。本文研究了斯提费尔流形在标准度量下的单射半径。斯提费尔流形$St(n,p)$是由维度为$n\times p$且列正交的矩形矩阵构成的集合，有时也被称为$\mathbb{R}^n$中正交$p$-框架的空间。基于标准曲率论证，Rentmeesters在2013年指出斯提费尔流形的单射半径上界为$\sqrt{\frac{4}{5}}\pi$。该界是否为紧界仍是未解决问题。通过单射半径基于测地线切割点的定义，我们得以通过研究测地线获取单射半径信息。具体而言，我们考虑了测地线的特殊变分——雅可比场的性质。借助这一方法，我们能够给出一个切割点的显式示例。此外，由于对切割点及作为其特例的共轭点进行测地线理论分析较为困难，我们通过数值实验探究了该上界的紧致性。