Inflection points and asymptotic lines on Lagrangean surfaces

TL;DR

This paper analyzes the local geometry of Lagrangean surfaces near inflection points, classifying asymptotic lines and establishing conditions for their existence based on topological invariants.

Contribution

It provides a detailed description of asymptotic lines near inflection points and proves the existence of inflection points on compact Lagrangean surfaces with non-zero Euler characteristic.

Findings

Inflection points correspond to specific cusp singularities in the discriminant curve.

In generic cases, asymptotic lines near inflection points follow two of three possible configurations.

Inflection points are guaranteed on compact Lagrangean surfaces with non-zero Euler characteristic.

Abstract

We describe the structure of the asymptotic lines near an inflection point of a Lagrangean surface, proving that in the generic situation it corresponds to two of the three possible cases when the discriminant curve has a cusp singularity. Besides being stable in general, inflection points are proved to exist on a compact Lagrangean surface whenever its Euler characteristic does not vanish.