Historical Comments on Monge's Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in ${\mathbb R}^3$

TL;DR

This paper reviews the historical development of principal configurations on surfaces, highlighting Monge's early contributions and their connection to the qualitative theory of differential equations, and proposes two open problems.

Contribution

It uncovers the historical links between Monge's 1796 work and modern differential equations, and introduces two open research problems in the field.

Findings

Identification of Monge's early elements of qualitative theory

Historical analysis of principal configurations on surfaces

Proposal of two open research problems

Abstract

This is an essay on the historical landmarks leading to the study of principal configurations on surfaces, their structural stability and further generalizations. Here it is pointed out that in the work of Monge, 1796, are found elements of the qualitative theory of differential equations, founded by Poincar\'e in 1881. Two open problems are proposed.