Geodesics in Jet Space

TL;DR

This paper explores the structure of geodesics in jet spaces modeled as Carnot groups, identifying which polynomial curves are globally minimizing and analyzing their geometric properties.

Contribution

It provides a partial characterization of globally minimizing geodesics in jet spaces beyond horizontal lifts, using Hamilton-Jacobi equations and period analysis.

Findings

Horizontal lifts are globally minimizing geodesics.

Certain polynomial curves are identified as potential minimizers.

Conjecture on the independence of cut time from initial position.

Abstract

The space $J^k$ of $k$-jets of a real function of one real variable $x$ admits the structure of Carnot group type. As such, $J^k$ admits a submetry (\sR submersion) onto the Euclidean plane. Horizontal lifts of Euclidean lines (which are the left-translates of horizontal one-parameter subgroups) are thus globally minimizing geodesics on $J^k$. All $J^k$-geodesics, minimizing or not, are constructed from degree $k$ polynomials in $x$ according to Anzaldo-Meneses and Monroy-Per\'ez, reviewed here. The constant polynomials correspond to the horizontal lifts of lines. Which other polynomials yield globally minimizers and what do these minimizers look like? We give a partial answer. Our methods include constructing an intermediate three-dimensional "magnetic" sub-Riemannian space lying between the jet space and the plane, solving a Hamilton-Jacobi (eikonal) equations on this space, and analyzing period asymptotics associated to period degenerations arising from two-parameter families of these polynomials. Along the way, we conjecture the independence of the cut time of any geodesic on jet space from the starting location on that geodesic.