Geodesic complexity of homogeneous Riemannian manifolds

TL;DR

This paper investigates the geodesic complexity of homogeneous Riemannian manifolds, providing bounds and exact values by analyzing cut loci structures, advancing understanding of motion planning in geometric contexts.

Contribution

It introduces new bounds and computes the geodesic complexity for specific homogeneous Riemannian manifolds using stratification and cut locus analysis.

Findings

Established new lower and upper bounds on geodesic complexity.

Computed geodesic complexity for certain classes of homogeneous manifolds.

Analyzed stratifications of cut loci to inform complexity bounds.

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. Methodically, we study properties of stratifications of cut loci and use results on their structures for certain homogeneous manifolds obtained by T. Sakai and others.