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.
@article{bwmeta1.element.doi-10_2478_s11533-012-0089-4,
author = {James Grant and Bradley Lackey},
title = {On Galilean connections and the first jet bundle},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1889-1895},
zbl = {1262.53011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0089-4}
}
James Grant; Bradley Lackey. On Galilean connections and the first jet bundle. Open Mathematics, Tome 10 (2012) pp. 1889-1895. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0089-4/
[1] Čap A., Slovák J., Souček V., Bernstein-Gelfand-Gelfand sequences, Annals of Math., 2001, 154(1), 97–113 http://dx.doi.org/10.2307/3062111
[2] Cartan É., Sur les variétés à connexion projective, Bull. Soc. Math. France, 1924, 52, 205–241 | Zbl 50.0500.02
[3] Cartan É., Observations sur le mémoire précédent, Math. Z., 1933, 37(1), 619–622 http://dx.doi.org/10.1007/BF01474603 | Zbl 0007.23101
[4] Chern S.-S., Sur la géométrie d’un système d’équations différentielles du second ordre, Bull. Sci. Math., 1939, 63, 206–212 | Zbl 65.1419.01
[5] Doubrov B., Komrakov B., Morimoto T., Equivalence of holonomic differential equations, Lobachevskii J. Math., 1999, 3, 39–71 | Zbl 0937.37051
[6] Kamran N., Lamb K.G., Shadwick W.F., The local equivalence problem for d 2y/dx 2 = F(x; y; dy/dx) and the Painlevé transcendents, J. Differential Geom., 1985, 22(2), 139–150 | Zbl 0594.34034
[7] Kosambi D.D., Parallelism and path-spaces, Math. Z., 1933, 37(1), 608–618 http://dx.doi.org/10.1007/BF01474602 | Zbl 0007.23004
[8] Kosambi D.D., Systems of differential equations of the second order, Quart. J. Math. Oxford Ser., 1935, 6, 1–12 http://dx.doi.org/10.1093/qmath/os-6.1.1 | Zbl 0011.08203
[9] Lackey B., Metric equivalence of path spaces, Nonlinear Studies, 2002, 7(2), 241–250 | Zbl 1001.53008
[10] Lie S., Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Teubner, Leipzig, 1891 | Zbl 43.0373.01
[11] Saunders D.J., The Geometry of Jet Bundles, London Math. Soc. Lecture Note Ser., 142, Cambridge University Press, Cambridge, 1989 http://dx.doi.org/10.1017/CBO9780511526411 | Zbl 0665.58002
[12] Sharpe R.W., Differential Geometry, Grad. Texts in Math., 166, Springer, New York, 1997
[13] Tresse A., Détermination des invariants ponctuels de l’équation différentielle ordinaire du second ordre y″ = ω(x; y; y′), Hirzel, Leipzig, 1896 | Zbl 27.0254.01