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

Piotr Mormul 1
  • 1 Warsaw University


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).

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] R.L. Bryant and L. Hsu: “Rigidity of integral curves of rank 2 distributions”, Invent. math., Vol. 114, (1993), pp. 435–461.

  • [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.

  • [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, (

  • [4] M. Gaspar: “Sobre la clasificacion de sistemas de Pfaff en bandera”, In: Proceedings of 10th Spanish-Portuguese Conference on Math., University of Murcia, 1985, pp. 67–74 (in Spanish).

  • [5] B. Jacquard: Le problème de la voiture à 2, 3, et 4 remorques, Preprint, DMI, ENS, Paris, 1993.

  • [6] F. Jean: “The car with N trailers: characterisation of the singular configurations”, ESAIM: Control, Optimisation and Calculus of Variations, Vol. 1, (1996), pp. 241–266, (http://www.edpsciences/cocv).

  • [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.

  • [8] F. Luca and J.-J. Risler: “The maximum of the degree of nonholonomy for the car with N trailers”, In: Proceedings of the 4th IFAC Symposium on Robot Control, Capri, 1994, pp. 165–170.

  • [9] R. Montgomery and M. Zhitomirskii: “Geometric approach to Goursat flags”, Ann. Inst. H. Poincaré—AN, Vol. 18, (2001), pp. 459–493.

  • [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, (

  • [12] E. Von Weber: “Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen”, Berichte Ges. Leipzig, Math-Phys. Classe, Vol. L, (1898), pp. 207–229.


Journal + Issues