Symmetries of 3-webs around a point

TL;DR

This work classifies symmetries of real-analytic planar 3-webs around a point, focusing on non-flat webs. It develops a normal form $f(x,y)=x+y+xy(x-y)g(x,y)$ to analyze simple and mirror symmetries and establishes precise antisymmetry conditions on $g$ that yield one or two simple or mirror symmetries. It also provides a constructive framework for circular symmetries, proving that non-flat 3-webs with such symmetry exist and presenting an explicit example built from an invariant polynomial $P$, along with a discussion linked to Gronwall's conjecture. The results advance understanding of the symmetry structure of 3-webs and supply concrete methods to generate and classify non-flat cases with specific symmetry types.

Abstract

Let W be a planar 3-web defined on a neighborhood of a point M. We call "symmetry of W around M" any local diffeomorphism which fixes M and maps each foliation of W to a (not necessarily the same) foliation of W. We say that it is a simple symmetry if it respects each foliation, a mirror symmetry if its respects one foliation and permutes the two other and a circular symmetry if it permutes circularly the three foliations. Hexagonal (i.e. flat) planar 3-webs have always the three types of symmetry. We study here the non-flat case. We give a classification of 3-webs which admits simple or mirror symmetries. We give a method to build all the 3-webs with a circular symmetry and we exhibit a precise non-flat example.

Theorems & Definitions (6)

• Proposition 2.1
• Theorem 2.2
• Theorem 3.1
• Lemma 4.1
• Lemma 4.2
• Theorem 4.3