Advanced Studies in Pure Mathematics

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

Rémi Langevin and Jean-Claude Sifre

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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.

Article information

Source
Geometry, Dynamics, and Foliations 2013: In honor of Steven Hurder and Takashi Tsuboi on the occasion of their 60th birthdays, T. Asuke, S. Matsumoto and Y. Mitsumatsu, eds. (Tokyo: Mathematical Society of Japan, 2017), 395-427

Dates
Received: 21 May 2014
Revised: 28 November 2014
First available in Project Euclid: 4 October 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1538671778

Digital Object Identifier
doi:10.2969/aspm/07210395

Mathematical Reviews number (MathSciNet)
MR3726721

Zentralblatt MATH identifier
1386.53028

Subjects
Primary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32]
Secondary: 65D17: Computer aided design (modeling of curves and surfaces) [See also 68U07] 53A30: Conformal differential geometry

Keywords
foliations of $\mathbb{S}^{3}$ Dupin cyclides quadrics Darboux cyclides

Citation

Langevin, Rémi; Sifre, Jean-Claude. Foliations of $\mathbb{S}^{3}$ by cyclides. Geometry, Dynamics, and Foliations 2013: In honor of Steven Hurder and Takashi Tsuboi on the occasion of their 60th birthdays, 395--427, Mathematical Society of Japan, Tokyo, Japan, 2017. doi:10.2969/aspm/07210395. https://projecteuclid.org/euclid.aspm/1538671778


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