No closed-form formula for Stiefel geodesics

Abstract

We use differential Galois theory to prove that given two arbitrary points $X$ and $Y$ on the Stiefel manifold, there is no closed-form analytic expression for the Riemannian logarithm $\log_X Y$, the geodesic between $X$ and $Y$, or the geodesic distance between $X$ and $Y$. In other words, these cannot be expressed in terms of elementary functions of the coordinates of $X$ and $Y$; in fact, our result goes further --- even if we allow for repeated integrals of elementary functions, there is still no such expression.