Geodesics in Jet Space

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