Foliations of $\mathbb{S}^{3}$ by cyclides

Abstract

Throughout the last 2–3 decades, there has been great interest in the extrinsic geometry of foliated Riemannian manifolds (see [2], [4] and [22]).

One approach is to build examples of foliations with reasonably simple singularities with leaves admitting some very restrictive geometric condition. For example (see [22], [23] and [17]), consider in particular foliations of $\mathbb{S}^{3}$ by totally geodesic or totally umbilical leaves with isolated singularities.

The article [14] provides families of foliations of $\mathbb{S}^{3}$ by Dupin cyclides with only one smooth curve of singularities. Quadrics and other families of cyclides like Darboux cyclides provide other examples. These foliations are built on solutions of a three contacts problem: we show that the surfaces of the considered family satisfying three imposed contact conditions, if they exist, form a one parameter family of surfaces which will be used to construct a foliation.

Finally we will study the four contact condition problem in the realm of Darboux–d'Alembert cyclides.