An encounter of classical differential geometry with dynamical systems in the realm of structural stability of principal curvature configurations

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 Gutiérrez and Sotomayor, 1982–1983, are recounted here. These 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 and evolving from the achievements of Poincaré, 1881, Andronov–Pontrjagin, 1937, and Peixoto, 1962. Some mathematical developments as well as open problems sprouting from the 1982–1983 works are mentioned on Sect. 10.