No Periodic Geodesics in $J^k(\mathbb{R},\mathbb{R}^n)$

Alejandro Bravo-Doddoli

Published 2022-05-12Version 1

The space of $k$-jets of $n$ real function of one real variable $x$ admits the structure of a Carnot group, which then has an associated Hamiltonian geodesic flow. As in any Hamiltonian flow, a natural question is the existence of periodic solutions. Does the space of $k$-jets have periodic geodesics? This study will demonstrate the integrability of \sR geodesic flow, characterize and classify the \sR geodesics in the space of $k$-jets, and show that they are never periodic.