Symmetries of 3-webs around a point

TL;DR

This paper classifies symmetries of planar 3-webs around a point, focusing on non-flat webs, and provides methods to construct webs with circular symmetry, expanding understanding of web symmetries beyond the hexagonal case.

Contribution

It offers a classification of non-flat 3-webs with simple or mirror symmetries and introduces a method to construct webs with circular symmetry.

Findings

Hexagonal webs have all three symmetry types.

Classification of non-flat webs with simple or mirror symmetries.

Method to construct webs with circular symmetry and a specific example.

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.