Dense lines of curvature on convex surfaces

Abstract:

The lines of curvature of a surface embedded in $\mathbb {R}^3$ together with umbilic points comprise its principal foliations. Monge described the principal foliations of the triaxial ellipsoid in the eighteenth century, but few examples beyond these and surfaces of revolution have been characterized since. This paper analyzes the principal foliations of perturbations of the ellipsoid. The results are surprising. Notably, surfaces with dense lines of curvature are prevalent A geometric construction establishes an intimate relation of the principal foliation on perturbed ellipsoids with nonvanishing vector fields on the two dimensional torus $T^2$, rather than the two sphere $S^2$. The dynamics of nonvanishing vector fields on $T^2$ with a global cross-section has been extensively studied. In generic one-parameter families, vector fields with dense, quasiperiodic trajectories occur at positive measure sets of parameters. This paper establishes a comparable result proving that large sets of embeddings of $S^2$ in $\mathbb {R}^3$ have dense lines of curvature.