Flat 3-webs of degree one on the projective plane

TL;DR

This paper investigates flat 3-webs of degree one on the projective plane, characterizing those whose Legendre transforms have zero curvature, using projective duality and differential equations.

Contribution

It provides a new characterization of degree 3 foliations with flat dual webs via the Legendre transform and projective duality.

Findings

Characterization of degree 3 foliations with flat Legendre transforms

Use of Legendre transform as a tool for dual web analysis

Insights into the geometry of flat 3-webs on the projective plane

Abstract

The aim of this work is to study global $3$-webs with vanishing curvature. We wish to investigate degree $3$ foliations for which their dual web is flat. The main ingredient is the Legendre transform, which is an avatar of classical projective duality in the realm of differential equations. We find a characterization of degree $3$ foliations whose Legendre transform are webs with zero curvature.