Dynamical Systems Seminar
The lines of curvature of a surface embedded in three space together with umbilic points comprise its principal foliations. These are analogous to the phase portrait of a vector field on the surface. This talk compares the two and describes significant differences. While computer visualizations of phase portraits are ubiquitous, few examples of principal foliations have been produced. One exception is the principal foliations of the triaxial ellipsoid described by Monge already in the eighteenth century and show in many differential geometry texts. Here, we display principal foliations of perturbations of the ellipsoid. Our numerical computations take into account that principal foliations are seldom orientable.
One of the key results about vector fields on the two sphere is the Poincare-Bendixson Theorem which precludes trajectories that are recurrent. However, Sotomayor and Guitierrez gave an example of a convex surface with a dense line of curvature. Here, we give a much larger set of examples by demonstrating a close relationship between lines of curvature on perturbations of the ellipsoid and vector fields on a torus.