No Periodic Geodesics in Jet Space
Alejandro Bravo-Doddoli

TL;DR
This paper proves that the space of k-jets of real functions, equipped with a sub-Riemannian structure, has no periodic geodesics, by explicitly constructing action-angle coordinates and analyzing the flow.
Contribution
It provides the first explicit construction of action-angle coordinates for the geodesic flow on jet spaces and shows the non-existence of periodic geodesics in this setting.
Findings
Geodesic flow on jet spaces is integrable.
No periodic geodesics exist in $J^k$.
Explicit action-angle coordinates are constructed.
Abstract
The space of -jets of a real function of one real variable 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 have periodic geodesics? This study will find the action-angle coordinates in for the geodesic flow and demonstrate that geodesics in are never periodic.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Advanced Differential Geometry Research · Black Holes and Theoretical Physics
