The Geometry of Quasi-Geodesics on Stiefel Manifolds

Abstract

Differential geometry supplies important mathematical tools for engineering applications. Geodesics play a crucial role in many iterative methods and, when there are explicit formulas to write them, they provide elegant and computationally efficient methods to solve real problems. Unfortunately, for some manifolds such formulas are hard to find. This is the case for the Stiefel manifold that plays an important role in computer vision and pattern recognition. Quasi-geodesics is another class of curves with constant geodesic curvature that have proved to be particularly useful to implement interpolating algorithms on Stiefel manifolds. The main objective of this paper is to reveal its interesting geometry, by showing their connection with solutions of a sub-Riemannian optimal control problem on a Lie group that acts on the manifold.