Alejandro Bravo-Doddoli
Published 2020-03-18Version 1
Carnot groups are subRiemannian manifolds. As such they admit geodesic flows, which are left-invariant Hamiltonian flows on their cotangent bundles. Some of these flows are integrable. Some are not. The space of k-jets for real-valued functions on the real line forms a Carnot group of dimension $k+2$. We show that its geodesic flow is integrable and that its geodesics generalize Euler's elastica, with the case $k=2$ corresponding to the elastica, as shown by Sachkov and Ardentov.
No Periodic Geodesics in Jet Space
No Periodic Geodesics in $J^k(\mathbb{R},\mathbb{R}^n)$
Chaotic subRiemannian geodesic flow in $J^2(\mathbb{R}^2,\mathbb{R})$
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