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Contact hamiltonians distinguishing locally certain Goursat systems

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For the first time in dimension 9, the Goursat distributions are not locally smoothly classified by their small growth vector at a point. As shown in [M1], in dimension 9 of the underlying manifold 93 different local behaviours are possible and four irregular pairs of them have coinciding small growth vectors. In the present paper we distinguish geometrically objects in three of those pairs. Smooth functions in three variables - contact hamiltonians in the terminology of Arnold, [A] - help to do that. One pair of models, however, resists this technique. Another example of similar resistance in dimension 10 is also given - through the exact classification in dimension 10 of one family of local pseudo-normal forms (with redundant real constants) for Goursat objects. The latter result is an harbinger of more general phenomena that will be treated in a subsequent paper.
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  • Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland
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  • [M1] P. Mormul, Local classification of rank-2 distributions satisfying the Goursat condition in dimension 9, preprint N°582 Inst.of Math., Polish Acad. Sci. (1998); submitted to the Proceedings of the conference 'Singularités et Géométrie Sous-Riemannienne', Chambéry, October 1997.
  • [M2] P. Mormul, Goursat distributions with one singular hypersurface - constants important in their Kumpera-Ruiz pseudo-normal forms, preprint N°185, Labo de Topologie, Univ. de Bourgogne, Dijon (1999).
  • [PR] W. Pasillas-Lépine and W. Respondek, On the geometry of Goursat structures, preprint (1999).
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