Wyniki wyszukiwania

1

Contact hamiltonians distinguishing locally certain Goursat systems

100%

Mormul P.

Banach Center Publications

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2000

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tom 51

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nr 1

219-230

EN

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.

2

Multi-dimensional Cartan prolongation and special k-flags

100%

Mormul P.

Banach Center Publications

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2004

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tom 65

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nr 1

157-178

EN

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].

3

Geometric classes of Goursat flags and the arithmetics of their encoding by small growth vectors

100%

Mormul P.

Open Mathematics

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2004

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tom 2

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nr 5

859-883

EN

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 encoded by words of length r over the alphabet {G,S,T} (Generic, Singular, Tangent) starting with two letters G and having letter(s) T only directly after an S, or directly after another T. It follows from [6] that the Goursat germs sitting in any fixed geometric class have, up to translations by rk D−2, one and the same small growth vector (at the reference point) that can be computed recursively in terms of the G,S,T code. In the present paper we give explicit solutions to the recursive equations of Jean and show how, thanks to a surprisingly neat underlying arithmetics, one can algorithmically read back the relevant geometric class from a given small growth vector. This gives a secondary, Gödel-like super-encoding of the geometric classes of Goursat objects (rather than just a 1-1 correspondence between those classes and small growth vectors).

4

Singularities of triples of vector fields on $R^{4}$: the focusing stratum

44%

Mormul P.

Studia Mathematica

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1988

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tom 91

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nr 3

241-273

Ograniczanie wyników

2 Banach Center Publications

1 Open Mathematics

1 Studia Mathematica

4 Mormul P.

2 2004

1 2000

1 1988

Contact hamiltonians distinguishing locally certain Goursat systems

Multi-dimensional Cartan prolongation and special k-flags

Geometric classes of Goursat flags and the arithmetics of their encoding by small growth vectors

Singularities of triples of vector fields on $R^{4}$: the focusing stratum