### Superfixed positions in the geometry of Goursat flags.

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Goursat distributions are subbundles, of codimension at least 2, in the tangent bundles to manifolds having the flag of consecutive Lie squares of ranks not depending on a point and growing-very slowly-always by 1. The length of a flag thus equals the corank of the underlying distribution. After the works of, among others, Bryant&Hsu (1993), Jean (1996), and Montgomery&Zhitomirskii (2001), the local behaviours of Goursat flags of any fixed length r≥2 are stratified into geometric classes...

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...

Since the mid-nineties it has gradually become understood that the Cartan prolongation of rank 2 distributions is a key operation leading locally, when applied many times, to all so-called Goursat distributions. That is those, whose derived flag of consecutive Lie squares is a 1-flag (growing in ranks always by 1). We first observe that successive generalized Cartan prolongations (gCp) of rank k + 1 distributions lead locally to all special k-flags: rank k + 1 distributions D with the derived...

We study the rank–2 distributions satisfying so-called Goursat condition (GC); that is to say, codimension–2 differential systems forming with their derived systems a flag. Firstly, we restate in a clear way the main result of[7] giving preliminary local forms of such systems. Secondly – and this is the main part of the paper – in dimension 7 and 8 we explain which constants in those local forms can be made 0, normalizing the remaining ones to 1. All constructed equivalences are explicit. ...

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