# Geodesics in Jet Space

*2022, Volume 27, Number 2, pp. 151-182*

Author(s):

**Bravo-Doddoli A., Montgomery R.**

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 (sub-Riemannian 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 [7–9],
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.

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