Journal of the Mathematical Society of Japan

Topological canal foliations

Gilbert HECTOR, Rémi LANGEVIN, and Paweł WALCZAK

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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.


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

Article information

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

foliation canal surface griddling


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

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  • [AHS] F. Alcalde, G. Hector and P. Schweitzer, Vanishing cycles and generalized Reeb components, preprint, 2017.
  • [BLW] A. Bartoszek, R. Langevin and P. Walczak, Special canal surfaces of 𝑆3, Bull. Braz. Math. Soc., 42 (2011) 301–320.
  • [BW] A. Bartoszek and P. Walczak, Foliations by surfaces of a peculiar class, Ann. Polon. Math., 94 (2008), 89–95.
  • [CSW] G. Cairns, R. W. Sharpe and L. Webb, Conformal invariants for curves in three dimensional space forms, Rocky Mountain J. Math., 24 (1994), 933–959.
  • [CC] A. Candel and L. Conlon, Foliations I, II, Amer. Math. Soc., Providence, 2000 and 2003.
  • [EMS] R. D. Edwards, K. C. Millet and D. Sullivan, Foliations with all leaves compact, Topology, 16 (1977), 13–32.
  • [E] D. B. A. Epstein, Periodic flows on three-manifolds, Annals of Math., 95 (1972), 66–82.
  • [H] G. Hector, Feuilletages en cylindres, In: Geometry and Topology, Rio de Janeiro, 1976, Lecture Notes in Math., 597, Springer, 1977, 252–270.
  • [HC] G. Hector and M. A. Chaouch, Dynamiques source-puits et flots transversalement affines, Contemp. Math., 498 (2009), 99–126.
  • [HH] G. Hector and U. Hirsch, Introduction to the geometry of foliations, Part A and B, Vieweg, 1981.
  • [Kn1] H. Kneser, Reguläre Kurvenscharen auf die Ringflachen, Math. Annalen, 91 (1923), 135–154.
  • [Kn2] H. Kneser, Die Deformationssätze der einfach zuzammenhängenden Flächer, Mat. Z., 23 (1926), 362–372.
  • [Kr] R. Krasauskas, Minimal rational parametrizations of canal surfaces, Computing, 79 (2007), 281–290.
  • [LW1] R. Langevin and P. Walczak, Conformal geometry of foliations, Geom. Dedicata, 132 (2008), 135–178.
  • [LW2] R. Langevin and P. Walczak, Canal foliations of 𝕊3, J. Math. Soc. Japan, 64 (2012), 659–682.
  • [MR] R. Moussu and R. Roussarie, Relations de conjugaison et de cobordisme entre certains feuilletages, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 143–168.
  • [N] S. P. Novikov, Topology of foliations, Trans. Moscow Math. Soc., 14 (1965), 268–304.
  • [PP] M. Peternell and H. Pottmann, Computing rational parametrization of canal surfaces, J. Symb. Comp., 23 (1997), 255–266.
  • [Ro] D. Rolfsen, Knots and links, Publish or Perish, Berkeley, 1975.
  • [Sm] S. Smale, Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc., 10 (1959), 621–626.
  • [Ze1] A. Zeghib, Laminations et hypersurfaces géodésiques des variétés hyperboliques, Ann. Sci. École Norm. Sup., 24 (1991), 171–188.
  • [Ze2] A. Zeghib, Sur les feuilletages géodésiques continus des variétés hyperboliques, Invent. Math., 114 (1993), 193–206.