Abstract

The study of the foliations defined on a surface by the lines of curvature, along which the surface bends extremally, has attracted the interest of mathematicians of several generations, including Euler, Monge, Dupin, Darboux and Caratheodory, to mention just a few
In 1982, the introduction of ideas from the qualitative theory of differential equations and dynamical systems, such as structural stability, bifurcation and genericity -- as in the work of Andronov, Pontrajgin, Peixoto, Anosov and Smale -- again to mention a few, gave new vigor to this classic geometric subject.
I will discuss some results obtained in collaboration with C. Gutierrez and R. Garcia, while focusing on the interaction between the classical ideas, that is, lines of curvature and their umbilic singular points, and the modern ideas mentioned above.