Journal of the Mathematical Society of Japan

Topological canal foliations

Gilbert HECTOR, Rémi LANGEVIN, and Paweł WALCZAK

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Abstract

Regular canal surfaces of $\mathbb{R}^3$ or $\mathbb{S}^3$ admit foliations by circles: the characteristic circles of the envelope. In order to build a foliation of $\mathbb{S}^3$ with leaves being canal surfaces, one has to relax the condition “canal” a little (“weak canal condition”) in order to accept isolated umbilics. Here, we define a topological condition which generalizes this “weak canal” condition imposed on leaves, and classify the foliations of compact orientable 3-manifolds we can obtain this way.

Note

The second and third authors were supported by the Polish NSC grant N$^\circ$ 6065/B/H03/2011/40.

Article information

Source
J. Math. Soc. Japan, Volume 71, Number 1 (2019), 43-63.

Dates
Received: 30 May 2017
First available in Project Euclid: 15 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1539590425

Digital Object Identifier
doi:10.2969/jmsj/78117811

Mathematical Reviews number (MathSciNet)
MR3909914

Zentralblatt MATH identifier
07056557

Subjects
Primary: 57R30: Foliations; geometric theory
Secondary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32]

Keywords
foliation canal surface griddling

Citation

HECTOR, Gilbert; LANGEVIN, Rémi; WALCZAK, Paweł. Topological canal foliations. J. Math. Soc. Japan 71 (2019), no. 1, 43--63. doi:10.2969/jmsj/78117811. https://projecteuclid.org/euclid.jmsj/1539590425


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