On Galilean connections and the first jet bundle

TitleOn Galilean connections and the first jet bundle
Publication TypeJournal Article
Year of Publication2012
AuthorsDE Grant, J, Lackey, B
JournalCentral European Journal of Mathematics
Date Published2012/10/01

We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations — sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order ordinary differential equations. We then show the converse — the “fundamental theorem” — that given such a coordinate system, and a system of second order ordinary differential equations, there exists regular Cartan connections yielding these, and such connections are completely determined by their torsion.