Geodesic complexity of homogeneous Riemannian manifolds

Abstract

We study the geodesic motion planning problem for complete Riemannian manifolds and investigate their geodesic complexity, an integer-valued isometry invariant introduced by D Recio-Mitter. Using methods from Riemannian geometry, we establish new lower and upper bounds on geodesic complexity and compute its value for certain classes of examples with a focus on homogeneous Riemannian manifolds. To achieve this, we study properties of stratifications of cut loci and use results on their structures for certain homogeneous manifolds obtained by T Sakai and others.