Abstract

Principal foliations of a surface embedded in three space are aligned with directions of maximal and minimal normal curvature. They are analogous to vector fields but are non-orientable. The singular points are umbilics where normal curvatures are constant in all directions. A century ago, Caratheodory stated the conjecture that compact convex surfaces have at least two umbilics. Several purported proofs of this conjecture have been presented, but there is still doubt about their veracity. This first part of this talk describes new results about this problem. Sotomayor and Gutierrez studied the analogy between principal foliations and two dimensional vector fields from a dynamical systems perspective. The second part of this talk describes results that extend their work, emphasizing the relationship of foliations with dense leaves to quasiperiodic flows on the two dimensional torus.