Abstract
The lines of curvature of a surface embedded in $\R^3$ comprise its principal foliations. Principal foliations of surfaces embedded in $\R^3$ resemble phase portraits of two dimensional vector fields, but there are significant differences in their geometry because principal foliations are not orientable. The Poincar\'e-Bendixson Theorem precludes flows on the two sphere $S^2$ with recurrent trajectories larger than a periodic orbit, but there are convex surfaces whose principal foliations are closely related to non-vanishing vector fields on the torus $T^2$. This paper investigates families of such surfaces that have dense lines of curvature at a Cantor set $C$ of parameters. It introduces discrete one dimensional return maps of a cross-section whose trajectories are the intersections of a line of curvature with the cross-section. The main result proved here is that the return map of a generic surface has \emph{breaks}; i.e., jump discontinuities of its derivative. Khanin and Vul discovered a qualitative difference between one parameter families of smooth diffeomorphisms of the circle and those with breaks: smooth families have positive Lebesgue measure sets of parameters with irrational rotation number and dense trajectories while families of diffeomorphisms with a single break do not. This paper discusses whether Lebesgue almost all parameters yield closed lines of curvature in families of embedded surfaces.
Highlights
Vaughan Jones’ generous support of the New Zealand Mathematics Research Institute, the 2016 Raglan Summer School on Continuation Methods in Dynamical Systems, helped launch the work described here
The primary objects of interest are principal foliations consisting of the lines of curvature on surfaces embedded in three space
Numerical methods for computing phase portraits of dynamical systems are a prominent part of dynamical systems theory, and they were the focus of the Raglan Summer School
Summary
Vaughan Jones’ generous support of the New Zealand Mathematics Research Institute, the 2016 Raglan Summer School on Continuation Methods in Dynamical Systems, helped launch the work described here. The primary objects of interest are principal foliations consisting of the lines of curvature on surfaces embedded in three space They are similar to the phase portraits of dynamical systems, but differ in that their tangent line fields are not orientable like vector fields. The simplest dynamical systems with large limit sets are non-vanishing vector fields on the two torus T 2 with a global cross-section Σ Their return maps σ : Σ → Σ are smooth diffeomorphisms of the circle. I describe how to produce circle homeomorphisms as return maps for principal foliations on connected surfaces with a minimal number of generic umbilic points This is followed by a review of relevant aspects of the theory of circle diffeomorphisms and circle diffeomorphisms with breaks. I apply the theory of circle diffeomorphisms with breaks to connected surfaces with only lemon umbilic points
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