No periodic geodesics in jet space

Alejandro Bravo-Doddoli

Vol. 322 (2023), No. 1, 11–19

DOI: 10.2140/pjm.2023.322.11

Abstract

The $J^k$ space of $k$-jets of a real function of one real variable $x$ admits the structure of a sub-Riemannian manifold, which then has an associated Hamiltonian geodesic flow, and it is integrable. As in any Hamiltonian flow, a natural question is the existence of periodic solutions. Does $J^k$ have periodic geodesics? This study will find the action-angle coordinates in $T^{\ast }{J}^k$ for the geodesic flow and demonstrate that geodesics in $J^k$ are never periodic.

Keywords

Carnot group, jet space, integrable system, Goursat distribution, sub-Riemannian geometry, Hamilton–Jacobi, periodic geodesics

Mathematical Subject Classification

Primary: 35R03, 53C17, 70Hxx

Milestones

Received: 9 April 2022

Revised: 30 November 2022

Accepted: 3 December 2022

Published: 3 May 2023