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

TL;DR

This paper proves that in the space of k-jets of real functions, the subRiemannian geodesic flow is integrable but does not admit any periodic geodesics, revealing fundamental geometric properties.

Contribution

It characterizes and classifies subRiemannian geodesics in jet spaces and demonstrates the non-existence of periodic geodesics in this setting.

Findings

Geodesic flow is integrable

Geodesics are never periodic

Provides classification of geodesics in jet spaces

Abstract

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 subRiemannian geodesic flow, characterize and classify the subRiemannian geodesics in the space of $k$-jets, and show that they are never periodic.