6533b7d8fe1ef96bd126b737

RESEARCH PRODUCT

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

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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.