Differential geometry/Dynamical systems
A Semple-type approach to a problem of Goursat: The multi-flag case
Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 921-925.

We consider the problem of classifying the orbits within a tower of fibrations with fibers diffeomorphic to projective planes and we generalize the tower of fiber bundles due to J. Semple. This tower, which was rediscovered by Montgomery and Zhitomirskii in the context of subriemannian geometry, admits a natural action of the diffeomorphism group of affine 3-space, and these orbits correspond to classes of Goursat multi-flags. We demonstrate that it is possible to classify many of these orbits by elementary means by appealing to some basic tools in projective geometry, and the combinatorics of spatial curves.

Nous nous intéressons au problème de classification des orbites dans une tour de fibration, où les fibres sont difféomorphes à des plans projectifs, généralisant ainsi les tours de fibrés projectif de J. Semple. Cette tour, redécouverte par R. Montgomery et M. Zhitomirskii dans le contexte de la géométrie sous-riemannienne, admet une action naturelle du groupe des difféomorphismes de lʼespace affine de dimension 3, et ces orbites correspondent à des classes de multi-drapeaux de Goursat. Nous démontrons quʼil est possible de classifier un grand nombre de ces orbites de manière élémentaire en utilisant des outils classiques de géométrie projective et la combinatoire des courbes spatiales.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.10.027
Castro, Alex L. 1; Howard, Wyatt C. 2

1 Departamento de Matemática, Pontíficia Universidade Católica do Rio de Janeiro, Brazil
2 Instituto National de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil
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Castro, Alex L.; Howard, Wyatt C. A Semple-type approach to a problem of Goursat: The multi-flag case. Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 921-925. doi : 10.1016/j.crma.2013.10.027. http://www.numdam.org/articles/10.1016/j.crma.2013.10.027/

[1] Arnold, V.I. Simple Singularities of Curves, Tr. Mat. Inst. Steklova, Volume 226 (1999), pp. 27-35

[2] Bryant, R.L.; Hsu, L. Rigidity of integral curves of rank 2 distributions, Invent. Math., Volume 114 (1993), pp. 435-461

[3] Bäcklund, A.V. Über Flächentransformationen, Math. Ann., Volume 9 (1875), pp. 297-320

[4] Castro, A.; Howard, W. A Monster Tower approach to Goursat multi-flags, Differ. Geom. Appl., Volume 30 (2012), pp. 405-427

[5] Castro, A.; Montgomery, R. Spatial curve singularities and the Monster/Semple Tower, Isr. J. Math. (2012), pp. 1-47

[6] Colley, S.; Kennedy, G. A higher-order contact formula for plane curves, Commun. Algebra, Volume 19 (1991), pp. 479-508

[7] Lejeune-Jalabert, M. Chains of points in the Semple Tower, Am. J. Math., Volume 128 (2006), pp. 1283-1311

[8] Montgomery, R.; Zhitomirskii, M. Points and curves in the Monster Tower, Mem. Am. Math. Soc., Volume 203 (2010) (x+137)

[9] Semple, J.G. Some investigations in the geometry of curve and surface elements, Proc. Lond. Math. Soc., Volume 3 (1954) no. 4, pp. 24-49

[10] Shibuya, K.; Yamaguchi, K. Drapeau theorem for differential systems, Differ. Geom. Appl., Volume 27 (2009), pp. 793-808

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