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.
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.
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@article{CRMATH_2005__341_3_169_0, author = {Goldberg, Vladislav V. and Lychagin, Valentin V.}, title = {On linearization of planar three-webs and {Blaschke's} conjecture}, journal = {Comptes Rendus. Math\'ematique}, pages = {169--173}, publisher = {Elsevier}, volume = {341}, number = {3}, year = {2005}, doi = {10.1016/j.crma.2005.06.017}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2005.06.017/} }
TY - JOUR AU - Goldberg, Vladislav V. AU - Lychagin, Valentin V. TI - On linearization of planar three-webs and Blaschke's conjecture JO - Comptes Rendus. Mathématique PY - 2005 SP - 169 EP - 173 VL - 341 IS - 3 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2005.06.017/ DO - 10.1016/j.crma.2005.06.017 LA - en ID - CRMATH_2005__341_3_169_0 ER -
%0 Journal Article %A Goldberg, Vladislav V. %A Lychagin, Valentin V. %T On linearization of planar three-webs and Blaschke's conjecture %J Comptes Rendus. Mathématique %D 2005 %P 169-173 %V 341 %N 3 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2005.06.017/ %R 10.1016/j.crma.2005.06.017 %G en %F CRMATH_2005__341_3_169_0
Goldberg, Vladislav V.; Lychagin, Valentin V. 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. http://archive.numdam.org/articles/10.1016/j.crma.2005.06.017/
[1] Linearizability of d-webs, , on two-dimensional manifolds, Selecta Math., Volume 10 (2004) no. 4, pp. 431-451
[2] Einführung in die Geometrie der Waben, Birkhäuser, Basel, 1955 (108 p)
[3] 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] Four-webs in the plane and their linearizability, Acta Appl. Math., Volume 80 (2004) no. 1, pp. 35-55
[5] On the Blaschke conjecture for 3-webs, 2004, submitted for publication, 52 p. (see also arXiv) | arXiv
[6] On the linearizability of 3-webs, Nonlinear Anal., Volume 47 (2001) no. 4, pp. 2643-2654
[7] Mayer brackets and solvability of PDEs, Differential Geom. Appl., Volume 17 (2002) no. 2–3, pp. 251-272
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