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Geodesic complexity of homogeneous Riemannian manifolds

Stephan Mescher and Maximilian Stegemeyer

Algebraic & Geometric Topology 23 (2023) 2221–2270
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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