# Affine curvature lines of surfaces in 3-space

## TL;DR

This paper investigates the behavior of affine curvature lines on surfaces in 3D, providing topological models near special points and describing generic behaviors around eigenvalue degeneracies.

## Contribution

It introduces a detailed analysis of affine principal lines, including topological models near affine umbilic points and behavior near points with double eigenvalues.

## Key findings

- Topological models for affine curvature lines near umbilic points
- Descriptions of generic behavior near points with double eigenvalues
- Analysis of affine curvature lines at parabolic points

## Abstract

In this work we study the affine principal lines of surfaces in 3-space. We consider the binary differential equation of the affine curvature lines and obtain the topological models of these curves near the affine umbilic points (elliptic and hyperbolic). We also describe the generic behavior of affine curvature lines in the neighborhood of points with double eigenvalues (but not umbilics) of the affine shape operator and parabolic points