On local classi cation of Goursat structures
نویسنده
چکیده
Goursat structures are the dual objects of E. Cartan's \special" systems (also called \en drapeau" in 7]). We prove that the growth vector is not a complete local invariant for a Goursat structure on R n , n 9; and that there exist innnitely many locally non-equivalent Goursat structures on R n , n 10. Sur la classiication locale des structures de Goursat Les structures de Goursat sont les objets duaux des syst emes \sp eciaux" consi-d er es par E. Cartan (dits \en drapeau" dans 7]). Nous d emontrons que le vecteur de croissance n'est pas un invariant local complet pour une structure de Goursat sur R n , n 9 ; et qu'il existe une innnit e de structures de Goursat localement non-equivalentes sur R n , n 10. Version frann caise abr eg ee Une distribution de rang k sur une vari et e M est un sous-br e vectoriel du br e tangent TM dont la bre type est de dimension k. D'une mani ere analogue, un syst eme de Pfaa de rang s est un sous-br e vectoriel du br e cotangent T M dont la bre type est de dimension s. On dit que deux distributions ou syst emes de Pfaa sont equivalents s'ils sont isomorphes en tant que br es vectoriels. Toute distribution D engendre une suite D 0 D 1 , d eenie par D 0 = D et D k+1 = D k +D 0 ; D k ]. On appelle vecteur de croissance au point p la suite (dimD k (p)) k2f0;:::;rg ; o u r = r(p) est le plus petit entier (eventuellement innni) pour lequel D r (p) = T p M. Une structure de Goursat sur une vari et e M de dimension n est une distribution D, de rang 2, pour laquelle en tout point p 2 M on a dimD (k)
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