Détermination du rang des tissus du plan et autres invariants géométriques

Let W(d) be a non singular d-web in the plane with d⩾3, presented by a first order differential equation of the type F(x,y,y′):=a0(x,y)⋅(y′)d+a1(x,y)⋅(y′)d−1+⋯+ad(x,y)=0, where ai∈C{x,y} and let (E,∇) 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 W(d). Our main result is an explicit determination of the rank of W(d). 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).