A canonical frame for nonholonomic rank two distributions of maximal class

Boris Doubrov; Igor Zelenko

doi:10.1016/j.crma.2006.02.010

Differential Geometry

A canonical frame for nonholonomic rank two distributions of maximal class

Presented by: Mikhaël Gromov

Boris Doubrov ¹; Igor Zelenko ²

¹ The Faculty of Applied Mathematics, Belorussian State University, F. Skaryny Ave. 4, Minsk, Belarus 220050, Belarus
² SISSA, via Beirut 2–4, 34014 Trieste, Italy

Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 589-594.

Abstract
Résumé

In 1910 E. Cartan constructed the canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in $R^{5}$ . We solve the analogous problems for rank 2 distributions in $R^{n}$ for arbitrary $n > 5$ . Our method is a kind of symplectification of the problem and it is completely different from the Cartan method of equivalence.

En 1910 E. Cartan a construit un repère canonique et a trouvé le cas le plus symétrique des distributions de rang 2 et non holonômes de manière maximale dans $R^{5}$ . Nous résolvons ici des problèmes analogues pour les distributions de rang 2 dans $R^{n}$ avec $n > 5$ arbitraire. Notre méthode est une sorte de symplectification du problème et est complètement différente de la méthode par équivalence de Cartan.

Received: 2005-12-07
Accepted: 2006-02-07
Published online: 2006-03-09

DOI: 10.1016/j.crma.2006.02.010

Author's affiliations:

Boris Doubrov ¹; Igor Zelenko ²

¹ The Faculty of Applied Mathematics, Belorussian State University, F. Skaryny Ave. 4, Minsk, Belarus 220050, Belarus
² SISSA, via Beirut 2–4, 34014 Trieste, Italy

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     title = {A canonical frame for nonholonomic rank two distributions of maximal class},
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Boris Doubrov; Igor Zelenko. A canonical frame for nonholonomic rank two distributions of maximal class. Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 589-594. doi : 10.1016/j.crma.2006.02.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.02.010/

References
Cited by

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[3] E. Cartan Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre, Œuvres completes, Partie II, vol. 2, Gauthier-Villars, Paris, 1953, pp. 927-1010

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[5] N. Tanaka On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J., Volume 6 (1979), pp. 23-84

[6] I. Zelenko, Variational approach to differential invariants of rank 2 vector distributions, Differential Geometry Appl., 30 pages, in press, , SISSA preprint 12/2004/M | arXiv

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