Cyclic Pursuit on Compact Manifolds

TL;DR

This paper investigates cyclic pursuit dynamics on compact Riemannian manifolds, proving that on nonpositively curved compact manifolds, pursuit agents either collide or form a closed geodesic.

Contribution

It establishes the conjecture that pursuit loops either collapse or converge to closed geodesics specifically on nonpositively curved compact manifolds.

Findings

Pursuit loops collapse in finite time or form closed geodesics.

The conjecture holds for nonpositively curved compact manifolds.

Provides insights into pursuit dynamics on curved spaces.

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 in finite time or converges to a closed geodesic. The main result is that this conjecture is valid for nonpositively curved compact manifolds.