Asian Journal of Mathematics

Volume 17 (2013)

Number 1

Rigid flat webs on the projective plane

Pages: 163 – 192



David Marín (Departament de Matemàtiques, Universitat Autònoma de Barcelona, Spain)

Jorge Vitório Pereira (Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brasil)


This paper studies global webs on the projective plane with vanishing curvature. The study is based on an interplay of local and global arguments. The main local ingredient is a criterium for the regularity of the curvature at the neighborhood of a generic point of the discriminant. The main global ingredient, the Legendre transform, is an avatar of classical projective duality in the realm of differential equations. We show that the Legendre transform of what we call reduced convex foliations are webs with zero curvature, and we exhibit a countable infinity family of convex foliations which give rise to a family of webs with zero curvature not admitting non-trivial deformations with zero curvature.


web geometry, Legendre transform, flat webs

2010 Mathematics Subject Classification

14C21, 32S65, 53A60

Published 21 March 2013