EMIS ELibM Electronic Journals Journal of Lie Theory
Vol. 15, No. 1, pp. 135–143 (2005)

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The only global contact transformations of order two or more are point transformations

Ricardo J. Alonso-Blanco and David Blázquez-Sanz

Ricardo J. Alonso-Blanco
David Blázquez-Sanz
Departamento de Matemáticas
Universidad de Salamanca
Plaza de la Merced 1-4
E-37008 Salamanca, Spain

Abstract: Let us consider $ J_m^kM $ as the space of $k$-jets of $m$-dimensional submanifolds of a smooth manifold $M$. Our purpose is to show that every contact transformation of $ J_m^kM $, $k\ge 2$, is induced by a diffeomorphism of $M$ (point transformations). It is also derived that a first order contact transformation can not be globally prolonged to higher order jets except when it is a point transformation. This holds true as well for jets of sections of a regular projection. The Legendre transformation gives us an example of this property. \endgraf {\eightsl Subject Matter Classification 2000}: 58A20 ; 58A3

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