Vol. 289, No. 1, 2017

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ISSN: 0030-8730
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
Authors
Dmitri Gekhtman
Department of Mathematics
California Institute of Technology
Pasadena, CA 91125
United States