An encounter in the realm of Structural Stability between a qualitative theory for geometric shapes and one for the integral foliations of differential equations

TL;DR

This paper explores the historical development and connections between geometric shape stability and integral foliations of differential equations, highlighting key theorems and their mathematical significance.

Contribution

It provides a historical and conceptual analysis linking geometric stability theory with the qualitative theory of differential equations on surfaces.

Findings

Historical links between shape stability and differential equations established.

Key theorems by Gutiérrez and Sotomayor identified as pivotal.

Mathematical developments from 1982-1983 works are discussed.

Abstract

This evocative essay focuses on some landmarks that led the author to the study of principal curvature configurations on surfaces in $\mathbb R^3$, their structural stability and generic properties. The starting point was an encounter with the book of D. Struik and the reading of the references to the works of Euler, Monge and Darboux found there. The concatenation of these references with the work of Peixoto, 1962, on differential equations on surfaces, was a crucial second step. The circumstances of the convergence toward the theorems of Guti\'errez and Sotomayor, 1982 - 1983, are recounted here. The above 1982 - 1983 theorems are pointed out as the first encounter between the line of thought disclosed from the works of Monge, 1796, Dupin, 1815, and Darboux, 1896, with that transpiring from the achievements of Poincar\'e, 1881, Andronov - Pontrjagin, 1937, and Peixoto, 1962. Some mathematical developments sprouting from the 1982 - 1983 works are mentioned on the final section of this essay.