Exercise 3.6.3 of Garcia and Sotomayor: a devil’s staircase of principal foliations

To Jorge Sotomayor Principal foliations are tangent to the directions of maximal and minimal normal curvatures on surfaces embedded in three space. Jorge Sotomayor initiated the qualitative study of structural stability for these foliations. This paper answers a question he raised in a set of lecture notes coauthored with Ronaldo Garcia about the existence of principal foliations with dense lines of curvature on simply connected surfaces. It demonstrates that there is a devil’s staircase of such foliations in the family of surfaces defined by Sotomayor and Garcia.