Differential geometry
On the linearizability of 3-webs: End of controversy
Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 97-99.

There are two theories describing the linearizability of 3-webs: one is developed in [10] and another in [8]. Unfortunately they cannot be both correct because on an explicit 3-web W0 they contradict: the first predicts that W0 is linearizable, while the second states that W0 is not linearizable. The essential question beyond this particular 3-web is: which theory describes correctly the linearizability condition? In this paper, we present a very short proof, due to J.-P. Dufour, that W0 is linearizable, confirming the result of [10].

Il existe deux théories décrivant la linéarisabilité des 3-tissus : l'une est développée dans [10], l'autre dans [8]. Malheureusement, elles ne peuvent pas être correctes toutes les deux, car sur un 3-tissu W0 elles se contredisent : la première prédit que le tissu W0 est linéarisable, tandis que la seconde affirme que W0 n'est pas linéarisable. La question essentielle au-delà de ce 3-tissu particulier est : quelle théorie décrit correctement la condition de linéarisabilité ? Dans cet article, nous présentons une preuve très courte, due à J.-P. Dufour, de ce que le tissu W0 est linéarisable, confirmant le résultat de [10].

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DOI: 10.1016/j.crma.2017.12.006
Muzsnay, Zoltán 1

1 Institute of Mathematics, University of Debrecen, H-4002 Debrecen, Pf. 400, Hungary
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Muzsnay, Zoltán. On the linearizability of 3-webs: End of controversy. Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 97-99. doi : 10.1016/j.crma.2017.12.006. http://www.numdam.org/articles/10.1016/j.crma.2017.12.006/

[1] Agafonov, S.I. Gronwall's conjecture for 3-webs with infinitesimal symmetries, 2015 | arXiv

[2] Agafonov, S.I. Counterexample to Gronwall's conjecture, 2017 | arXiv

[3] Blaschke, W. Einführung in die Geometrie der Waben, Birkhäuser Verlag, Basel und Stuttgart, 1955

[4] Bol, G. Topologische Fragen der Differentialgeometrie 31. Geradlinige Kurvengewebe, Abh. Math. Semin. Univ. Hamb., Volume 8 (1931) no. 1, pp. 264-270

[5] Chern, S.-S. Eine Invariantentheorie der Dreigewebe aus r-dimensionalen Mannigfaltigkeiten im R2r, Abh. Math. Semin. Univ. Hamb., Volume 11 (1935) no. 1, pp. 333-358

[6] Goldberg, V.V. 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

[7] Goldberg, V.V.; Lychagin, V.V. On linearization of planar three-webs and Blaschke's conjecture, C. R. Acad. Sci. Paris, Ser. I, Volume 341 (2005) no. 3, pp. 169-173

[8] Goldberg, V.V.; Lychagin, V.V. On the Blaschke conjecture for 3-webs, J. Geom. Anal., Volume 16 (2006) no. 1, pp. 69-115

[9] Griffiths, P.A. Variations on a theorem of Abel, Invent. Math., Volume 35 (1976), pp. 321-390

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

[11] Grifone, J.; Muzsnay, Z.; Saab, J. Linearizable 3-webs and the Gronwall conjecture, Publ. Math. (Debr.), Volume 71 (2007) no. 1–2, pp. 207-227

[12] Muzsnay, Z. On the problem of linearizability of a 3-web, Nonlinear Anal., Volume 68 (2008) no. 6, pp. 1595-1602

[13] Nagy, P.T. Invariant tensor fields and the canonical connection of a 3-web, Aequ. Math., Volume 35 (1988) no. 1, pp. 31-44

[14] Pereira, J.V.; Pirio, L. An Invitation to Web Geometry, IMPA Monographs, vol. 2, Springer, Cham, Switzerland, 2015

[15] Wang, J.S. On the Gronwall conjecture, J. Geom. Anal., Volume 22 (2012) no. 1, pp. 38-73

Cited by Sources:

This work is partially supported by the EFOP-3.6.2-16-2017-00015 project and by the EFOP-3.6.1-16-2016-00022 project. The projects have been supported by the European Union, co-financed by the European Social Fund.