Blaschke's asymptotic lines of surfaces in R3

TL;DR

This paper investigates the behavior of Blaschke's asymptotic lines on surfaces in three-dimensional space, focusing on their differential equations near special points like affine cusps, umbilics, and parabolic sets.

Contribution

It provides a detailed analysis of the differential equations governing affine asymptotic lines in various regions of surfaces, including elliptic, hyperbolic, and parabolic zones, near critical points.

Findings

Characterization of affine asymptotic lines near affine cusp points.

Description of lines near affine umbilic points.

Analysis of lines near Euclidean parabolic and flat umbilic points.

Abstract

In this paper we consider the Blaschke's asymptotic lines (also called affine asymptotic lines) of regular surfaces in 3-space. We study the binary differential equations defining Blaschke's asymptotic lines in the elliptic and hyperbolic regions of the surface near affine cusp points and to at affine umbilic points. We also describe the affine asymptotic lines near the Euclidean parabolic set including the Euclidean flat umbilic points.