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2004 | 65 | 1 | 157-178
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Multi-dimensional Cartan prolongation and special k-flags

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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 flag ℱ being a k-flag possessing a corank 1 involutive subflag preserving the Lie square of ℱ. (Note that 1-flags are always special.)
Secondly, we show that special k-flags are effectively nilpotentizable (or: weakly nilpotent) in the sense that local polynomial pseudo-normal forms for such D resulting naturally from sequences of gCp's give local nilpotent bases for D. Moreover, the nilpotency orders of the generated real Lie algebras can be explicitly computed by means of simple linear algebra (for k = 1 this was done earlier in [M1], [M3]). For k = 2 we also transform our linear algebra formulas into recursive ones that resemble a bit Jean's formulas [Je] for nonholonomy degrees of Goursat germs.
Additionally it is shown that, when all parameters appearing in a local form for a special k-flag vanish, then such a distribution germ is also strongly nilpotent in the sense of [AGau] and [M1].
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  • Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland
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