Abstract
In this paper we give a new definition of the classical contact elements of a smooth manifold M as ideals of its ring of smooth functions: they are the kernels of Weil's near points. Ehresmann's jets of cross-sections of a fibre bundle are obtained as a particular case. The tangent space at a point of a manifold of contact elements of M is shown to be a quotient of a space of derivations from the same ring C ∞(M) into certain finite-dimensional local algebras. The prolongation of an ideal of functions from a Weil bundle to another one is the same ideal, when its functions take values into certain Weil algebras; following the same idea vector fields are prolonged, without any considerations about local one-parameter groups. As a consequence, we give an algebraic definition of Kuranishi's fundamental identification on Weil bundles, and study their affine structures, as a generalization of the classical results on spaces of jets of cross-sections.
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References
R. J. Alonso Blanco: D-modules, contact valued calculus and Poincaré-Cartan form. Submitted for publication.
C. Ehresmann: Les prolongements d'une varieté différentiable. I: Calcul del jets, prolongement principal; II: L'espace des jets d'ordre r de V n dans V m . C. R. Acad. Sci. Paris 233 (1951), 598–600, 777–779.
H. Goldschmidt: Integrability criteria for systems of non-linear partial differential equations. J. Differential Geom. 1 (1967), 269–307.
D. R. Grigore and D. Krupka: Invariants of velocities, and higher order Grassmann bundles. J. Geom. Phys 24 (1998), 244–264.
N. Jacobson: Lie Algebras. John Wiley &; Sons, Inc., New York, 1962.
I. Kolář: An infinite dimensional motivation in higher order geometry. Proc. Conf. Aug. 28–Sep. 1 1995. Brno, Czech Republic, 1996, pp. 151–159.
I. Kolář, P. W. Michor and J. Slovák: Natural Operations in Differential Geometry. Springer-Verlag, New York, 1993.
M. Kuranishi: Lectures on Involutive Systems. Publicações Soc. Math., São Paulo, 1969.
S. Lie: Theorie der Transformationsgruppen. Chelsea Publishing, New York, 1970.
A. Morimoto: Prolongation of connections to bundles of infinitely near points. J. Differential Geom. 11 (1976), 479–498.
J. Muñoz, F. J. Muriel and J. Rodríguez: Integrability of lie equations and pseudogroups. To appear in J. Math. Anal. Appl.
V. V. Shurygin: Manifolds over algebras and their application to the geometry of jet bundles. Russian Math. Surveys 48 (1993), 75–104.
A. Weil: Théorie des points proches sur les variétés différentiables. Colloque de Géometrie Différentielle, C.N.R.S. (1953), 111–117.
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Muñoz, J., Rodriguez, J. & Muriel, F.J. Weil bundles and jet spaces. Czechoslovak Mathematical Journal 50, 721–748 (2000). https://doi.org/10.1023/A:1022408527395
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DOI: https://doi.org/10.1023/A:1022408527395