Gekhtman, Dmitri (2017) Cyclic pursuit on compact manifolds. Pacific Journal of Mathematics, 289 (1). pp. 153-168. ISSN 0030-8730. doi:10.2140/pjm.2017.289.153. https://resolver.caltech.edu/CaltechAUTHORS:20170712-131427876
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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.
| Item Type: | Article | ||||||||||||
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| Additional Information: | © 2017 Mathematical Sciences Publishers. Received: 4 October 2016; Revised: 18 December 2016; Accepted: 29 December 2016; Published: 12 May 2017. This research was conducted mostly at the SUMMER@ICERM Undergraduate Summer Research Program in 2012. I would like to thank Tarik Aougab and Sergei Tabachnikov for their mentorship. I would like to thank Francisc Bozgan for pointing out the application of Jensen’s inequality in Section 3. I would like to thank Anton Petrunin for a MathOverflow answer which helped with the proof of Proposition 8.1. | ||||||||||||
| Subject Keywords: | cyclic pursuit, curve shortening, closed geodesics, nonpositive curvature | ||||||||||||
| Issue or Number: | 1 | ||||||||||||
| Classification Code: | MSC2010: primary 53B21; secondary 37D40 | ||||||||||||
| DOI: | 10.2140/pjm.2017.289.153 | ||||||||||||
| Record Number: | CaltechAUTHORS:20170712-131427876 | ||||||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20170712-131427876 | ||||||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||
| ID Code: | 79015 | ||||||||||||
| Collection: | CaltechAUTHORS | ||||||||||||
| Deposited By: | Tony Diaz | ||||||||||||
| Deposited On: | 12 Jul 2017 21:25 | ||||||||||||
| Last Modified: | 15 Nov 2021 17:44 |
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