Geodesic complexity of homogeneous Riemannian manifolds

Mescher, Stephan; Stegemeyer, Maximilian

doi:10.2140/agt.2023.23.2221

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Geodesic complexity of homogeneous Riemannian manifolds

Stephan Mescher and Maximilian Stegemeyer

Algebraic & Geometric Topology 23 (2023) 2221–2270

DOI: 10.2140/agt.2023.23.2221

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.

Keywords

geodesic complexity, topological complexity, sectional category, cut locus

Mathematical Subject Classification

Primary: 55M30

Secondary: 53C22

References

Publication

Received: 25 May 2021

Revised: 29 November 2021

Accepted: 9 January 2022

Published: 25 July 2023

Authors

Stephan Mescher
Institut für Mathematik Martin-Luther-Universität Halle-Wittenberg Halle Germany

Maximilian Stegemeyer
Mathematisches Institut Universität Leipzig Leipzig Germany

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Volume 23, issue 5 (2023)