Geodesic complexity via fibered decompositions of cut loci

TL;DR

This paper introduces a new method to estimate the geodesic complexity of Riemannian manifolds by decomposing their cut loci, providing bounds and exact values for specific symmetric spaces.

Contribution

It develops a novel approach using fibered decompositions of cut loci to bound and compute geodesic complexity, including for complex and quaternionic projective spaces.

Findings

Established a new upper bound for geodesic complexity.

Computed geodesic complexity for complex projective spaces.

Provided estimates for homogeneous manifolds.

Abstract

The geodesic complexity of a Riemannian manifold is a numerical isometry invariant that is determined by the structure of its cut loci. In this article we study decompositions of cut loci over whose components the tangent cut loci fiber in a convenient way. We establish a new upper bound for geodesic complexity in terms of such decompositions. As an application, we obtain estimates for the geodesic complexity of certain classes of homogeneous manifolds. In particular, we compute the geodesic complexity of complex and quaternionic projective spaces with their standard symmetric metrics.