Cyclic pursuit on compact manifolds

Dmitri Gekhtman

Vol. 289 (2017), No. 1, 153–168

DOI: 10.2140/pjm.2017.289.153

Abstract

We study a form of cyclic pursuit on Riemannian manifolds with positive injectivity radius. We conjecture that on a compact manifold, the piecewise geodesic loop formed by connecting consecutive pursuit agents either collapses to a point in finite time or converges to a closed geodesic. The main result is that this conjecture is valid for nonpositively curved compact manifolds.

Keywords

cyclic pursuit, curve shortening, closed geodesics, nonpositive curvature

Mathematical Subject Classification 2010

Primary: 53B21

Secondary: 37D40

Milestones

Received: 4 October 2016

Revised: 18 December 2016

Accepted: 29 December 2016

Published: 12 May 2017