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Comptes Rendus Mathématique
Volume 341, n° 4
pages 247-252 (août 2005)
Doi : 10.1016/j.crma.2005.07.007
Received : 10 January 2005 ;  accepted : 9 July 2005
Détermination du rang des tissus du plan et autres invariants géométriques
Determination of the rank of planar webs and other geometrical invariants
 

Olivier Ripoll
LaBAG, UMR 5467, 351, cours de la Libération, 33405 Talence, France 

Résumé

Soit   un d -tissu non singulier du plan présenté par une équation différentielle du premier ordre à coefficients dans   de la forme  , avec   et dont on note   sa connexion associée. Nous montrons que la trace de la courbure de   est la somme des courbures de Blaschke des 3-tissus extraits. En outre nous indiquons comment la courbure rend compte de la linéarisabilité du tissu  . Notre résultat principal est un procédé explicite de détermination pour d quelconque du rang de  , à partir des coefficients de F . En application, nous retrouvons également des résultats connus en géométrie des tissus. Pour citer cet article : O. Ripoll, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

The full text of this article is available in PDF format.
Abstract

Let   be a non singular d -web in the plane with  , presented by a first order differential equation of the type  , where   and let   be the connection associated with F . We show that the trace of its curvature is the sum of the Blaschke curvatures of extracted 3-webs of  . Our main result is an explicit determination of the rank of  . We also recover some well known results in web geometry. To cite this article: O. Ripoll, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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