  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.
