# Géométrie des tissus du plan et équations différentielles

Abstract : Let $\mathcal{W}(d)$ be a non singular planar $d$-web, implicitly presented by a differential equation $F$ and let $(E,\nabla)$ be the connection associated to $F$. New invariants are updated ; In particular, we show that $(E,\nabla)$ is entirely determined by a fondamental $1$-form and the linearization polynome. We notice that the connection gives informations about the linearization of the web. Studying the trace of the curvature, we show that the determinant bundle of $(E,\nabla)$ is isomorphic to the tensor product of the line bundles associated to extracted $3$-webs. We then give a generalisation of Thomsen's construction of the hexagon, related to the trace. We also give an explicit way of determination of the rank of any $d$-web $\mathcal{W}(d)$. Some well known results in web geometry are proven and we indicate some new perspectives, in particular for the study of exceptional webs.
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https://tel.archives-ouvertes.fr/tel-00011928
Contributor : Olivier Ripoll <>
Submitted on : Sunday, March 12, 2006 - 9:38:05 PM
Last modification on : Thursday, January 11, 2018 - 6:21:22 AM
Long-term archiving on: : Saturday, April 3, 2010 - 10:47:38 PM

### Identifiers

• HAL Id : tel-00011928, version 1

### Citation

Olivier Ripoll. Géométrie des tissus du plan et équations différentielles. Mathématiques [math]. Université Sciences et Technologies - Bordeaux I, 2005. Français. ⟨tel-00011928⟩

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