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

Open Mathematics (2004)

- Volume: 2, Issue: 5, page 859-883
- ISSN: 2391-5455

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topPiotr Mormul. "Geometric classes of Goursat flags and the arithmetics of their encoding by small growth vectors." Open Mathematics 2.5 (2004): 859-883. <http://eudml.org/doc/268896>.

@article{PiotrMormul2004,

abstract = {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).},

author = {Piotr Mormul},

journal = {Open Mathematics},

keywords = {58A17},

language = {eng},

number = {5},

pages = {859-883},

title = {Geometric classes of Goursat flags and the arithmetics of their encoding by small growth vectors},

url = {http://eudml.org/doc/268896},

volume = {2},

year = {2004},

}

TY - JOUR

AU - Piotr Mormul

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

JO - Open Mathematics

PY - 2004

VL - 2

IS - 5

SP - 859

EP - 883

AB - 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).

LA - eng

KW - 58A17

UR - http://eudml.org/doc/268896

ER -

## References

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- [2] E. Cartan: “Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes”, Bull. Soc. Math. France, Vol. XLII, (1914), pp. 12–48. Zbl45.0472.04
- [3] M. Cheaito and P. Mormul: “Rank-2 distributions satisfying the Goursat condition: all their local models in dimension 7 and 8”, ESAIM: Control, Optimisation and Calculus of Variations, Vol. 4, (1999), pp. 137–158, (http://www.edpsciences.org/cocv) http://dx.doi.org/10.1051/cocv:1999107 Zbl0957.58002
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- [7] A. Kumpera and C. Ruiz: “Sur l'équivalence locale des systèmes de Pfaff en drapeau”, In: F. Gherardelli (Ed): Monge-Ampère Equations and Related Topics, Florence, 1980; Ist. Alta Math. F. Severi, Rome, 1982, pp. 201–248. Zbl0516.58004
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- [9] R. Montgomery and M. Zhitomirskii: “Geometric approach to Goursat flags”, Ann. Inst. H. Poincaré-AN, Vol. 18, (2001), pp. 459–493. http://dx.doi.org/10.1016/S0294-1449(01)00076-2 Zbl1013.58004
- [10] P. Mormul: “Local classification of rank-2 distributions satisfying the Goursat condition in dimension 9”, In: P. Orro and F. Pelletier (Eds): Singularités et géométrie sous-riemannienne, Chambéry, 1997; Travaux en cours, Vol. 62, Hermann, Paris, 2000, pp. 89–119.
- [11] W. Pasillas-Lépine and W. Respondek: “On the geometry of Goursat structures”, ESAIM: Control, Optimisation and Calculus of Variations, Vol. 6, (2001), pp. 119–181, (http://www.edpsciences.org/cocv). http://dx.doi.org/10.1051/cocv:2001106 Zbl0966.58002
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