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2012 | 10 | 5 | 1889-1895
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On Galilean connections and the first jet bundle

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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.
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Bibliografia
  • [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
  • [3] Cartan É., Observations sur le mémoire précédent, Math. Z., 1933, 37(1), 619–622 http://dx.doi.org/10.1007/BF01474603
  • [4] Chern S.-S., Sur la géométrie d’un système d’équations différentielles du second ordre, Bull. Sci. Math., 1939, 63, 206–212
  • [5] Doubrov B., Komrakov B., Morimoto T., Equivalence of holonomic differential equations, Lobachevskii J. Math., 1999, 3, 39–71
  • [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
  • [7] Kosambi D.D., Parallelism and path-spaces, Math. Z., 1933, 37(1), 608–618 http://dx.doi.org/10.1007/BF01474602
  • [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
  • [9] Lackey B., Metric equivalence of path spaces, Nonlinear Studies, 2002, 7(2), 241–250
  • [10] Lie S., Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Teubner, Leipzig, 1891
  • [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
  • [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
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bwmeta1.element.doi-10_2478_s11533-012-0089-4
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