Comptes Rendus
no. 3
Differential Geometry
On linearization of planar three-webs and Blaschke's conjecture
[Sur la linéarisation de 3-tissus plans et la conjecture de Blaschke]

Présenté par : Charles-Michel Marle

Vladislav V. Goldberg1 ; Valentin V. Lychagin2
1 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
2 Department of Mathematics and Statistics, University of Tromso, N-9037 Tromso, Norway
Comptes Rendus. Mathématique, Volume 341 (2005) no. 3, pp. 169-173.

Nous présentons des invariants relatifs différentiels d'ordre huit et neuf pour un 3-tissu plan non parallélisable dont l'annulation est nécessaire et suffisante pour que la 3-tissu soit linéarisable. Ceci apporte une solution a la conjecture de Blaschke pour le problème de linéarisation des 3-tissus.

We find relative differential invariants of orders eight and nine for a planar nonparallelizable 3-web such that their vanishing is necessary and sufficient for a 3-web to be linearizable. This solves the Blaschke conjecture for 3-webs.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.06.017
Vladislav V. Goldberg 1 ; Valentin V. Lychagin 2

1 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
2 Department of Mathematics and Statistics, University of Tromso, N-9037 Tromso, Norway 
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     title = {On linearization of planar three-webs and {Blaschke's} conjecture},
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Vladislav V. Goldberg; Valentin V. Lychagin. On linearization of planar three-webs and Blaschke's conjecture. Comptes Rendus. Mathématique, Volume 341 (2005) no. 3, pp. 169-173. doi : 10.1016/j.crma.2005.06.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.06.017/

[1] M.A. Akivis; V.V. Goldberg; V.V. Lychagin Linearizability of d-webs, d⩾4, on two-dimensional manifolds, Selecta Math., Volume 10 (2004) no. 4, pp. 431-451

[2] W. Blaschke Einführung in die Geometrie der Waben, Birkhäuser, Basel, 1955 (108 p)

[3] V.V. Goldberg On a linearizability condition for a three-web on a two-dimensional manifold, Peniscola 1988 (Lecture Notes in Math.), Volume vol. 1410, Springer, Berlin (1989), pp. 223-239

[4] V.V. Goldberg Four-webs in the plane and their linearizability, Acta Appl. Math., Volume 80 (2004) no. 1, pp. 35-55

[5] V.V. Goldberg; V.V. Lychagin On the Blaschke conjecture for 3-webs, 2004, submitted for publication, 52 p. (see also arXiv) | arXiv

[6] J. Grifone; Z. Muzsnay; J. Saab On the linearizability of 3-webs, Nonlinear Anal., Volume 47 (2001) no. 4, pp. 2643-2654

[7] B. Kruglikov; V. Lychagin Mayer brackets and solvability of PDEs, Differential Geom. Appl., Volume 17 (2002) no. 2–3, pp. 251-272

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